Examples/MeshGenToolbox/doc/reqApp.tex

+\documentclass[12pt,titlepage]{article}
+
+\include{Preamble}
+
+\author{Wen Yu}
+
+\begin{document}
+
+\begin{titlepage}
+
+% Fonts for the title page
+\newcommand{\texttl}{\textsf} %font for the title (Emil's version is \textsf)
+\newcommand{\textau}{\textsf} %font for the authors, date, etc. (Emils version is \textsc)
+
+\begin{center}
+ 
+\HRule \\[0.4cm]
+{ \huge \bfseries \texttl{Software Requirements Specification for a Parallel Mesh Generation Toolbox}}\\[0.4cm]
+ 
+\HRule \\[2cm]
+
+\textau{\LARGE Wen Yu}\\[2cm]
+ 
+\textau{\LARGE {September 2008}}
+ 
+\vfill
+ 
+% Bottom of the page
+%\includegraphics[width=0.27\textwidth]{logo.pdf}
+\hfill
+\textau{\LARGE
+\raisebox{2pt}{\begin{tabular}[b]{r}
+\raisebox{8pt}{Computing and Software} \\
+McMaster University \end{tabular}}
+}
+\end{center}
+
+\end{titlepage}
+
+\tableofcontents
+
+\newpage
+
+\section{Reference Material}
+\subsection{Table of Symbols, Abbreviations and Acronyms}
+
+\subsubsection{Symbols}
+\begin{tabular}{p{\firstCol}p{\secondCol}} 
+$\Omega$ & a closed bounded domain in $\mathbb{R}^{2}$  \\
+$\Omega^{*}$ & a mesh covering the domain bounded by $\Omega$\\
+$K$ & a simple shape, such as a line segment in 1D, a triangle or a quadrilateral in 2D, or a tetrahedron or hexahedron
+in 3D\\
+$M^{\mbox{IN}}$ & an input mesh\\
+$M^{\mbox{OUT}}$ & an output mesh\\
+$I$ & instructions on how a mesh should be \\
+ & refined/coarsened
+\end{tabular}
+%\newpage
+
+\subsubsection{Abbreviations and Acronyms}
+\begin{tabular}{p{\firstCol}p{\secondCol}} 
+1D & One Dimensional Space\\
+2D & Two Dimensional Space\\
+3D & Three Dimensional Space\\
+FEA & Finite Element Analysis\\
+HPC & High Performance Computing\\
+PDE & Partial Differential Equation\\
+PMGT & Parallel Mesh Generation Toolbox\\
+SHARCNET & Shared Hierarchical Academic Research Computing Network\\
+SRS & Software Requirements Specification\\
+AOMD & Algorithm Oriented Mesh Database\\
+\end{tabular}
+\newpage
+
+\subsection{Index of Requirements}
+CoarseningMesh, \pageref{CoarseningMesh}\\
+Conformal, \pageref{Conformal}\\\\
+DomainDimension, \pageref{DomainDimension}\\\\
+ElmShape, \pageref{ElmShape}\\
+ElmTopology, \pageref{ElmTopology}\\
+ElmUniqueID, \pageref{ElmUniqueID}\\
+Exception, \pageref{Exception}\\\\
+Help, \pageref{Help}\\\\
+InputDefinition, \pageref{InputDefinition}\\\\
+LookAndFeel, \pageref{LookAndFeel}\\\\
+Maintainability, \pageref{Maintainability}\\
+MeshType, \pageref{MeshType}\\\\
+OutElmOrder, \pageref{OutElmOrder}\\
+OutputStorage, \pageref{OutputStorage}\\
+OutVertexOrder, \pageref{OutVertexOrder}\\\\
+Performance, \pageref{Performance}\\
+Portability, \pageref{Portability}\\
+Precision, \pageref{Precision}\\\\
+RCInstruction, \pageref{RCInstruction}\\
+RefiningMesh, \pageref{RefiningMesh}\\
+RefiningOrCoarsening, \pageref{RefiningOrCoarsening}\\\\
+Usability, \pageref{Usability}\\\\
+VertexUniqueID, \pageref{VertexUniqueID}
+%\printindex
+\newpage
+
+\section{Introduction \label{AreqSecIntro}}
+
+This section gives an overview of the Software Requirements Specification (SRS) for a Parallel Mesh Generation Toolbox (PMGT). First, the purpose of the document is provided. Second, the scope of PMGT is identified. Third, some terminology for software engineering and mesh generation are defined. Finally, the organization of the document is summarized. The Table of Symbols, Abbreviation and Acronyms, and Index of Requirement are given at the beginning of the SRS.
+
+\subsection{Purpose of the Document}
+This SRS provides a black-box description of PMGT. The intended audience of the SRS is the development team and the users of PMGT.
+
+\subsection{Scope of the Software Product}
+PMGT provides a library that will be embedded into a larger application, such as a finite element analysis (FEA) program. 
+\begin{itemize}
+\item The input of PMGT is an existing mesh $M^{\mbox{IN}}$ with instructions $I$ provided by the user on how the mesh should be refined/coarsened.
+\item PMGT refines/coarsens $M^{\mbox{IN}}$ according to the supplied instructions $I$ on how the mesh should be refined/coarsened.
+\item PMGT will take advantage of parallel computation.
+\item The output of PMGT is a refined/coarsened mesh $M^{\mbox{OUT}}$.
+\end{itemize}
+Note that depending on the given instruction, PMGT can either refine or coarsen the given mesh, but cannot do both at the same time. That is, any individual transition from $M^{\mbox{IN}}$ to $M^{\mbox{OUT}}$ will only do one of refining or coarsening. The embedding application will have access to reading the mesh information, such as information on the position of vertices and on the vertices that define a given element. However, the application cannot directly change any mesh data, except for the information indicating which elements should be refined/coarsened. 
+
+\subsection{Terminology Definition}
+This subsection provides the definitions for terminology used in the SRS. There are two classes of terminology. One relates to software engineering, and the other relates to mesh generation. The definitions are listed in alphabetical order.
+\subsubsection{Software Engineering Related Terminology\label{AreqSEterm}}
+\begin{description}
+
+\item[Constraint:] A statement that expresses measurable bounds for an element or function of the system. That is, a constraint is a factor that is imposed on the solution by force or compulsion and may limit or modify the design changes. \citep{IEEE1998}
+
+\item[Context:] The boundaries between the system that we intend to build and the people, organizations, other system and pieces of technology that have a direct interface with the system. \citep{Volere2001}
+
+\item[Functional Requirements:] Functional requirements define precisely what input are expected by the software, what outputs will be generated by the software, and the details of relationships that exist between those inputs and outputs. In short, functional requirements describe all aspects of interface between the software and its environment (that is, hardware, humans, and other software). \citep{Davis1990}
+
+\item[Goal:] Goals capture, at different levels of abstraction, the various objectives the system under
+consideration should achieve. \citep{Lamsweerde2001}
+
+\item[Non-functional Requirements:] Non-functional requirements define the overall qualities or attributes to be exhibited by the resulting software system. \citep{Davis1990}
+
+\item[Requirements:] A software requirement is: \begin {inparaenum}[\itshape i\upshape)]
+\item a condition or capability needed by a user to solve a problem or achieve an objective;
+\item a condition or capability that must be met or possessed by a system or system component to satisfy a contract,
+standard, specification, or other formally imposed document; or, 
+\item a documented representation of a condition or capability as in the above two definitions.
+\end{inparaenum} \citep{IEEE2000}
+
+\item[Software Engineering:] Software Engineering is the application of a systematic, disciplined, quantifiable approach to the development, operation, and maintenance of software. \citep{IEEE1990}
+
+\item[Software Requirements Specification:] A Software Requirements Specification (SRS) is a document containing a complete description of what the software will do without describing how it will do it. \citep{Davis1990}
+
+\item[System:] An interdependent group of people, objects, and procedures constituted to achieve defined objectives or some operational role by performing specified functions. \citep{IEEE1998}
+
+\item[System Context:] System Context documents the relationships between the system being specified and other human and computer systems. \citep{Sommerville1992}
+
+\item[User:]The person, or persons, who operate or interact directly with the product. \citep{IEEE2000}
+
+\end{description}
+
+\subsubsection{Mesh Generation Related Terminology\label{AreqMGterm}}
+\begin{description}
+
+%\item [Aspect Ratio:] The aspect ratio of a simplex is defined as its maximum side length divided by its minimum altitude. \citep{Mitchell2000}
+%The aspect ratio of a convex body is the ratio between its longest dimension and its shortest dimension.
+
+\item [Cell:] Another name for an element, as defined in page \pageref{AreqElmDef}.
+
+\item[Conformal Mesh:] A conformal mesh is a mesh (defined on page \pageref{def_mesh}) following the definition of a mesh, with the addition of the
+following property:\\
+The intersection of two elements in the mesh $\Omega^{*}$ is either the empty set, a vertex, an edge or a face (when the
+dimension is 3). \citep{FreyAndGeorge2000}
+
+\item [Connectivity:] There are two types of connectivity, one for the mesh and one for a mesh element:
+\begin{enumerate}
+\item ``The connectivity of a mesh is the definition of the connection between its vertices.''~\citep{FreyAndGeorge2000}
+\item ``The connectivity of a mesh element is the definition of the connections between the vertices at the element
+level.''~\citep{FreyAndGeorge2000}
+\end{enumerate}
+
+\item [Domain:] The area or volume that is to be discretized.  The domain is sometimes referred to as the
+computational domain. \citep{Smith2004}
+
+
+\item [Edge:] An edge is a line segment between two vertices.
+
+\item [Element \label{AreqElmDef}:] The original domain is discretized into smaller, usually simpler, shapes called elements. 
+The typical shapes for elements in 1D is a line, in 2D is a triangle or a quadrilateral, and in 3D a tetrahedron or a
+hexahedron.  Elements are also called cells. \citep{Smith2004}
+
+\item [Embedding Application:] The software that uses PMGT.
+
+\item [Face:] A face is a maximal connected subset of the plane without vertices inside the subset. In 2D, a face is a cell.\citep{FreyAndGeorge2000}
+
+\item [Hybrid Mesh:] A mesh is said to be hybrid if it includes some elements with a different spatial dimension. \citep{FreyAndGeorge2000}
+
+\item[Mesh:] \label{def_mesh}
+In \citet{Smith2004}, a mesh is defined as follows:\\
+Let $\Omega$ be a closed bounded domain in $\mathbb{R}$ or $\mathbb{R}^2$ or $\mathbb{R}^3$
+and let $K$ be an element.  A mesh of $\Omega$, denoted by $\Omega^{*}$, has the following properties:
+\begin{enumerate}
+\item $\Omega \approx \cup(K | K \in \Omega^{*} : K)$, where $\cup$ is first closed and then opened
+\item the length of every element $K$, of dimension 1, in $\Omega^{*}$ is greater than zero
+\item the interior of every element $K$, of dimension 2 or greater, in $\Omega^{*}$ is nonempty
+\item the intersection of the interior of two elements is empty
+\end{enumerate}
+The only difference between above definition and the definition given by \citet{FreyAndGeorge2000} is that equality (=) had been changed to approximate equality ($\approx$).
+
+\item [Mesh Generation:] The automatic mesh generation problem is that of attempting to define a set of elements to best describe a geometric domain, subject to various element size and shape criteria. \citep{Smith2004}
+
+\item [Mixed Mesh:] A mesh is said to be mixed if it includes some elements of a different geometric nature. \citep{FreyAndGeorge2000}
+
+\item [Structured Mesh:] The mesh in which the local organization of the grid points and the form of the grid cells do not depend on their position but are defined by a general rule.  There is a pattern to the topology that repeats. 
+\citet{FreyAndGeorge2000} say, ``a mesh is called structured if its connectivity is of the finite difference type.'' 
+They go on to remark, ``Peculiar meshes other than quad or hex meshes could have a structured connectivity.  For
+instance, one can consider a classical grid of quads where each of them are subdivided into two triangles using the same
+subdivision pattern.''
+
+\item [Topology:] ``The topology of a mesh element is the definition of this element in terms of its faces and edges,
+these last two being defined in terms of the element's vertices.''~\citep{FreyAndGeorge2000}\\
+The topology of a mesh is the set of topologies of its constitute mesh elements.
+
+\item [Unstructured Mesh:] The mesh whose element connectivity of the neighbouring grid vertices varies from point to
+point.  Any mesh that is not structured is an unstructured mesh. \citep{Smith2004}
+
+\item [Vertices:] The locations that define the shape of the cells.  In 1D the vertices are the end-points of the
+elements.  For 2D and 3D elements the vertices correspond to the location in space that defines the intersection of
+the edges of an element. \citep{Smith2004}
+
+\end{description}
+
+
+\subsection{Organization of the Document}
+This SRS follows the template introduced by~\cite{Lai2004}. Lai's template targets an SRS for scientific computing software. In particular, the example shown is for engineering mechanics software, such as software to analyze beams. In the current work, Lai's template is modified to fit PMGT, which is a more general purpose software. For example, the instanced model section of Lai's template is removed since PMGT is not designed for solving a specific physical problem.
+ 
+Section \ref{AreqSecIntro} (this section) is an introduction to the SRS. 
+The rest of the document is arranged as follows. Section \ref{AreqSecGSD} provides the general information about the system. Section \ref{AreqSecSSR} is the major part of the SRS. All functional requirements and non-functional requirements of the software are presented in this section. Section \ref{AreqSecOSI} discusses some other system issues. Section \ref{AreqSecTM} gives a traceability matrix that summaries the association of each requirement with goals, assumptions, theoretical models and data definitions introduced in \ref{AreqSecSSR}. This SRS also contains the list of possible changes in the requirements and values of auxiliary constants. The references are listed at the end of this document.  
+
+\section{General System Description \label{AreqSecGSD}}
+
+This section describes the general information about the system. The interfaces between the system and its environment are defined first. Then the characteristics of potential users are discussed. At end of this section, some system constraints are described.
+
+\subsection{System Context}
+The software to be built is a library tool that will be called by other applications. There is no direct interaction between the system and the end users. Users of the embedding application, such as an FEA program, provide some parameters directly to the FEA program. Some of these parameters are passed to PMGT by the FEA program. The interface between PMGT and the embedding application should only show what PMGT can do and hide the information about how to do it. Therefore, users who are not experts in mesh generation or in parallel processing will be able to use this toolbox.  %The characteristics of users of PMGT will be described in the Section \ref{userchar}.
+\begin{figure}[ht]
+\centering
+\epsfig{file=syscon.eps, width=\textwidth}
+\caption{System Context Diagram}
+\label{AreqSyscon}
+\end{figure}
+
+Figure~\ref{AreqSyscon} shows the context that PMGT will normally fit into.
+A circle represents an external entity outside the system, an embedding application in this case. The rectangle is the system itself. Arrows represent the data flows between them.
+
+The \emph{input}: $M^{\mbox{IN}}\times I$ 
+
+The \emph{output}: $M^{\mbox{OUT}}$.
+
+PMGT has the following function:
+\begin{quote}
+A mesh $M^{\mbox{IN}}$ and some refining/coarsening instructions $I$ are given. PMGT generates a refined/coarsened mesh $M^{\mbox{OUT}}$ according to the instructions $I$. 
+\end{quote}
+%\begin{quotation}
+
+%\end{quotation}
+
+\subsection {User Characteristics}\label{AreqUserchar}
+The target user group of PMGT includes both software designers, who intend to embed this library in their applications, and theoreticians, who are involved in parallel mesh generation. A user of PMGT is expected to be familiar with the notion/knowledge of mesh creation. PMGT is a library used by other applications. Therefore, users should not be novices in terms of software design. The prerequisite software design knowledge are equivalent to that of a senior undergraduate student in science or engineering who took an introductory course on programming. For example, they should be comfortable with compilation of the programming language in which PMGT is written, be familiar with embedding a library in their software, $etc.$
+
+\subsection{System Constraints}
+This system is intended to be built on the Shared Hierarchical Academic Research Computing Network (SHARCNET). 
+SHARCNET is structured as a ``cluster of clusters'' across South Central Ontario, designed to meet the computational needs of researchers in a diverse number of research areas and to facilitate the development of leading-edge tools for high performance computing (HPC) grids.
+
+Large production clusters, located at the Universities of Western Ontario, Guelph and McMaster, house over 400 HP/Compaq
+Alpha processors and large symmetric multiprocessor computers. Windsor and Wilfrid Laurier host smaller development
+clusters (8 processors), which enable researchers to develop and test code before moving to one of the larger clusters.
+A glance of SHARCNET systems is shown in Table \ref{AreqSharcnet}. Note that the network is constantly being updated. Detailed information can be found at \citet{sharcnet2006}.
+\begin{table}[htbp]
+\centering
+\begin{tabular}{p{0.2\textwidth}|p{0.2\textwidth}|p{0.1\textwidth}|p{0.1\textwidth}|p{0.25\textwidth}} 
+\hline \hline
+System & Make & Type & CPUs & OS \\
+\hline
+bala & Compaq & Cluster & 8 & Red Hat Linux 7.2\\
+cat & Unknown & Cluster & 162 & Red Hat Linux 8 \\
+goblin & Sun & Cluster & 56 & Fedora Core 2 \\
+hammerhead & Compaq & Cluster & 112 & Red Hat Linux 7.2 \\
+idra & Compaq SC & Cluster & 128 & Tru64 \\
+mako & HP & Cluster & 16 & Fedora Core 2 \\
+tiger & Compaq & Cluster & 8 & Red Hat Linux 7.2 \\
+typhon & Compaq & SMP & 16 & Tru64 \\
+wobbe & Unknown & Cluster & 193 & Red Hat Linux 8 \\
+TOTALS & & & 699 \\
+\hline \hline
+\end{tabular} 
+\caption{A Glance at the SHARCNET System}
+\label{AreqSharcnet}
+\end{table}
+
+\section{Specific System Requirements \label{AreqSecSSR}} 
+
+This section describes the system requirements in detail. After the problem is clearly and unambiguously stated, some solution characteristics are specified. Non-functional requirements are also included in this section. The symbol $:=$ is used to indicate type definition. The notation for set building and expressions used in this section follows \citet{Gries1993}. To define the notation, first let $x$ be a list of dummies, $t$ a type, $R$ a predicate, $E$ an expression, $*$ an operator, and $P$ a predicate. Notation $\{ x:t\ |\ R:E\}$ represents a set of values that result from evaluating $E[x:=v]$ in the state for each value $v$ in $t$ such that $R[x:=v]$ holds in that state. Expression $(*x:t\ |\ R:P)$ denotes the application of operator $*$ to the values $P$ for all $x$ in $t$ for which range $R$ is true. 
+
+\subsection{Problem Description \label{AreqSecPD}}
+The problems (goals) specified in this subsection represent ideal general models. The problems are simplified by introducing some assumptions, which are listed in Section \ref{AreqSolution}.
+
+\subsubsection{Background Overview}
+Many physical problems of importance to scientists and engineers are modeled as a set of Partial Differential Equations (PDEs). In most practical cases, it is necessary to solve the PDEs numerally. Numerical methods to solve PDEs frequently require that the domain of interest be divided into a mesh, which is a set of small, simple elements that cover the computational domain. In some applications, a single mesh is generated and used  many times; in this case the processing time spent on mesh construction is not critical and a relatively slow, sequential algorithm suffices \citep{Ruppert1993}. However, some applications need adaptive meshing, which requires that the meshes be generated once and then modified many times. For instance, adaptive meshing is used for reliable Finite Element Analysis (FEA) using a posterrori error estimation \citep{Zienkiewicz2005}. The increased mesh interaction for adaptive meshing means an increased need for speed of managing the mesh data which suggest employing parallel processing techniques. Although generating a mesh using multiple processors is complicated, it can offer considerable speed-up over sequential processing. In addition, some FEA applications are implemented on multiple processors. If the adaptive mesh can be generated in multiple processors as well, the mesh data can remain on the local processors. Potentially, time to be used will be significantly reduced. 
+
+\subsubsection{Goal Statements}
+There are two related goals for PMGT.
+\begin{description}
+\item[\refstepcounter{goalnum}\gthegoalnum \label{gRefining}:] Given a mesh $M^{\mbox{IN}}$ and instructions $I$ on how to refine the mesh, PMGT should generate a refined mesh $M^{\mbox{OUT}}$ according to the instructions $I$.
+\item[\refstepcounter{goalnum}\gthegoalnum \label{gCoarsening}:]Given a mesh $M^{\mbox{IN}}$ and instructions $I$ on how to coarsen the mesh, PMGT should generate a coarsened mesh $M^{\mbox{OUT}}$ according to the instructions $I$.
+\end{description}
+
+\subsection{Solution Characteristics Specification\label{AreqSolution}}
+The goals stated in the last section are too general to achieve. In this section, the assumptions are specified first to reduce the scope of the software. Second, the theoretical models for the goals are described. Third, data definitions are given to assist with defining the theoretical models. Finally, the system behaviour is summarized.     
+\subsubsection{Assumptions}
+\begin{description}
+\item [\refstepcounter{assnum} \atheassnum \label{aDimension}:]PMGT focuses on a 2D domain.
+\item[\refstepcounter{assnum} \atheassnum \label{aClosed}:]The input and output meshes are bounded.
+\item [\refstepcounter{assnum} \atheassnum \label{aUnstructured}:]The input and output meshes are unstructured.
+\item[\refstepcounter{assnum} \atheassnum \label{aConformal}:]The input and output meshes are conformal.
+\item[\refstepcounter{assnum} \atheassnum \label{aElmShape}:]The elements of input and output meshes are triangles.
+\item[\refstepcounter{assnum} \atheassnum \label{aValidMesh}:] The initial mesh is valid.
+\end{description}
+
+\subsubsection{Theoretical Model}
+The theoretical models corresponding to the goals given in Section \ref{AreqSecPD} describes the relationship between the input mesh ($M^{\mbox{IN}}$) and the output mesh ($M^{\mbox{OUT}}$). The meshes are assumed to be embedded in a 2D space. 
+\begin{description}
+\item[\refstepcounter{tmnum} \tmthetmnum \label{tmRefining}:] Refining Mesh
+\begin{description}
+\item [\emph{Input}:] $M^{\mbox{IN}}$: {\tt MeshT}, $I$: {\tt RCinstructionT}
+\item [\emph{Output}:] $M^{\mbox{OUT}}$: {\tt MeshT}
+\end{description}
+The following behavior is specified: 
+\begin{itemize}
+\item {\it Refined}($M^{\mbox{OUT}},\ M^{\mbox{IN}}$)
+\end{itemize}
+That is, the output mesh is a refined version of the input mesh.
+\item[\refstepcounter{tmnum} \tmthetmnum \label{tmCoarsening}:] Coarsening Mesh
+\begin{description}
+\item [\emph{Input}:] $M^{\mbox{IN}}$: {\tt MeshT}, $I$: {\tt RCinstructionT}
+\item [\emph{Output}:] $M^{\mbox{OUT}}$: {\tt MeshT}
+\end{description}
+The following behavior is specified: 
+\begin{itemize}
+\item {\it Coarsened}($M^{\mbox{OUT}},\ M^{\mbox{IN}}$)
+\end{itemize}
+That is, the output mesh is a coarsened version of the input mesh.
+\end{description}
+
+\subsubsection{Data Definitions}
+The data definitions below are organized so that a definition listed in the beginning may be used to define a data item  listed after it.
+\begin{description} 
+\item [VertexT (\refstepcounter{ddnum} \dtheddnum \label{dVertexT}):] A vertex is represented by two real numbers, which are its $x$ coordinate and $y$ coordinate. More formally,\\
+{\tt VertexT} := tuple of ($x: \mathbb{R},\  y: \mathbb{R}$).
+
+\item [EdgeT (\refstepcounter{ddnum} \dtheddnum \label{dEdgeT}):] An edge is represented by a set of {\tt VertexT}. More formally,\\
+{\tt EdgeT} := set of {\tt VertexT}.
+
+\item [ValidEdge (\refstepcounter{ddnum} \dtheddnum \label{dValidEdge}):] An edge is valid if the edge is a line segment (that is, the set has two elements). More formally, \\
+{\it ValidEdge}: {\tt EdgeT} $\rightarrow \mathbb{B}$\\
+{\it ValidEdge}($e$: {\tt EdgeT}) $\equiv \# e = 2$
+
+\item [CellT (\refstepcounter{ddnum} \dtheddnum \label{dCellT}):] A cell is represented by a set of $VertexT$. More formally, \\
+{\tt CellT} := set of {\tt VertexT}
+
+\item [Area (\refstepcounter{ddnum} \dtheddnum \label{dArea}):] The area of a triangle whose apexes are elements of a cell. More formally, \\
+{\it Area}: {\tt CellT} $\rightarrow \mathbb{R}$\\
+{\it Area}($c$: {\tt CellT}) $\equiv \Sigma v1,v2,v3$: {\tt VertexT} $|\ v1\in c \wedge v2\in c \wedge v3\in c \\
+\wedge v1\neq v2 \wedge v2\neq v3 \wedge v3\neq v1:\\
+\frac{1}{12}* |v1.x*v2.y - v2.x*v1.y +\\
+v2.x*v3.y - v3.x*v2.y +\\
+v1.x*v3.y - v3.x*v1.y |$
+
+\item [ValidCell (\refstepcounter{ddnum} \dtheddnum \label{dValidCell}):] A cell is valid if the cell is a triangle (that is , the set has three elements) and the area of the triangle is greater than zero. More formally, \\
+{\it ValidCell}: {\tt CellT} $\rightarrow \mathbb{B}$\\
+{\it ValidCell}($c$: {\tt CellT}) $\equiv \# c = 3 \wedge$ {\it Area}($c$) $\geq 0$
+
+\item [MeshT (\refstepcounter{ddnum} \dtheddnum \label{dMeshT}):] A mesh is represented by a set of cells. More formally,\\
+{\tt MeshT} := set of {\tt CellT}.
+
+\item [OnEdge (\refstepcounter{ddnum} \dtheddnum \label{dOnEdge}):] Checks if a vertex is on the line segment between  two vertices (exclusive) of an edge. More formally,\\
+{\it OnEdge}: {\tt VertexT} $\times$ {\tt EdgeT} $\rightarrow \mathbb{B}$\\
+{\it OnEdge}($v$: {\tt VertexT}, $e$: {\tt EdgeT}) $\equiv \exists\ v1,\ v2$: {\tt VertexT} $|\\
+ v1 \in e \wedge v2 \in e \wedge v1 \neq v2 \wedge v \neq v1 \wedge v \neq v2:\\
+(v1.x < v.x \leq v2.x \wedge \\(v.y-v1.y)/(v.x-v1.x)=(v2.y-v1.y)/(v2.x-v1.x))$
+
+\item [BelongToCell (\refstepcounter{ddnum} \dtheddnum \label{dBelongToCell}):] Checks if an edge belongs to a cell. More formally, \\
+{\it BelongToCell}: {\tt VertexT} $\times$ {\tt CellT} $\rightarrow \mathbb{B}$\\
+{\it BelongeToCell}($e$: {\tt EdgeT}, $c$: {\tt CellT}) $\equiv \forall\ v$: {\tt VertexT} $|\ v\in e: v\in c$
+
+\item [Inside (\refstepcounter{ddnum} \dtheddnum \label{dInside}):] Checks if a point (of type VertexT) is inside of a cell. The {\em inside} checking is false if the point is on an edge of the cell or the point is a vertex of the cell. (The algorithm to check if a point is inside a polygon is from \citet{Blackpawn}.) More formally,\\
+{\it Inside}: {\tt VertexT} $\times$ {\tt CellT} $\rightarrow \mathbb{B}$\\
+{\it Inside}($v$: {\tt VertexT}, $c$: {\tt CellT}) $\equiv \exists\ v1,\ v2,\ v3$: {\tt VertexT} $|\\
+v1\in c \wedge v2\in c \wedge v3\in c \wedge v1\neq v2 \wedge v2\neq v3 \wedge v3 \neq v1:\\
+((v.y-v1.y)*(v2.x-v1.x)-(v.x-v1.x)*(v2.y-v1.y))*\\
+((v.y-v2.y)*(v3.x-v2.x)-(v.x-v2.x)*(v3.y-v2.y)) > 0 \wedge \\
+((v.y-v2.y)*(v3.x-v2.x)-(v.x-v2.x)*(v3.y-v2.y))*\\
+((v.y-v3.y)*(v1.x-v3.x)-(v.x-v3.x)*(v1.y-v3.y)) > 0$
+
+\item [Vertices (\refstepcounter{ddnum} \dtheddnum \label{dVertices}):] A set of all vetices of the mesh. More formally,\\
+{\it Vertices}: {\tt MeshT} $\rightarrow$ set of {\tt VertexT}\\
+{\it Vertices}($m$: {\tt MeshT}) $\equiv$ \{$v$: {\tt VertexT} $|$ ($\forall\ c$: {\tt CellT} $|\ c\ \in m :v\ \in c):v\}$
+
+\item [Edges (\refstepcounter{ddnum} \dtheddnum \label{dEdges}):] A set of all edges of the mesh. More formally, \\
+{\it Edges}: {\tt MeshT} $\rightarrow $ set of $EdgeT$\\
+{\it Edges}($m$: {\tt MeshT}) $\equiv \{v1,\ v2$: {\tt VertexT} $|\ (\forall c$: {\tt CellT} $|\ c \in m:\\ v1 \in c \wedge v2 \in c \wedge v1 \neq v2):\{v1, v2\}\}$
+ 
+\item [BoundaryEdges (\refstepcounter{ddnum} \dtheddnum \label{dBoundaryEdges}):]A set of edges are boundary edges if they form a boundary of a mesh. More formally,\\
+{\it BoundaryEdges}: {\tt MeshT} $\rightarrow $ set of $EdgeT$\\
+{\it BoundaryEdges}($m$: {\tt MeshT}) $\equiv \{ b$: {\tt EdgeT} $|\ b\in$ {\it Edges}($m$) $\wedge \\
+(\# \{c$: {\tt CellT} $|\ c\in m \wedge$ {\it BelongToCell}({\it b, c}): $c\}=1$): $b\}$
+
+\item [BoundaryVertices (\refstepcounter{ddnum} \dtheddnum \label{dBoundaryVertices}):] A set of boundary vertices of the mesh. More formally,\\
+{\it BoundaryVertices}: {\tt MeshT} $\rightarrow$ set of {\tt VertexT}\\
+{\it BoundaryVertices}($m$: {\tt MeshT}) $\equiv \\
+\{v$: {\tt VertexT} $|\ v \in$ {\it BoundaryEdges}($m$): $v\}$
+
+\item [Bounded (\refstepcounter{ddnum} \dtheddnum \label{dBounded}):] A mesh is bounded if the boundary edges form a closed polygon(all vertices of boundary edges belong to exactly two boundary edges). More formally, \\
+{\it Bounded}: {\tt MeshT} $\rightarrow \mathbb{B}$\\
+{\it Bounded}($m$: {\tt MeshT}) $\equiv \forall v$: {\tt VertexT} $|\ v\in$ {\it BoundaryVertices}($m$): \\
+($\# \{e$: {\tt EdgeT} $|\ e \in$ {\it BoundaryEdge}($m$) $\wedge v\in e: e\}=2)$
+
+\item [Conformal (\refstepcounter{ddnum} \dtheddnum \label{dConformal}):] In 2D, a mesh is conformal if the intersection of any two cells is either a vertex or an edge or empty. More formally,\\
+{\it Conformal}: {\tt MeshT} $\rightarrow \mathbb{B}$\\
+{\it Conformal}($m$: {\tt MeshT}) $\equiv \forall c1, c2$: {\tt CellT} $|\ c1\in m \wedge c2\in m \wedge c1\neq c2 : \\
+(\exists e$: {\tt EdgeT} $|\ e\in$ {\it Eeges}($m$): $(\exists v:$ {\tt VertexT} $|\ v\in$ {\it Vertices}($m$):\\
+$(c1\cap c2 = e \vee c1\cap c2 = v \vee c1\cap c2 = \emptyset) \wedge (\neg$ {\it OnEdge(v, e))  ))}
+
+\item [NoInteriorIntersect (\refstepcounter{ddnum} \dtheddnum \label{dNoInteriorIntersect}):] NoInteriorIntersect is true if a point in space (of type VertexT) is inside only one cell of the mesh. More formally,\\
+{\it NoInteriorIntersect}: {\tt MeshT} $\rightarrow \mathbb{B}$\\
+{\it NoInteriorIntersect}($m$: {\tt MeshT}) $\equiv \forall c1, c2$: {\tt CellT} $|\\ c1\in m \wedge c2\in m \wedge c1 \neq c2:
+ (\forall v$: {\tt VertexT} $|$ {\it Inside(v, c1)}: $\neg$ {\it Inside(v,c2))}
+
+\item [ValidMesh (\refstepcounter{ddnum} \dtheddnum \label{dValidMesh}):] A mesh is valid if the mesh is bounded, conformal, and any point is only inside one cell. More formally, \\
+{\it ValidMesh}: {\tt MeshT} $\rightarrow \mathbb{B}$\\
+{\it ValidMesh}($m$: {\tt MeshT}) $\equiv (\forall e$: {\tt EdgeT} $|\ e\in$ {\it Edges(m): ValidEdge(e))} \\
+$\wedge\ (\forall c$: {\tt CellT} $|\ c\in m$: {\it ValidCell(c))} $\wedge$\\
+{\it Bounded(m)} $\wedge$ {\it Conformal(m)} $\wedge$ {\it NoInteriorIntersect(m)} 
+
+\item [CoveringUp (\refstepcounter{ddnum} \dtheddnum \label{dCoveringUp}):] True if two meshes covering up each other, that is, if all endpoints of the boundary edges of one mesh are on the boundary edges or are end points of the boundary edges of another mesh. More formally, \\
+{\it CoveringUp}: {\tt MeshT} $\times$ {\tt MeshT} $\rightarrow \mathbb{B}$\\
+{\it CoveringUp}($m1,\ m2$: {\tt MeshT}) $\equiv \forall v1,\ v2$: {\tt VertexT}, $| \\
+ v1 \in$ {\it BoundaryVertice(m1)} $\wedge\ v2\in$ {\it BoundaryVertices(m2)}: \\
+ $(\exists b1,\ b2$: {\tt EdgeT} $|\ b1\in$ {\it BoundaryEdges(m1)} $\wedge b2\in$ {\it BoundaryEdges(m2):
+ (OnEdge(v1, b2)} $\vee v1\in b2) \wedge$ {\it (OnEdge(v2, b1)} $\vee v2\in b1))$
+
+\item [InstructionT (\refstepcounter{ddnum} \dtheddnum \label{dInstructionT}):] The type of instructions is defined as:\\
+{\tt InstructionT} := \{{\tt REFINE, COARSEN, NOCHANGE}\}
+
+\item [CellInstructionT (\refstepcounter{ddnum} \dtheddnum \label{dCellInstructionT}):] The type of instructions on a cell is defined as:\\
+{\tt CellInstructionT}:= tuple of ({\it cell}: {\tt CellT}, {\it instr}: {\tt InstructionT})\\
+(For each cell, there is an instruction for refining, coarsening, or nochange.) 
+
+\item [RCinstructionT (\refstepcounter{ddnum} \dtheddnum \label{dRCInstructionT}):] The type of instructions on a mesh is defined as:\\
+{\tt RCinstructionT}:= tuple of ({\it rORc}: {\tt InstructionT}, {\it cInstr}: set of {\tt CellInstructionT})\\
+(For each mesh, there is an instruction on whole mesh, and there are set of instruction on each cell.)
+
+\item [Refined (\refstepcounter{ddnum} \dtheddnum \label{dRefined}):] True if a mesh $M'$ is a refined mesh of a mesh $M$. More formally\\
+{\it Refined}: {\tt MeshT} $\times$ {\tt MeshT} $\times$ {\tt RCinstructionT} $\rightarrow \mathbb{B}$\\
+{\it Refined}({\it m', m}: {\tt MeshT}, {\it rc}: {\tt RCinstructionT}) $\equiv$ \\
+{\it rc.rORc} = {\tt REFINE} $\wedge$ {\it ValidMesh(m)} $\wedge$ {\it ValidMesh(m')} $\wedge$ \\
+{\it CoveringUp(m', m)} $\wedge \# m' \geq \#m$
+
+\item [Coarsened (\refstepcounter{ddnum} \dtheddnum \label{dCoarsened}):] True if a mesh $M'$ is a coarsened mesh of a mesh $M$. More formally \\
+{\it Coarsened}: {\tt MeshT} $\times$ {\tt MeshT} $\times$ {\tt RCinstructionT} $\rightarrow \mathbb{B}$\\
+{\it Coarsened}({\it m', m}: {\tt MeshT}, {\it rc}: {\tt RCinstructionT}) $\equiv$ \\
+{\it rc.rORc} = {\tt COARSEN} $\wedge$ {\it ValidMesh(m)} $\wedge$ {\it ValidMesh(m')} $\wedge$ \\
+{\it CoveringUp(m', m)} $\wedge \# m' \leq \# m$
+
+\end{description}
+
+\subsubsection{System Behaviour}
+System Behaviour, shown through functional requirements, defines what the software should do. The functional requirements, as well as nonfunctional requirements in Section \ref{nonFunReq}, partially come from \citet{Smith2004}. \citet{Smith2004} listed all requirements that are common for mesh generation systems. They also considered the difference between meshes in term of variabilities. However, the mesh generations analyzed by \citet{Smith2004} are targeted at full FEA applications. PMGT only manages the geometric information about the mesh, not other FEA related information, such as boundary condition and material property. Hence, only commonalities that is meaningful for PMGT are selected. Variabilities with parameters of variation that are suitable for PMGT are also considered. Other part of the requirements are obtained from Dr. Smith.
+
+New functional requirements, \emph{RCInstruction} (\fref{RCInstruction}) and \emph{Help} (\fref{Help}) are added. \fref{RCInstruction} is unique to PMGT and \fref{Help} facilities the non-functional requirements \emph{Usability} (\nref{Usability}).
+
+We specify both functional requirements and non-functional requirements in the tables. In each table, the field \emph{Description} gives a brief description of this requirement. It tells what PMGT should do to fulfill this requirement. There are two potential sources, shown in the \emph{Source} field, for each requirement. One source is from \citet{Smith2004}, and the other comes from Dr.~Smith. If the requirement is from \citet{Smith2004}, then this field will show the commonality number, with a prefix $C$ and the associated variability, shown by a prefix $V$. Where applicable, \emph{Related Data Definitions} and \emph{Related Theoretical Models} gives the numbers of related data definitions and the numbers of related theoretical models, respectively. These two field only appear for functional requirements. The \emph{Binding Time} field either shows scope time or run time. \emph{Scope time} means that this requirement is determined when the SRS is written. \emph{Run time} means that this requirement is determined when the system is running. \emph{History} records the time of creating and changing of the requirements. 
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{RefiningMesh} \\ 
+{\bf Requirements Name} & \mi{RefiningMesh}\\ \hline
+Description & PMGT should have capabilities for refining an existing mesh.\\
+& {\it I.rORc} = {\tt REFINE} $\wedge$\\
+& {\it Refined}($M^{\mbox{OUT}},\ M^{\mbox{IN}})$\\
+Source & C1, V3\\ 
+Related Data Definitions & \dref{dInstructionT}, \dref{dRCInstructionT}, \dref{dRefined}\\
+Related Theoretical Models & \tmref{tmRefining}\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2005. Change the name from ``ImprovingMesh'' to ``RefiningMesh''.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{CoarseningMesh} \\ 
+{\bf Requirements Name} & \mi{CoarseningMesh}\\ \hline
+Description & PMGT should have capabilities for coarsening an existing mesh.\\
+& {\it I.rORc} = {\tt COARSEN} $\wedge $\\
+& {\it Coarsened}$(M^{\mbox{OUT}}, M^{\mbox{IN}})$\\
+Source & C1, V3\\ 
+Related Data Definitions & \dref{dInstructionT}, \dref{dRCInstructionT}, \dref{dCoarsened}\\
+Related Theoretical Models & \tmref{tmCoarsening}\\
+Binding Time & Scope time\\
+History & Created -- October, 2006.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{RefiningOrCoarsening} \\ 
+{\bf Requirements Name} & \mi{RefiningOrCoarsening}\\ \hline
+Description & PMGT can either refine a given mesh to a refined mesh, or coarsen a mesh to a coarsened mesh. However, PMGT cannot do both refining and coarsening at the same time.\\
+Source & C1, V3\\ 
+Related Data Definitions & \dref{dInstructionT}, \dref{dRCInstructionT}, \dref{dRefined}, \dref{dCoarsened}\\
+Related Theoretical Models &  \tmref{tmRefining}, \tmref{tmCoarsening}\\
+Binding Time & Run time\\
+History & Created -- October, 2006.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{MeshType} \\ 
+{\bf Requirements Name} & \mi{MeshType}\\ \hline
+Description & The mesh generated by PMGT is unstructured.\\
+Source & C1, V6\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{ElmShape} \\ 
+{\bf Requirements Name} & \mi{ElmShape}\\ \hline
+Description & The shape of the elements in both input and output meshes are triangles.\\
+& $\forall c1, c2$: {\tt CellT} $|$\\
+& $c1\in M^{\mbox{IN}} \wedge c2\in M^{\mbox{OUT}}:$\\
+& $\# c1=3\wedge$ {\it Area}$(c1)>0 \wedge $\\
+& $\# c2 = 3 \wedge$ {\it Area}$(c2)>0$\\
+Source & C1, V9\\ 
+Related Data Definitions & \dref{dCellT}, \dref{dArea}\\
+Related Theoretical Models & \tmref{tmRefining}, \tmref{tmCoarsening} \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+& Modified -- October, 2006. Take out the requirement for generating quadrilateral meshes.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{DomainDimension} \\ 
+{\bf Requirements Name} & \mi{DomainDimension}\\ \hline
+Description & The computational domain is in 2D space.\\
+Source & C1, V13\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models &\tmref{tmRefining}, \tmref{tmCoarsening} \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{Conformal} \\ 
+{\bf Requirements Name} & \mi{Conformal}\\ \hline
+Description & Both input and output meshes are conformal.\\
+& {\it Conformal}$(M^{\mbox{IN}}) \ \wedge$ \\
+& {\it Conformal}$(M^{\mbox{OUT}})$\\
+Source & C1, V18\\ 
+Related Data Definitions & \dref{dConformal} \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+& Modified -- October, 2006. Take out the requirement for generating non-conformal meshes.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{InputDefinition} \\ 
+{\bf Requirements Name} & \mi{InputDefinition}\\ \hline
+Description & The input of PMGT should be provided by the embedding application.\\
+Source & C8\\ 
+Related Data Definitions & \dref{dMeshT}, \dref{dInstructionT}, \dref{dRCInstructionT} \\
+Related Theoretical Models & \tmref{tmRefining}, \tmref{tmCoarsening} \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified--October 2005. Change the name from ``Input'' to ``InputDefinition'' to clarify that this requirements is about the source of the input. \\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{RCInstruction} \\ 
+{\bf Requirements Name} & \mi{RCInstruction}\\ \hline
+Description & The Instruction on how to refine/coarsen a mesh includes the instruction of whether to refine or coarsen the mesh and an individual instruction for each element of the the mesh to indicate refining, coarsening, or no change.\\
+Source & Dr. Smith\\ 
+Related Data Definitions & \dref{dInstructionT}, \dref{dCellInstructionT}, \dref{dRCInstructionT} \\
+Related Theoretical Models & \tmref{tmRefining}, \tmref{tmCoarsening} \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{OutputStorage} \\ 
+{\bf Requirements Name} & \mi{OutputStorage}\\ \hline
+Description & The output of PMGT is stored in memory or in files or in both memory and files.\\
+Source & C12, Dr. Smith\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Run time\\
+History & Created -- June, 2005.\\
+ & Modified -- October 2005. Change the name from ``Output'' to ``OutputStorage'' to clarify that this requirements is about the storage of the output.\\
+ & Modified -- October 2006. Add the requirement of storing the output mesh in files or both memory and files.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{VertexUniqueID} \\ 
+{\bf Requirements Name} & \mi{VertexUniqueID}\\ \hline
+Description & Each vertex in the output file has a unique identifier.\\
+Source & C2\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{ElmUniqueID} \\ 
+{\bf Requirements Name} & \mi{ElmUniqueID}\\ \hline
+Description & Each element in the output file has a unique identifier.\\
+Source & C3\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{ElmTopology} \\ 
+{\bf Requirements Name} & \mi{ElmTopology}\\ \hline
+Description & The topology of an element in the output file is given by the connectivity of its set of vertices.\\
+Source & C4\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{OutElmOrder} \\ 
+{\bf Requirements Name} & \mi{OutElmOrder}\\ \hline
+Description & The element information in output files is listed in ascending order.\\
+Source & C13, V34\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{OutVertexOrder} \\ 
+{\bf Requirements Name} & \mi{OutVertexOrder}\\ \hline
+Description & The vertex information, such as the coordinates, in output files is listed in ascending order.\\
+Source & C14, V35\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{funnum} \fthefunnum \label{Help} \\ 
+{\bf Requirements Name} & \mi{Help}\\ \hline
+Description & Helps on documenting the interface and the functionality of each function should be provided.\\
+Source & Dr. Smith\\ 
+Related Data Definitions & N/A \\
+Related Theoretical Models & N/A \\
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+ & Modified -- October 2005. Add the requirement of documenting functionality of each function.\\
+ & Modified -- October, 2006. Field for ``Related Data Definitions'' and ``Related Theoretical Models'' are added.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\subsection{Non-functional Requirements \label{nonFunReq}}
+All non-functional requirements listed in \citet{Smith2004} are selected except for C16, which is solution tolerance, since a mesh refined/coarsened by different algorithms may have different solutions, but all of these solutions can still be valid. All potential output meshes are valid as long as the output meshes are covering/covered up meshes of the original mesh, and they are refined/coarsened according to the RCInstruction. The resulting mesh is difficult to measure in terms of \emph{solution tolerance}. Three new non-functional requirements, which are \emph{LookAndFeel} (\nref{LookAndFeel}), \emph{Usability} (\nref{Usability}), and \emph{Maintainability} (\nref{Maintainability}), are added. These requirements are mentioned in \citet{Lai2004}. 
+
+PMGT is difficult to validate. One reason is that the solution for refining/coarsening a mesh is unknown, as mentioned above. The other reason is that it is difficult to write validatable requirements, especially for nonfunctional requirements. For example, what is the proper way for specifying the requirement of \emph{Usability} (\nref{Usability}) of PMGT? On the one hand, that \emph{the software should easy to use} is not validatable. On the other hand, that \emph{a person should be able to use the software in two days} is validatable. However, the measurement, \emph{two days}, often lacks a justifiable rationale. 
+
+The approach to validate this kind of requirements are to compare it with other software with similar functionality. Phrases that are in italics and capitalized, such as \emph{MANPROP}, represent constant defined in Section \ref{SecAC}. Usually, these constants come from other applications with similar functionalities. For example, the \emph{Usability} requirement of PMGT is presented as follows:
+\begin{quotation}
+This system should be easy to use. Users with the background specified in Section~\ref{AreqUserchar} should take \emph{LEARNTIME} to reproduce an example mesh, which is specified by the test case \cref{ATCrm} in the Appendix \ref{Atest}.
+\end{quotation}
+First, more general requirement is given. Then, a suggestion to reproduce an example mesh is specified. The constant \emph{LEARNTIME} is defined as the time to produce the same mesh for users with the same background using AOMD. 
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{nfnum} \nthenfnum \label{Performance} \\ 
+{\bf Requirements Name} & \mi{Performance}\\ \hline
+Description & Refining/coarsening a mesh using multiple processors should be faster than when using a single processor. In addition, the performance of PMGT should be comparable with that of similar applications. The execution time to refine an example mesh, which is specified by the test case \cref{ATCrm} in the Appendix \ref{Atest}.\\
+Source & C15, V39\\ 
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{nfnum} \nthenfnum \label{Precision} \\ 
+{\bf Requirements Name} & \mi{Precision}\\ \hline
+Description & The number of decimal digits should agree with the IEEE standard for floating-point numbers.\\
+Source & C17, V41\\ 
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005.\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{nfnum} \nthenfnum \label{Exception} \\ 
+{\bf Requirements Name} & \mi{Exception}\\ \hline
+Description & Run-time exception handling should check at least the following exceptions: division by zero, redundant vertices, redundant edges, redundant cells.\\
+Source & C18, V42\\ 
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{nfnum} \nthenfnum \label{Portability} \\ 
+{\bf Requirements Name} & \mi{Portability}\\ \hline
+Description & PMGT should build on a platform with access to SHARCNET or on a the system that has similar architecture to SHARCNET. The memory capacity should be \emph{MEMCAP}.\\
+Source & C19, V37, V38\\ 
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{nfnum} \nthenfnum \label{LookAndFeel} \\ 
+{\bf Requirements Name} & \mi{LookAndFeel}\\ \hline
+Description & PMGT should follow the programming conventions of the language in which the application is coded in.\\
+Source & Dr. Smith\\ 
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{nfnum} \nthenfnum \label{Usability} \\ 
+{\bf Requirements Name} & \mi{Usability}\\ \hline
+Description & This system should be easy to use. Users with the background specified in Section~\ref{AreqUserchar} should take \emph{LEARNTIME} to reproduce an example mesh, which is specified in the Appendex \ref{Atest}. \\
+Source & Dr. Smith\\ 
+%Depending on & None\\
+Binding Time & Scope time\\
+History & Created -- June, 2005\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Requirements Number} & \refstepcounter{nfnum} \nthenfnum \label{Maintainability} \\ 
+{\bf Requirements Name} & \mi{Maintainability}\\ \hline
+Description & The system should be developed in the way that the effort spent to maintain the system or to add in features would be minimum. The redevelopment time to add a new algorithm to coarsen meshes in PMGT should be \emph{MANPROP}.\\
+Source & Dr. Smith\\ 
+Binding Time & Scope time\\
+History & Created -- June, 2005\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\section{Other System Issues \label{AreqSecOSI}}
+This section includes some other supporting information that might contribute to the success or failure of the system development. The following factors are considered:
+\begin{itemize}
+\item Open issues are statements of factors that are uncertain and might make significant difference to the system.
+  \item Off-the-shell solutions are existing systems and/or components bought or borrowed. They could be the potential solutions.
+  \item Waiting rooms provide a blueprint of how the system will be extended.
+\end{itemize}
+
+\subsection{Open Issues}
+There are no open issues for PMGT at this stage.
+\subsection{Off-the-shelf Solutions}
+The following programs may be used in PMGT.
+\begin{itemize}
+\item AOMD: a mesh management library (or database) that is able to provide a variety of services for mesh users \citep{AOMD2006}. 
+\end{itemize}
+
+\subsection{Waiting Rooms}
+Here, we list the possible changes that can affect the extension of the system. These changes are related to the assumptions specified in Section \ref{AreqSolution}.
+\begin{enumerate}
+\item PMGT may produce both structured and unstructured meshes.
+\item PMGT may produce both conformal and nonconformal meshes.
+\item The elements of input and output mesh may be of a shape other than triangles.
+\item The system may deal with invalid input mesh.
+\item The system may accommodate a mixed mesh.
+\item The system may accommodate a hybrid mesh.
+\item The system may deal with a 3D problem domain.
+\end{enumerate}
+\section{Traceability Matrix \label{AreqSecTM}}
+The traceability matrix defined in this section gives a big picture of the associations among goals, assumptions, data definitions, theoretical models, and functional requirements. Goals are ideal general models. After assumptions are applied, these goals are restricted to problems that can be solved by PMGT. Data definitions and theoretical models are used to describe the requirements. The matrix is too big to fit one page. For the sake of clarity, it is split into three parts in five tables, which are Table \ref{AreqTM11}, Table \ref{AreqTM12}, Table \ref{AreqTM21}, Table \ref{AreqTM22}, and Table \ref{AreqTM3}. In addition, only items that have a relation with items in the same part are listed. If there is a \checkmark in a cell, it means that if the goal, or the assumption, or the theoretical model, or the data definition, or the requirement in the corresponding column changes, the assumption, or the data definition, or the theoretical model, or the requirement in the corresponding row should also change.
+
+\begin{table}[ht!]
+\centering
+\begin{tabular}{|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|}
+\hline
+&\gref{gRefining} & \gref{gCoarsening} & \aref{aDimension} & \aref{aClosed} & \aref{aUnstructured} & \aref{aConformal} & \aref{aElmShape} & \aref{aValidMesh}& \tmref{tmRefining} & \tmref{tmCoarsening}\\
+\hline\hline
+\aref{aDimension}& \checkmark & \checkmark & \checkmark &  &  &  &  &  & &  \\\hline %a1
+\aref{aClosed}& \checkmark & \checkmark &  & \checkmark &  &  &  &  &  &  \\\hline %a2
+\aref{aUnstructured}& \checkmark & \checkmark &  &  & \checkmark &  &  & & &  \\\hline %a3
+\aref{aConformal}& \checkmark & \checkmark &  &  &  & \checkmark &  &  & & \\\hline %a4
+\aref{aElmShape}& \checkmark & \checkmark & \checkmark &  &  &  & \checkmark &  &  &\\\hline %a5
+\aref{aValidMesh}& \checkmark & \checkmark &  &  &   &   &   & \checkmark &  &\\\hline %a5
+\hline
+\dref{dVertexT}& \checkmark & \checkmark & \checkmark &  &  &  &  &  & & \\\hline %d1
+\dref{dEdgeT}& \checkmark & \checkmark &  &  &  &  &  &  &  &\\\hline%d2
+\dref{dValidEdge}& \checkmark & \checkmark &  &  &  &  &  &  &  &\\\hline%d2
+\dref{dCellT}& \checkmark & \checkmark & \checkmark &  &  &  & \checkmark & & &  \\\hline%d3
+\dref{dArea}& \checkmark & \checkmark & \checkmark &  &  &  & \checkmark & & &  \\\hline%d3
+\dref{dValidCell}& \checkmark & \checkmark & \checkmark &  &  &  & \checkmark & & &  \\\hline%d3
+\dref{dMeshT}& \checkmark & \checkmark &  &  &  &  &  &  &  & \\\hline%d4
+\dref{dOnEdge}& \checkmark & \checkmark & \checkmark &  &  &  &  &  &  & \\\hline%d4
+\dref{dBelongToCell}& \checkmark & \checkmark &  \checkmark&  &  &  &  &  &  & \\\hline%d4
+\dref{dInside}& \checkmark & \checkmark & \checkmark &  &  &  & \checkmark &  &  & \\\hline%d4
+\dref{dVertices}& \checkmark & \checkmark &  &  &  &  &  &  &  & \\\hline%d4
+\dref{dEdges}& \checkmark & \checkmark & \checkmark &  &  &  &  & \checkmark &  & \\\hline%d4
+\dref{dBoundaryEdges}& \checkmark & \checkmark & \checkmark &  &  &  &  &  &  &\\\hline%d5
+\dref{dBoundaryVertices}& \checkmark & \checkmark & \checkmark &  &  &  &  &  &  &\\\hline%d5
+\dref{dBounded}& \checkmark & \checkmark & \checkmark & \checkmark &  &  &  &  &  &\\\hline%d6
+\dref{dConformal}& \checkmark & \checkmark & \checkmark &  &  & \checkmark &  &  &  &\\\hline%d7
+\dref{dNoInteriorIntersect}& \checkmark & \checkmark & \checkmark &\checkmark  &   &  & &  &  &\\\hline%d3
+\dref{dValidMesh}& \checkmark & \checkmark &  &  & &  &  &  &  &\\\hline%d3
+\dref{dInstructionT}& \checkmark & \checkmark &  &  &  &  &  &  &  &\\\hline%d3
+\dref{dCellInstructionT}& \checkmark & \checkmark &  &  &  &  &  &  &  &\\\hline%d11
+\dref{dRCInstructionT}& \checkmark & \checkmark &  &  &  &  &  &  &  &\\\hline%d11
+\dref{dCoveringUp}& \checkmark & \checkmark & \checkmark & \checkmark &  &  &  &  & &\\\hline%d8
+\dref{dRefined}& \checkmark &  &  &  &  &  &  &  &  &\\\hline%d9
+\dref{dCoarsened}&  & \checkmark &  &  &  &  &  &  &  &\\\hline%d10
+\hline
+\tmref{tmRefining} & \checkmark &  & \checkmark &  &  &  &  &  & \checkmark& \\\hline
+\tmref{tmCoarsening}&  & \checkmark & \checkmark &  &  &  &  &  & &\checkmark\\\hline
+\end{tabular}
+\caption{Traceability Matrix (PART I): Goals, Assumptions, Theoretical Models, Data Definitions, and Requirements (I)}
+\label{AreqTM11}
+\end{table}
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|}
+\hline
+&\gref{gRefining} & \gref{gCoarsening} & \aref{aDimension} & \aref{aClosed} & \aref{aUnstructured} & \aref{aConformal} & \aref{aElmShape} & \aref{aValidMesh}& \tmref{tmRefining} & \tmref{tmCoarsening}\\
+\hline\hline
+\fref{RefiningMesh}& \checkmark &  &  &  &  &  & & & \checkmark & \\\hline%f2
+\fref{CoarseningMesh}&  & \checkmark &  &  &  &  & & &  &\checkmark \\\hline%f3
+\fref{RefiningOrCoarsening}& \checkmark & \checkmark &  &  &  &  & & & \checkmark & \checkmark\\\hline%f1
+\fref{MeshType}&  &  &  &  & \checkmark &  &  &  & & \\\hline%f4
+\fref{ElmShape}& \checkmark & \checkmark & \checkmark &  &  &  & \checkmark & & \checkmark & \checkmark \\\hline%f5
+\fref{DomainDimension}& \checkmark & \checkmark & \checkmark &  &  &  & & & \checkmark & \checkmark \\\hline%f6
+\fref{Conformal}& \checkmark & \checkmark &  &  &  & \checkmark &  & & & \\\hline%f7
+\fref{InputDefinition}& \checkmark & \checkmark &  &  &  &  & & & \checkmark & \checkmark\\\hline%f11
+\fref{RCInstruction}& \checkmark & \checkmark &  &  &  &  &  && \checkmark & \checkmark \\\hline%f12
+\fref{OutputStorage}& \checkmark & \checkmark &  &  &  &  & & &  &  \\\hline%f13
+\fref{Help} & \checkmark & \checkmark &  &  &  &  & & &  &  \\\hline%f16
+\end{tabular}
+\caption{Traceability Matrix (PART I): Goals, Assumptions, Theoretical Models, Data Definitions, and Requirements (II)}
+\label{AreqTM12}
+\end{table}
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{|p{0.05\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|}
+\hline
+&\dref{dVertexT} & \dref{dEdgeT} & \dref{dValidEdge} & \dref{dCellT} & \dref{dArea} & \dref{dValidCell} & \dref{dMeshT} & \dref{dOnEdge}& \dref{dBelongToCell}& \dref{dInside} & \dref{dVertices} & \dref{dEdges}   \\
+\hline\hline
+\dref{dVertexT}& \checkmark &  &  &  &  &  &  &  &  &  &  &  \\\hline %d1
+\dref{dEdgeT}& \checkmark & \checkmark &  &  &  &  &  &  &  &  &  &   \\\hline%d2
+\dref{dValidEdge}&  & \checkmark & \checkmark &  &  &  &  &  &  &  &  &   \\\hline%d2
+\dref{dCellT}& \checkmark &  &  & \checkmark &  &  &  &  &  &  &  &  \\\hline%d3
+\dref{dArea}& \checkmark &  &  &  & \checkmark &  &  &  &  &  &  &  \\\hline%d3
+\dref{dValidCell}&  &  &  & \checkmark & \checkmark & \checkmark &  &  &  &  &  &  \\\hline%d3
+\dref{dMeshT}&  &  &  & \checkmark &  &  & \checkmark &  &  &  &  &   \\\hline%d4
+\dref{dOnEdge}& \checkmark & \checkmark &  &  &  &  &  & \checkmark &  &  &  &   \\\hline%d4
+\dref{dBelongToCell}& \checkmark & \checkmark &  & \checkmark &  &  &  &  &  \checkmark&  &  &   \\\hline%d4
+\dref{dInside}& \checkmark &  &  & \checkmark &  &  &  &  &  & \checkmark &  &   \\\hline%d4
+\dref{dVertices}& \checkmark &  &  & \checkmark &  &  & \checkmark &  &  &  & \checkmark &   \\\hline%d4
+\dref{dEdges}& \checkmark & \checkmark &  & \checkmark &  &  & \checkmark &  &  &  &  &  \checkmark \\\hline%d4
+\dref{dBoundaryEdges}&  & \checkmark & & \checkmark  &  &  &  &  & \checkmark &  &  &  \checkmark \\\hline%d5
+\dref{dBoundaryVertices}& \checkmark &  &   &  &  &  & \checkmark &  &  &  &  &   \\\hline%d5
+\dref{dBounded}& \checkmark & \checkmark &  &  &  &  & \checkmark &  &  &  &  &  \\\hline%d6
+\dref{dConformal}& \checkmark &  \checkmark&  & \checkmark &  &  & \checkmark & & & & \checkmark & \checkmark\\\hline%d7
+\dref{dNoInteriorIntersect}& \checkmark &  &  & \checkmark &  &  & \checkmark &  &  & \checkmark &  &  \\\hline%d6
+\dref{dValidMesh}&  &  & \checkmark &  &  & \checkmark & \checkmark &  &  &  &  &  \\\hline%d6
+\dref{dCoveringUp}& \checkmark & \checkmark &   &  &   &   & \checkmark & \checkmark &  &  &  &   \\\hline%d8
+\dref{dInstructionT}&  &  &  &  &  &  &  &  &  &  &  &  \\\hline%d10
+\dref{dCellInstructionT}&  &  &  & \checkmark &  &  &  &  &  &  &  &  \\\hline%d11
+\dref{dRCInstructionT}&  &  &  &  &  &  &  &  &  &  &  &  \\\hline%d11
+\dref{dRefined}&  &  &  &  &  &  &  &  &   &  &  &   \\\hline%d9
+\dref{dCoarsened}&  &  &  &  &  &  &  &  &   &  &   & \\\hline%d10
+\hline
+\tmref{tmRefining}&  &  &  &  &  &  & \checkmark &  &  &  &  &  \\\hline%f2
+\tmref{tmCoarsening}&  &  &  &  &  &  & \checkmark &  &  &  &  &   \\\hline%f3
+\hline
+\fref{RefiningMesh}&  &  &  &  &  &  &  &  &  &  &  &  \\\hline%f2
+\fref{CoarseningMesh}&  &  &  &  &  &  &  &  &  &  &  &   \\\hline%f3
+\fref{RefiningOrCoarsening}&  &  &  &  &  &  &  &  &  &  &  & \checkmark  \\\hline%f1
+\fref{ElmShape}&  &  &  &  \checkmark &  & \checkmark &  &  &  &  &  &   \\\hline%f5
+\fref{Conformal}&  &  &  &  &  &  &  &  &  &  &  &   \\\hline%f7
+\fref{InputDefinition}&  &  &  &  & &  & \checkmark  &  &  &  & & \\\hline%f11
+\fref{RCInstruction}&  &  &  &  &  &  &  &  &  &  &  &  \\\hline%f12
+\end{tabular}
+\caption{Traceability Matrix (PART II): Data Definitions and Requirements (I)}
+\label{AreqTM21}
+\end{table}
+
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{|p{0.05\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|p{0.04\textwidth}|}
+\hline
+&\dref{dBoundaryEdges} & \dref{dBoundaryVertices} & \dref{dBounded} & \dref{dConformal} & \dref{dNoInteriorIntersect} & \dref{dValidMesh} & \dref{dCoveringUp} & \dref{dInstructionT}& \dref{dCellInstructionT}& \dref{dRCInstructionT} & \dref{dRefined} & \dref{dCoarsened}   \\
+\hline\hline
+\dref{dBoundaryEdges}& \checkmark &   &   &  &  &  &  &  &  &  &  &   \\\hline%d5
+\dref{dBoundaryVertices}& \checkmark & \checkmark  &   &  &  &  &  &  &  &  &  &   \\\hline%d5
+\dref{dBounded}& \checkmark & \checkmark & \checkmark &  &  &  &  &  &  &  &  &  \\\hline%d6
+\dref{dConformal}&   &   &  &  \checkmark &  &  &   & & & &   &  \\\hline%d7
+\dref{dNoInteriorIntersect}&  &  &  &   & \checkmark &  &   &  &  &   &  &  \\\hline%d6
+\dref{dValidMesh}&  &  & \checkmark & \checkmark & \checkmark & \checkmark  &   &  &  &  &  &  \\\hline%d6
+\dref{dCoveringUp}& \checkmark & \checkmark &   &  &   &   & \checkmark &  &  &  &  &   \\\hline%d8
+\dref{dInstructionT}&   &   &  &  &  &  &  & \checkmark &  &  &  &  \\\hline%d10
+\dref{dCellInstructionT}&  &  &  &   &  &  &  & \checkmark & \checkmark &  &  &  \\\hline%d11
+\dref{dRCInstructionT}&  &  &  &  &  &  &  & \checkmark & \checkmark & \checkmark &  &  \\\hline%d11
+\dref{dRefined}&  &  &  &  &  & \checkmark & \checkmark & \checkmark &   &  \checkmark& \checkmark &   \\\hline%d9
+\dref{dCoarsened}&  &  &   &  &  & \checkmark & \checkmark & \checkmark &   & \checkmark &   & \checkmark\\\hline%d10
+\hline
+\tmref{tmRefining}&  &  &  &  &  &  &  &  &  & \checkmark &  \checkmark&  \\\hline%f2
+\tmref{tmCoarsening}&  &  &  &  &  &  &  &  &  & \checkmark &  &  \checkmark \\\hline%f3
+\hline
+\fref{RefiningMesh}&  &  &  &  &  &  &  & \checkmark &  & \checkmark & \checkmark &  \\\hline%f2
+\fref{CoarseningMesh}&  &  &  &  &  &  &  & \checkmark &  & \checkmark &  & \checkmark  \\\hline%f3
+\fref{RefiningOrCoarsening}&  &  &  &  &  &  &  & \checkmark &  & \checkmark & \checkmark & \checkmark  \\\hline%f1
+\fref{ElmShape}&  &  &  &   &  &  &  &  &  &  &  &   \\\hline%f5
+\fref{Conformal}&  &  &  & \checkmark &  &  &  &  &  &   &  &   \\\hline%f7
+\fref{InputDefinition}&  &  &  &  &  &  &  & \checkmark &  & \checkmark & & \\\hline%f11
+\fref{RCInstruction}&  &  &  &  &  &  &  & \checkmark & \checkmark & \checkmark &  &  \\\hline%f12
+\end{tabular}
+\caption{Traceability Matrix (PART II): Data Definitions and Requirements (II)}
+\label{AreqTM22}
+\end{table}
+
+
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|p{0.05\textwidth}|}
+\hline
+& \fref{RefiningMesh}& \fref{CoarseningMesh}& \fref{DomainDimension}&\fref{InputDefinition}&\fref{OutputStorage}& \nref{Usability}\\
+\hline\hline
+\fref{RefiningOrCoarsening}& \checkmark & \checkmark &  &  &  &   \\\hline%f1
+\fref{ElmShape}&  &  & \checkmark &  &  & \\\hline%f5
+\fref{RCInstruction}&  &  &  & \checkmark &  &  \\\hline%f12
+\fref{VertexUniqueID}&  &  &  &  & \checkmark & \\\hline%f8
+\fref{ElmUniqueID}&  &  &  &  & \checkmark &  \\\hline%f9
+\fref{ElmTopology}&  &  &  &  & \checkmark &  \\\hline%f10
+\fref{OutElmOrder}&  &  &  &  & \checkmark &  \\\hline%f14
+\fref{OutVertexOrder}&  &  &  &  & \checkmark &  \\\hline%f15
+\fref{Help} &  &  &  &  &  &  \checkmark \\\hline%f16
+\end{tabular}
+\caption{Traceability Matrix (PART III): Requirements }
+\label{AreqTM3}
+\end{table}
+
+
+\section{List of Possible Changes in the Requirements \label{AreqSecPC}}
+The system might evolve to accommodate the following changes in the future. These changes will add additional goals to the software library.
+\begin{enumerate}
+\item The input of PMGT may include material properties.
+\item The input of PMGT may include boundary conditions.
+\end{enumerate}
+
+\section{Values of Auxiliary Constants \label{SecAC}}
+The constants given in this section are used to validate some nonfunctional requirements. The compatible software chosen is AOMD. However, other software can also be used as long as the other software has the required functionalities to validate the given requirement.
+
+\begin{tabular}{p{\firstCol}p{\secondCol}} 
+\emph{LEARNTIME} & The time that reproduce the same example as that specified in nonfunctional requirement \nref{Usability} using AOMD.\\
+\emph{MANPROP} &  The redevelopment time to add the same algorithm as that specified in the nonfunctional requirement \nref{Maintainability}, using AOMD. If the algorithm is already in AOMD, the the time that AOMD took to add it.\\
+\emph{RSPTIME} & The execution time to refine the same mesh as that specified in nonfunctional requirement \nref{Performance} using AOMD. \\ 
+\emph{MEMCAP} & The typical memory capacity of a machine on SHARCNET.\\ 
+\end{tabular}
+
+\newpage
+
+\bibliography{WenRef}
+\bibliographystyle{plainnat}
+
+\end{document}
+

Examples/MeshGenToolbox/doc/speedup.eps

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+  /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly
+  vradius add translate hradius vradius scale 0 0 1 180 270 arc 
+  tMatrix setmatrix lrx hradius sub uly vradius add translate
+  hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix
+  lrx hradius sub lry vradius sub translate hradius vradius scale
+  0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub
+  translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix
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+  sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix
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+  sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix
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+  ind1 get exch ind2 get sub exch } bdef
+/tri_to_matrix {
+  2 0 asub 3 1 asub 4 0 asub 5 1 asub
+  dup 0 get exch 1 get 7 -1 roll astore } bdef
+/compute_transform {
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+  currentfile exch readhexstring pop
+  /bmap xdef pop pop} bdef
+/it {gs np dtri aload pop moveto lineto lineto cp c
+  cols rows 8 compute_transform 
+  {bmap} image gr}bdef
+/il {newpath moveto lineto stroke}bdef
+currentdict end def
+%%EndProlog
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+gs 1004 3428 4789 958 MR c np
+  36   36 1077 4312 FO
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+3214 -469 1071 -2263 357 -755 1077 4312 4 MP stroke
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+1041 4312 mt 1113 4312 L
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+%%Trailer
+%%EOF

Examples/MeshGenToolbox/doc/syscon.eps

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+		begin
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+			{
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+				{
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+			{
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+						ct_Vars /vm currentglobal put
+						({ResourceStatus} stopped) 0 () /SubFileDecode filter cvx exec
+							{ cleartomark false }
+							{ { 3 2 roll pop true } { cleartomark false } ifelse }
+						ifelse
+						ct_Vars /vm get setglobal
+						end
+					}
+				}
+				{ { resourcestatus } }
+			ifelse bind def
+			/CIDFont /Category ct_resourcestatus
+				{ pop pop }
+				{
+				currentglobal  true setglobal
+				/Generic /Category findresource
+				dup length dict copy
+				dup /InstanceType /dicttype put
+				/CIDFont exch /Category defineresource pop
+				setglobal
+				}
+			ifelse
+			ct_UseNativeCapability?
+				{
+				/CIDInit /ProcSet findresource begin
+				12 dict begin
+				begincmap
+				/CIDSystemInfo 3 dict dup begin
+				  /Registry (Adobe) def
+				  /Ordering (Identity) def
+				  /Supplement 0 def
+				end def
+				/CMapName /Identity-H def
+				/CMapVersion 1.000 def
+				/CMapType 1 def
+				1 begincodespacerange
+				<0000> <FFFF>
+				endcodespacerange
+				1 begincidrange
+				<0000> <FFFF> 0
+				endcidrange
+				endcmap
+				CMapName currentdict /CMap defineresource pop
+				end
+				end
+				}
+			if
+			}
+			{
+			/ct_Category 2 dict begin
+			/CIDFont  10 dict def
+			/ProcSet	2 dict def
+			currentdict
+			end
+			def
+			/defineresource
+				{
+				ct_Category 1 index 2 copy known
+					{
+					get
+					dup dup maxlength exch length eq
+						{
+						dup length 10 add dict copy
+						ct_Category 2 index 2 index put
+						}
+					if
+					3 index 3 index put
+					pop exch pop
+					}
+					{ pop pop /defineresource /undefined ct_raise }
+				ifelse
+				} bind def
+			/findresource
+				{
+				ct_Category 1 index 2 copy known
+					{
+					get
+					2 index 2 copy known
+						{ get 3 1 roll pop pop}
+						{ pop pop /findresource /undefinedresource ct_raise }
+					ifelse
+					}
+					{ pop pop /findresource /undefined ct_raise }
+				ifelse
+				} bind def
+			/resourcestatus
+				{
+				ct_Category 1 index 2 copy known
+					{
+					get
+					2 index known
+					exch pop exch pop
+						{
+						0 -1 true
+						}
+						{
+						false
+						}
+					ifelse
+					}
+					{ pop pop /findresource /undefined ct_raise }
+				ifelse
+				} bind def
+			/ct_resourcestatus /resourcestatus load def
+			}
+		ifelse
+		/ct_CIDInit 2 dict
+			begin
+			/ct_cidfont_stream_init
+				{
+					{
+					dup (Binary) eq
+						{
+						pop
+						null
+						currentfile
+						ct_Level2?
+							{
+								{ cid_BYTE_COUNT () /SubFileDecode filter }
+							stopped
+								{ pop pop pop }
+							if
+							}
+						if
+						/readstring load
+						exit
+						}
+					if
+					dup (Hex) eq
+						{
+						pop
+						currentfile
+						ct_Level2?
+							{
+								{ null exch /ASCIIHexDecode filter /readstring }
+							stopped
+								{ pop exch pop (>) exch /readhexstring }
+							if
+							}
+							{ (>) exch /readhexstring }
+						ifelse
+						load
+						exit
+						}
+					if
+					/StartData /typecheck ct_raise
+					}
+				loop
+				cid_BYTE_COUNT ct_CID_STR_SIZE le
+					{
+					2 copy cid_BYTE_COUNT string exch exec
+					pop
+					1 array dup
+					3 -1 roll
+					0 exch put
+					}
+					{
+					cid_BYTE_COUNT ct_CID_STR_SIZE div ceiling cvi
+					dup array exch 2 sub 0 exch 1 exch
+						{
+						2 copy
+						5 index
+						ct_CID_STR_SIZE
+						string
+						6 index exec
+						pop
+						put
+						pop
+						}
+					for
+					2 index
+					cid_BYTE_COUNT ct_CID_STR_SIZE mod string
+					3 index exec
+					pop
+					1 index exch
+					1 index length 1 sub
+					exch put
+					}
+				ifelse
+				cid_CIDFONT exch /GlyphData exch put
+				2 index null eq
+					{
+					pop pop pop
+					}
+					{
+					pop /readstring load
+					1 string exch
+						{
+						3 copy exec
+						pop
+						dup length 0 eq
+							{
+							pop pop pop pop pop
+							true exit
+							}
+						if
+						4 index
+						eq
+							{
+							pop pop pop pop
+							false exit
+							}
+						if
+						}
+					loop
+					pop
+					}
+				ifelse
+				} bind def
+			/StartData
+				{
+				mark
+					{
+					currentdict
+					dup /FDArray get 0 get /FontMatrix get
+					0 get 0.001 eq
+						{
+						dup /CDevProc known not
+							{
+							/CDevProc 1183615869 internaldict /stdCDevProc 2 copy known
+								{ get }
+								{
+								pop pop
+								{ pop pop pop pop pop 0 -1000 7 index 2 div 880 }
+								}
+							ifelse
+							def
+							}
+						if
+						}
+						{
+						 /CDevProc
+							 {
+							 pop pop pop pop pop
+							 0
+							 1 cid_temp /cid_CIDFONT get
+							 /FDArray get 0 get
+							 /FontMatrix get 0 get div
+							 7 index 2 div
+							 1 index 0.88 mul
+							 } def
+						}
+					ifelse
+					/cid_temp 15 dict def
+					cid_temp
+						begin
+						/cid_CIDFONT exch def
+						3 copy pop
+						dup /cid_BYTE_COUNT exch def 0 gt
+							{
+							ct_cidfont_stream_init
+							FDArray
+								{
+								/Private get
+								dup /SubrMapOffset known
+									{
+									begin
+									/Subrs SubrCount array def
+									Subrs
+									SubrMapOffset
+									SubrCount
+									SDBytes
+									ct_Level2?
+										{
+										currentdict dup /SubrMapOffset undef
+										dup /SubrCount undef
+										/SDBytes undef
+										}
+									if
+									end
+									/cid_SD_BYTES exch def
+									/cid_SUBR_COUNT exch def
+									/cid_SUBR_MAP_OFFSET exch def
+									/cid_SUBRS exch def
+									cid_SUBR_COUNT 0 gt
+										{
+										GlyphData cid_SUBR_MAP_OFFSET cid_SD_BYTES ct_GetInterval
+										0 cid_SD_BYTES ct_cvnsi
+										0 1 cid_SUBR_COUNT 1 sub
+											{
+											exch 1 index
+											1 add
+											cid_SD_BYTES mul cid_SUBR_MAP_OFFSET add
+											GlyphData exch cid_SD_BYTES ct_GetInterval
+											0 cid_SD_BYTES ct_cvnsi
+											cid_SUBRS 4 2 roll
+											GlyphData exch
+											4 index
+											1 index
+											sub
+											ct_GetInterval
+											dup length string copy put
+											}
+										for
+										pop
+										}
+									if
+									}
+									{ pop }
+								ifelse
+								}
+							forall
+							}
+						if
+						cleartomark pop pop
+						end
+					CIDFontName currentdict /CIDFont defineresource pop
+					end end
+					}
+				stopped
+					{ cleartomark /StartData ct_reraise }
+				if
+				} bind def
+			currentdict
+			end def
+		/ct_saveCIDInit
+			{
+			/CIDInit /ProcSet ct_resourcestatus
+				{ true }
+				{ /CIDInitC /ProcSet ct_resourcestatus }
+			ifelse
+				{
+				pop pop
+				/CIDInit /ProcSet findresource
+				ct_UseNativeCapability?
+					{ pop null }
+					{ /CIDInit ct_CIDInit /ProcSet defineresource pop }
+				ifelse
+				}
+				{ /CIDInit ct_CIDInit /ProcSet defineresource pop null }
+			ifelse
+			ct_Vars exch /ct_oldCIDInit exch put
+			} bind def
+		/ct_restoreCIDInit
+			{
+			ct_Vars /ct_oldCIDInit get dup null ne
+				{ /CIDInit exch /ProcSet defineresource pop }
+				{ pop }
+			ifelse
+			} bind def
+		/ct_BuildCharSetUp
+			{
+			1 index
+				begin
+				CIDFont
+					begin
+					Adobe_CoolType_Utility /ct_BuildCharDict get
+						begin
+						/ct_dfCharCode exch def
+						/ct_dfDict exch def
+						CIDFirstByte ct_dfCharCode add
+						dup CIDCount ge
+							{ pop 0 }
+						if
+						/cid exch def
+							{
+							GlyphDirectory cid 2 copy known
+								{ get }
+								{ pop pop nullstring }
+							ifelse
+							dup length FDBytes sub 0 gt
+								{
+								dup
+								FDBytes 0 ne
+									{ 0 FDBytes ct_cvnsi }
+									{ pop 0 }
+								ifelse
+								/fdIndex exch def
+								dup length FDBytes sub FDBytes exch getinterval
+								/charstring exch def
+								exit
+								}
+								{
+								pop
+								cid 0 eq
+									{ /charstring nullstring def exit }
+								if
+								/cid 0 def
+								}
+							ifelse
+							}
+						loop
+			} def
+		/ct_SetCacheDevice
+			{
+			0 0 moveto
+			dup stringwidth
+			3 -1 roll
+			true charpath
+			pathbbox
+			0 -1000
+			7 index 2 div 880
+			setcachedevice2
+			0 0 moveto
+			} def
+		/ct_CloneSetCacheProc
+			{
+			1 eq
+				{
+				stringwidth
+				pop -2 div -880
+				0 -1000 setcharwidth
+				moveto
+				}
+				{
+				usewidths?
+					{
+					currentfont /Widths get cid
+					2 copy known
+						{ get exch pop aload pop }
+						{ pop pop stringwidth }
+					ifelse
+					}
+					{ stringwidth }
+				ifelse
+				setcharwidth
+				0 0 moveto
+				}
+			ifelse
+			} def
+		/ct_Type3ShowCharString
+			{
+			ct_FDDict fdIndex 2 copy known
+				{ get }
+				{
+				currentglobal 3 1 roll
+				1 index gcheck setglobal
+				ct_Type1FontTemplate dup maxlength dict copy
+					begin
+					FDArray fdIndex get
+					dup /FontMatrix 2 copy known
+						{ get }
+						{ pop pop ct_defaultFontMtx }
+					ifelse
+					/FontMatrix exch dup length array copy def
+					/Private get
+					/Private exch def
+					/Widths rootfont /Widths get def
+					/CharStrings 1 dict dup /.notdef
+						<d841272cf18f54fc13> dup length string copy put def
+					currentdict
+					end
+				/ct_Type1Font exch definefont
+				dup 5 1 roll put
+				setglobal
+				}
+			ifelse
+			dup /CharStrings get 1 index /Encoding get
+			ct_dfCharCode get charstring put
+			rootfont /WMode 2 copy known
+				{ get }
+				{ pop pop 0 }
+			ifelse
+			exch
+			1000 scalefont setfont
+			ct_str1 0 ct_dfCharCode put
+			ct_str1 exch ct_dfSetCacheProc
+			ct_SyntheticBold
+				{
+				currentpoint
+				ct_str1 show
+				newpath
+				moveto
+				ct_str1 true charpath
+				ct_StrokeWidth setlinewidth
+				stroke
+				}
+				{ ct_str1 show }
+			ifelse
+			} def
+		/ct_Type4ShowCharString
+			{
+			ct_dfDict ct_dfCharCode charstring
+			FDArray fdIndex get
+			dup /FontMatrix get dup ct_defaultFontMtx ct_matrixeq not
+				{ ct_1000Mtx matrix concatmatrix concat }
+				{ pop }
+			ifelse
+			/Private get
+			Adobe_CoolType_Utility /ct_Level2? get not
+				{
+				ct_dfDict /Private
+				3 -1 roll
+					{ put }
+				1183615869 internaldict /superexec get exec
+				}
+			if
+			1183615869 internaldict
+			Adobe_CoolType_Utility /ct_Level2? get
+				{ 1 index }
+				{ 3 index /Private get mark 6 1 roll }
+			ifelse
+			dup /RunInt known
+				{ /RunInt get }
+				{ pop /CCRun }
+			ifelse
+			get exec
+			Adobe_CoolType_Utility /ct_Level2? get not
+				{ cleartomark }
+			if
+			} bind def
+		/ct_BuildCharIncremental
+			{
+				{
+				Adobe_CoolType_Utility /ct_MakeOCF get begin
+				ct_BuildCharSetUp
+				ct_ShowCharString
+				}
+			stopped
+				{ stop }
+			if
+			end
+			end
+			end
+			end
+			} bind def
+		/BaseFontNameStr (BF00) def
+		/ct_Type1FontTemplate 14 dict
+			begin
+			/FontType 1 def
+			/FontMatrix  [0.001 0 0 0.001 0 0] def
+			/FontBBox  [-250 -250 1250 1250] def
+			/Encoding ct_cHexEncoding def
+			/PaintType 0 def
+			currentdict
+			end def
+		/BaseFontTemplate 11 dict
+			begin
+			/FontMatrix  [0.001 0 0 0.001 0 0] def
+			/FontBBox  [-250 -250 1250 1250] def
+			/Encoding ct_cHexEncoding def
+			/BuildChar /ct_BuildCharIncremental load def
+			ct_Clone?
+				{
+				/FontType 3 def
+				/ct_ShowCharString /ct_Type3ShowCharString load def
+				/ct_dfSetCacheProc /ct_CloneSetCacheProc load def
+				/ct_SyntheticBold false def
+				/ct_StrokeWidth 1 def
+				}
+				{
+				/FontType 4 def
+				/Private 1 dict dup /lenIV 4 put def
+				/CharStrings 1 dict dup /.notdef <d841272cf18f54fc13> put def
+				/PaintType 0 def
+				/ct_ShowCharString /ct_Type4ShowCharString load def
+				}
+			ifelse
+			/ct_str1 1 string def
+			currentdict
+			end def
+		/BaseFontDictSize BaseFontTemplate length 5 add def
+		/ct_matrixeq
+			{
+			true 0 1 5
+				{
+				dup 4 index exch get exch 3 index exch get eq and
+				dup not
+					{ exit }
+				if
+				}
+			for
+			exch pop exch pop
+			} bind def
+		/ct_makeocf
+			{
+			15 dict
+				begin
+				exch /WMode exch def
+				exch /FontName exch def
+				/FontType 0 def
+				/FMapType 2 def
+				/FontMatrix matrix def
+				/bfCount 1 index /CIDCount get 256 idiv 1 add
+					dup 256 gt { pop 256} if def
+				/Encoding
+					256 array 0 1 bfCount 1 sub { 2 copy dup put pop } for
+					bfCount 1 255 { 2 copy bfCount put pop } for
+					def
+				/FDepVector bfCount dup 256 lt { 1 add } if array def
+				BaseFontTemplate BaseFontDictSize dict copy
+					begin
+					/CIDFont exch def
+					CIDFont /FontBBox known
+						{ CIDFont /FontBBox get /FontBBox exch def }
+					if
+					CIDFont /CDevProc known
+						{ CIDFont /CDevProc get /CDevProc exch def }
+					if
+					currentdict
+					end
+				BaseFontNameStr 3 (0) putinterval
+				0 1 bfCount dup 256 eq { 1 sub } if
+					{
+					FDepVector exch
+					2 index BaseFontDictSize dict copy
+						begin
+						dup /CIDFirstByte exch 256 mul def
+						FontType 3 eq
+							{ /ct_FDDict 2 dict def }
+						if
+						currentdict
+						end
+					1 index  16
+					BaseFontNameStr  2 2 getinterval cvrs pop
+					BaseFontNameStr exch definefont
+					put
+					}
+				for
+				ct_Clone?
+					{ /Widths 1 index /CIDFont get /GlyphDirectory get length dict def }
+				if
+				FontName
+				currentdict
+				end
+			definefont
+			ct_Clone?
+				{
+				gsave
+				dup 1000 scalefont setfont
+				ct_BuildCharDict
+					begin
+					/usewidths? false def
+					currentfont /Widths get
+						begin
+						exch /CIDFont get /GlyphDirectory get
+							{
+							pop
+							dup charcode exch 1 index 0 2 index 256 idiv put
+							1 index exch 1 exch 256 mod put
+							stringwidth 2 array astore def
+							}
+						forall
+						end
+					/usewidths? true def
+					end
+				grestore
+				}
+				{ exch pop }
+			ifelse
+			} bind def
+		/ct_ComposeFont
+			{
+			ct_UseNativeCapability?
+				{
+				2 index /CMap ct_resourcestatus
+					{ pop pop exch pop }
+					{
+					/CIDInit /ProcSet findresource
+						begin
+						12 dict
+							begin
+							begincmap
+							/CMapName 3 index def
+							/CMapVersion 1.000 def
+							/CMapType 1 def
+							exch /WMode exch def
+							/CIDSystemInfo 3 dict dup
+								begin
+								/Registry (Adobe) def
+								/Ordering
+								CMapName ct_mkocfStr100 cvs
+								(Adobe-) search
+									{
+									pop pop
+									(-) search
+										{
+										dup length string copy
+										exch pop exch pop
+										}
+										{ pop (Identity)}
+									ifelse
+									}
+									{ pop  (Identity)  }
+								ifelse
+								def
+								/Supplement 0 def
+								end def
+							1 begincodespacerange
+							<0000> <FFFF>
+							endcodespacerange
+							1 begincidrange
+							<0000> <FFFF> 0
+							endcidrange
+							endcmap
+							CMapName currentdict /CMap defineresource pop
+							end
+						end
+					}
+				ifelse
+				composefont
+				}
+				{
+				3 2 roll pop
+				0 get /CIDFont findresource
+				ct_makeocf
+				}
+			ifelse
+			} bind def
+		/ct_MakeIdentity
+			{
+			ct_UseNativeCapability?
+				{
+				1 index /CMap ct_resourcestatus
+					{ pop pop }
+					{
+					/CIDInit /ProcSet findresource begin
+					12 dict begin
+					begincmap
+					/CMapName 2 index def
+					/CMapVersion 1.000 def
+					/CMapType 1 def
+					/CIDSystemInfo 3 dict dup
+						begin
+						/Registry (Adobe) def
+						/Ordering
+						CMapName ct_mkocfStr100 cvs
+						(Adobe-) search
+							{
+							pop pop
+							(-) search
+								{ dup length string copy exch pop exch pop }
+								{ pop (Identity) }
+							ifelse
+							}
+							{ pop (Identity) }
+						ifelse
+						def
+						/Supplement 0 def
+						end def
+					1 begincodespacerange
+					<0000> <FFFF>
+					endcodespacerange
+					1 begincidrange
+					<0000> <FFFF> 0
+					endcidrange
+					endcmap
+					CMapName currentdict /CMap defineresource pop
+					end
+					end
+					}
+				ifelse
+				composefont
+				}
+				{
+				exch pop
+				0 get /CIDFont findresource
+				ct_makeocf
+				}
+			ifelse
+			} bind def
+		currentdict readonly pop
+		end
+	end
+%%EndResource
+/currentpacking where{pop currentpacking true setpacking}if
+%%BeginResource: procset pdfvars 6.0 1
+%%Copyright: Copyright 1987-2002 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: definition of dictionary of variables used by PDF & PDFText procsets
+userdict /PDF 162 dict put
+userdict /PDFVars 89 dict dup begin put
+/docSetupDone false def
+/InitAll 0 def
+/TermAll 0 def
+/DocInitAll 0 def
+/DocTermAll 0 def
+/_pdfEncodings 2 array def
+/_pdf_str1 1 string def
+/_pdf_i 0 def
+/_pdf_na 0 def
+/_pdf_showproc 0 def
+/_italMtx [1 0 .212557 1 0 0] def
+/_italMtx_WMode1 [1 -.212557 0 1 0 0] def
+/_italMtxType0 [1 0 .1062785 1 0 0] def
+/_italMtx_WMode1Type0 [1 -.1062785 0 1 0 0] def
+/_basefont 0 def
+/_basefonto 0 def
+/_pdf_oldCIDInit null def
+/_pdf_FontDirectory 30 dict def
+/_categories 10 dict def
+/_sa? true def
+/_ColorSep5044? false def
+/nulldict 0 dict def
+/_processColors 0 def
+/overprintstack null def
+/_defaulttransfer currenttransfer def
+/_defaultflatness currentflat def
+/_defaulthalftone null def
+/_defaultcolortransfer null def
+/_defaultblackgeneration null def
+/_defaultundercolorremoval null def
+/_defaultcolortransfer null def
+PDF begin
+[/c/cs/cm/d/d0/f/h/i/j/J/l/m/M/n/q/Q/re/ri/S/sc/sh/Tf/w/W
+/applyInterpFunc/applystitchFunc/domainClip/encodeInput
+/initgs/int/limit/rangeClip
+/defineRes/undefineRes/findRes/setSA/pl
+/? /! /| /: /+ /GetGlyphDirectory
+/pdf_flushFilters /pdf_readstring /pdf_dictOp /pdf_image /pdf_maskedImage
+/pdf_shfill /pdf_sethalftone
+] {null def} bind forall
+end
+end
+%%EndResource
+PDFVars begin PDF begin
+%%BeginResource: procset pdfutil 6.0 1
+%%Copyright: Copyright 1993-2001 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: Basic utilities used by other PDF procsets
+/bd {bind def} bind def
+/ld {load def} bd
+/bld {
+dup length dict begin
+{ null def } forall
+bind
+end
+def
+} bd
+/dd { PDFVars 3 1 roll put } bd
+/xdd { exch dd } bd
+/Level2?
+systemdict /languagelevel known
+{ systemdict /languagelevel get 2 ge } { false } ifelse
+def
+/Level1? Level2? not def
+/Level3?
+systemdict /languagelevel known
+{systemdict /languagelevel get 3 eq } { false } ifelse
+def
+/getifknown {
+2 copy known { get true } { pop pop false } ifelse
+} bd
+/here {
+currentdict exch getifknown
+} bd
+/isdefined? { where { pop true } { false } ifelse } bd
+%%EndResource
+%%BeginResource: procset l2compat 6.0 1
+%%Copyright: Copyright 1987-2003 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: Level 1 emulation of level 2 operators
+/setcmykcolor isdefined? not
+{
+/setcmykcolor {
+1 sub 4 1 roll
+3 {
+3 index add neg dup 0 lt { pop 0 } if
+3 1 roll
+} repeat
+setrgbcolor
+pop
+} bd
+} if
+/rectclip isdefined? not
+{
+/rectclip { newpath re clip newpath } bd
+} if
+/rectfill isdefined? not
+{
+/rectfill { gsave newpath re fill grestore } bd
+} if
+/sethalftone isdefined? not
+{
+/sethalftone {
+begin
+HalftoneType 1 eq
+{ Frequency Angle /SpotFunction load setscreen }
+if
+end
+} bd
+} if
+Level1?
+{
+/pdf_show {show} bd
+/xshow
+{
+PDFVars /_pdf_showproc /pdf_show load put
+pdf_xshow
+} bd
+/yshow
+{
+PDFVars /_pdf_showproc /pdf_show load put
+pdf_yshow
+} bd
+/xyshow
+{
+PDFVars /_pdf_showproc /pdf_show load put
+pdf_xyshow
+} bd
+} if
+/getrampcolor {
+cvi
+/indx exch def
+[
+0 1 NumComp 1 sub {
+dup
+Samples exch get
+dup type /stringtype eq { indx get } if
+exch
+Scaling exch get aload pop
+3 1 roll
+mul add
+} for
+]
+L1setcolor
+} bd
+/sssetbackground { L1setcolor } bd
+%%EndResource
+%%BeginResource: procset pdf 6.0 1
+%%Copyright: Copyright 1998-2003 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: General operators for PDF, common to all Language Levels.
+/cm { matrix astore concat } bd
+/d /setdash ld
+/f /fill ld
+/h /closepath ld
+/i {dup 0 eq {pop _defaultflatness} if setflat} bd
+/j /setlinejoin ld
+/J /setlinecap ld
+/M /setmiterlimit ld
+/n /newpath ld
+/S /stroke ld
+/w /setlinewidth ld
+/W /clip ld
+/sg /setgray ld
+/initgs {
+0 setgray
+[] 0 d
+0 j
+0 J
+10 M
+1 w
+false setSA
+/_defaulttransfer load settransfer
+0 i
+/RelativeColorimetric ri
+newpath
+} bd
+/int {
+dup 2 index sub 3 index 5 index sub div 6 -2 roll sub mul
+exch pop add exch pop
+} bd
+/limit {
+dup 2 index le { exch } if pop
+dup 2 index ge { exch } if pop
+} bd
+/domainClip {
+Domain aload pop 3 2 roll
+limit
+} [/Domain] bld
+/applyInterpFunc {
+0 1 DimOut 1 sub
+{
+dup C0 exch get exch
+dup C1 exch get exch
+3 1 roll
+1 index sub
+3 index
+N exp mul add
+exch
+currentdict /Range_lo known
+{
+dup Range_lo exch get exch
+Range_hi exch get
+3 2 roll limit
+}
+{
+pop
+}
+ifelse
+exch
+} for
+pop
+} [/DimOut /C0 /C1 /N /Range_lo /Range_hi] bld
+/encodeInput {
+NumParts 1 sub
+0 1 2 index
+{
+dup Bounds exch get
+2 index gt
+{ exit }
+{ dup
+3 index eq
+{ exit }
+{ pop } ifelse
+} ifelse
+} for
+3 2 roll pop
+dup Bounds exch get exch
+dup 1 add Bounds exch get exch
+2 mul
+dup Encode exch get exch
+1 add Encode exch get
+int
+} [/NumParts /Bounds /Encode] bld
+/rangeClip {
+exch dup Range_lo exch get
+exch Range_hi exch get
+3 2 roll
+limit
+} [/Range_lo /Range_hi] bld
+/applyStitchFunc {
+Functions exch get exec
+currentdict /Range_lo known {
+0 1 DimOut 1 sub {
+DimOut 1 add -1 roll
+rangeClip
+} for
+} if
+} [/Functions /Range_lo /DimOut] bld
+/pdf_flushfilters
+{
+aload length
+{ dup status
+1 index currentfile ne and
+{ dup flushfile closefile }
+{ pop }
+ifelse
+} repeat
+} bd
+/pdf_readstring
+{
+1 index dup length 1 sub get
+exch readstring pop
+exch pdf_flushfilters
+} bind def
+/pdf_dictOp
+{
+3 2 roll
+10 dict copy
+begin
+_Filters dup length 1 sub get def
+currentdict exch exec
+_Filters pdf_flushfilters
+end
+} [/_Filters] bld
+/pdf_image {{image} /DataSource pdf_dictOp} bd
+/pdf_imagemask {{imagemask} /DataSource pdf_dictOp} bd
+/pdf_shfill {{sh} /DataSource pdf_dictOp} bd
+/pdf_sethalftone {{sethalftone} /Thresholds pdf_dictOp} bd
+/pdf_maskedImage
+{
+10 dict copy begin
+/miDict currentdict def
+/DataDict DataDict 10 dict copy def
+DataDict begin
+/DataSource
+_Filters dup length 1 sub get
+def
+miDict image
+_Filters pdf_flushfilters
+end
+end
+} [/miDict /DataDict /_Filters] bld
+/RadialShade {
+40 dict begin
+/background exch def
+/ext1 exch def
+/ext0 exch def
+/BBox exch def
+/r2 exch def
+/c2y exch def
+/c2x exch def
+/r1 exch def
+/c1y exch def
+/c1x exch def
+/rampdict exch def
+gsave
+BBox length 0 gt {
+newpath
+BBox 0 get BBox 1 get moveto
+BBox 2 get BBox 0 get sub 0 rlineto
+0 BBox 3 get BBox 1 get sub rlineto
+BBox 2 get BBox 0 get sub neg 0 rlineto
+closepath
+clip
+newpath
+} if
+c1x c2x eq
+{
+c1y c2y lt {/theta 90 def}{/theta 270 def} ifelse
+}
+{
+/slope c2y c1y sub c2x c1x sub div def
+/theta slope 1 atan def
+c2x c1x lt c2y c1y ge and { /theta theta 180 sub def} if
+c2x c1x lt c2y c1y lt and { /theta theta 180 add def} if
+}
+ifelse
+gsave
+clippath
+c1x c1y translate
+theta rotate
+-90 rotate
+{ pathbbox } stopped
+{ 0 0 0 0 } if
+/yMax exch def
+/xMax exch def
+/yMin exch def
+/xMin exch def
+grestore
+xMax xMin eq yMax yMin eq or
+{
+grestore
+end
+}
+{
+/max { 2 copy gt { pop } {exch pop} ifelse } bind def
+/min { 2 copy lt { pop } {exch pop} ifelse } bind def
+rampdict begin
+40 dict begin
+background length 0 gt { background sssetbackground gsave clippath fill grestore } if
+gsave
+c1x c1y translate
+theta rotate
+-90 rotate
+/c2y c1x c2x sub dup mul c1y c2y sub dup mul add sqrt def
+/c1y 0 def
+/c1x 0 def
+/c2x 0 def
+ext0 {
+0 getrampcolor
+c2y r2 add r1 sub 0.0001 lt
+{
+c1x c1y r1 360 0 arcn
+pathbbox
+/aymax exch def
+/axmax exch def
+/aymin exch def
+/axmin exch def
+/bxMin xMin axmin min def
+/byMin yMin aymin min def
+/bxMax xMax axmax max def
+/byMax yMax aymax max def
+bxMin byMin moveto
+bxMax byMin lineto
+bxMax byMax lineto
+bxMin byMax lineto
+bxMin byMin lineto
+eofill
+}
+{
+c2y r1 add r2 le
+{
+c1x c1y r1 0 360 arc
+fill
+}
+{
+c2x c2y r2 0 360 arc fill
+r1 r2 eq
+{
+/p1x r1 neg def
+/p1y c1y def
+/p2x r1 def
+/p2y c1y def
+p1x p1y moveto p2x p2y lineto p2x yMin lineto p1x yMin lineto
+fill
+}
+{
+/AA r2 r1 sub c2y div def
+AA -1 eq
+{ /theta 89.99 def}
+{ /theta AA 1 AA dup mul sub sqrt div 1 atan def}
+ifelse
+/SS1 90 theta add dup sin exch cos div def
+/p1x r1 SS1 SS1 mul SS1 SS1 mul 1 add div sqrt mul neg def
+/p1y p1x SS1 div neg def
+/SS2 90 theta sub dup sin exch cos div def
+/p2x r1 SS2 SS2 mul SS2 SS2 mul 1 add div sqrt mul def
+/p2y p2x SS2 div neg def
+r1 r2 gt
+{
+/L1maxX p1x yMin p1y sub SS1 div add def
+/L2maxX p2x yMin p2y sub SS2 div add def
+}
+{
+/L1maxX 0 def
+/L2maxX 0 def
+}ifelse
+p1x p1y moveto p2x p2y lineto L2maxX L2maxX p2x sub SS2 mul p2y add lineto
+L1maxX L1maxX p1x sub SS1 mul p1y add lineto
+fill
+}
+ifelse
+}
+ifelse
+} ifelse
+} if
+c1x c2x sub dup mul
+c1y c2y sub dup mul
+add 0.5 exp
+0 dtransform
+dup mul exch dup mul add 0.5 exp 72 div
+0 72 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+1 index 1 index lt { exch } if pop
+/hires exch def
+hires mul
+/numpix exch def
+/numsteps NumSamples def
+/rampIndxInc 1 def
+/subsampling false def
+numpix 0 ne
+{
+NumSamples numpix div 0.5 gt
+{
+/numsteps numpix 2 div round cvi dup 1 le { pop 2 } if def
+/rampIndxInc NumSamples 1 sub numsteps div def
+/subsampling true def
+} if
+} if
+/xInc c2x c1x sub numsteps div def
+/yInc c2y c1y sub numsteps div def
+/rInc r2 r1 sub numsteps div def
+/cx c1x def
+/cy c1y def
+/radius r1 def
+newpath
+xInc 0 eq yInc 0 eq rInc 0 eq and and
+{
+0 getrampcolor
+cx cy radius 0 360 arc
+stroke
+NumSamples 1 sub getrampcolor
+cx cy radius 72 hires div add 0 360 arc
+0 setlinewidth
+stroke
+}
+{
+0
+numsteps
+{
+dup
+subsampling { round } if
+getrampcolor
+cx cy radius 0 360 arc
+/cx cx xInc add def
+/cy cy yInc add def
+/radius radius rInc add def
+cx cy radius 360 0 arcn
+eofill
+rampIndxInc add
+}
+repeat
+pop
+} ifelse
+ext1 {
+c2y r2 add r1 lt
+{
+c2x c2y r2 0 360 arc
+fill
+}
+{
+c2y r1 add r2 sub 0.0001 le
+{
+c2x c2y r2 360 0 arcn
+pathbbox
+/aymax exch def
+/axmax exch def
+/aymin exch def
+/axmin exch def
+/bxMin xMin axmin min def
+/byMin yMin aymin min def
+/bxMax xMax axmax max def
+/byMax yMax aymax max def
+bxMin byMin moveto
+bxMax byMin lineto
+bxMax byMax lineto
+bxMin byMax lineto
+bxMin byMin lineto
+eofill
+}
+{
+c2x c2y r2 0 360 arc fill
+r1 r2 eq
+{
+/p1x r2 neg def
+/p1y c2y def
+/p2x r2 def
+/p2y c2y def
+p1x p1y moveto p2x p2y lineto p2x yMax lineto p1x yMax lineto
+fill
+}
+{
+/AA r2 r1 sub c2y div def
+AA -1 eq
+{ /theta 89.99 def}
+{ /theta AA 1 AA dup mul sub sqrt div 1 atan def}
+ifelse
+/SS1 90 theta add dup sin exch cos div def
+/p1x r2 SS1 SS1 mul SS1 SS1 mul 1 add div sqrt mul neg def
+/p1y c2y p1x SS1 div sub def
+/SS2 90 theta sub dup sin exch cos div def
+/p2x r2 SS2 SS2 mul SS2 SS2 mul 1 add div sqrt mul def
+/p2y c2y p2x SS2 div sub def
+r1 r2 lt
+{
+/L1maxX p1x yMax p1y sub SS1 div add def
+/L2maxX p2x yMax p2y sub SS2 div add def
+}
+{
+/L1maxX 0 def
+/L2maxX 0 def
+}ifelse
+p1x p1y moveto p2x p2y lineto L2maxX L2maxX p2x sub SS2 mul p2y add lineto
+L1maxX L1maxX p1x sub SS1 mul p1y add lineto
+fill
+}
+ifelse
+}
+ifelse
+} ifelse
+} if
+grestore
+grestore
+end
+end
+end
+} ifelse
+} bd
+/GenStrips {
+40 dict begin
+/background exch def
+/ext1 exch def
+/ext0 exch def
+/BBox exch def
+/y2 exch def
+/x2 exch def
+/y1 exch def
+/x1 exch def
+/rampdict exch def
+gsave
+BBox length 0 gt {
+newpath
+BBox 0 get BBox 1 get moveto
+BBox 2 get BBox 0 get sub 0 rlineto
+0 BBox 3 get BBox 1 get sub rlineto
+BBox 2 get BBox 0 get sub neg 0 rlineto
+closepath
+clip
+newpath
+} if
+x1 x2 eq
+{
+y1 y2 lt {/theta 90 def}{/theta 270 def} ifelse
+}
+{
+/slope y2 y1 sub x2 x1 sub div def
+/theta slope 1 atan def
+x2 x1 lt y2 y1 ge and { /theta theta 180 sub def} if
+x2 x1 lt y2 y1 lt and { /theta theta 180 add def} if
+}
+ifelse
+gsave
+clippath
+x1 y1 translate
+theta rotate
+{ pathbbox } stopped
+{ 0 0 0 0 } if
+/yMax exch def
+/xMax exch def
+/yMin exch def
+/xMin exch def
+grestore
+xMax xMin eq yMax yMin eq or
+{
+grestore
+end
+}
+{
+rampdict begin
+20 dict begin
+background length 0 gt { background sssetbackground gsave clippath fill grestore } if
+gsave
+x1 y1 translate
+theta rotate
+/xStart 0 def
+/xEnd x2 x1 sub dup mul y2 y1 sub dup mul add 0.5 exp def
+/ySpan yMax yMin sub def
+/numsteps NumSamples def
+/rampIndxInc 1 def
+/subsampling false def
+xStart 0 transform
+xEnd 0 transform
+3 -1 roll
+sub dup mul
+3 1 roll
+sub dup mul
+add 0.5 exp 72 div
+0 72 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+1 index 1 index lt { exch } if pop
+mul
+/numpix exch def
+numpix 0 ne
+{
+NumSamples numpix div 0.5 gt
+{
+/numsteps numpix 2 div round cvi dup 1 le { pop 2 } if def
+/rampIndxInc NumSamples 1 sub numsteps div def
+/subsampling true def
+} if
+} if
+ext0 {
+0 getrampcolor
+xMin xStart lt
+{ xMin yMin xMin neg ySpan rectfill } if
+} if
+/xInc xEnd xStart sub numsteps div def
+/x xStart def
+0
+numsteps
+{
+dup
+subsampling { round } if
+getrampcolor
+x yMin xInc ySpan rectfill
+/x x xInc add def
+rampIndxInc add
+}
+repeat
+pop
+ext1 {
+xMax xEnd gt
+{ xEnd yMin xMax xEnd sub ySpan rectfill } if
+} if
+grestore
+grestore
+end
+end
+end
+} ifelse
+} bd
+/currentdistillerparams where { pop currentdistillerparams /CoreDistVersion get 5000 lt}{true}ifelse
+{
+/PDFMark5 {cleartomark} bd
+}
+{
+/PDFMark5 {pdfmark} bd
+}ifelse
+/ReadByPDFMark5
+{
+2 dict begin
+/makerString exch def string /tmpString exch def
+{
+currentfile tmpString readline pop
+makerString anchorsearch
+{
+pop pop cleartomark exit
+}
+{
+3 copy /PUT PDFMark5 pop 2 copy (\n) /PUT PDFMark5
+} ifelse
+}loop
+end
+}bd
+%%EndResource
+%%BeginResource: procset sep_ops 6.0 1
+%%Copyright: Copyright 1987-2001 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: Support for Separations in Level 1, following the conventions of Tech Note 5044
+userdict /sep_ops 50 dict dup begin put
+/bdef {bind def} bind def
+/xdef {exch def} bdef
+/colorimagebuffer {
+0 1 2 index length 1 sub {
+dup 2 index exch get 255 exch sub 2 index 3 1 roll put
+} for
+} bdef
+/addprocs {
+[ 3 1 roll
+/exec load
+dup 3 1 roll
+] cvx
+} bdef
+/L1? {
+systemdict /languagelevel known {
+systemdict /languagelevel get 2 lt
+}{
+true
+} ifelse
+} bdef
+/colorexists {
+statusdict /processcolors known {
+statusdict /processcolors get exec
+}{
+/deviceinfo where {
+pop deviceinfo /Colors known {
+deviceinfo /Colors get
+statusdict /processcolors {
+deviceinfo /Colors known {
+deviceinfo /Colors get
+}{
+1
+} ifelse
+} put
+}{
+1
+} ifelse
+}{
+1
+} ifelse
+} ifelse
+1 gt
+} bdef
+/colorplate colorexists { 0 } { 5 } ifelse def
+/negativecolorplate false def
+/setcmykcolor where {
+pop
+gsave
+1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
+0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
+0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
+0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub
+4 {4 copy} repeat
+grestore
+1 dict begin
+/foureq {
+4 index eq 8 1 roll
+4 index eq 8 1 roll
+4 index eq 8 1 roll
+4 index eq 8 1 roll
+pop pop pop pop and and and
+} def
+1 0 0 0 foureq {/colorplate 1 store} if
+0 1 0 0 foureq {/colorplate 2 store} if
+0 0 1 0 foureq {/colorplate 3 store} if
+0 0 0 1 foureq {/colorplate 4 store} if
+0 0 0 0 foureq {/colorplate 6 store} if
+end
+} if
+0 systemdict /currenttransfer get exec exec
+1 systemdict /currenttransfer get exec exec
+2 copy
+eq
+{
+/colorplate 6 store
+pop
+/negativecolorplate exch 0.5 lt store
+}
+{
+gt /negativecolorplate exch store
+}
+ifelse
+/findcmykcustomcolor where { pop }
+{
+/findcmykcustomcolor {
+[ 6 1 roll ] readonly
+} bdef
+} ifelse
+/setoverprint where {
+pop
+}{
+/setoverprint {
+pop
+} bdef
+} ifelse
+/currentoverprint where {
+pop
+}{
+/currentoverprint {
+false
+} bdef
+} ifelse
+/setcustomcolor where {
+pop
+}{
+L1? {
+/setcustomcolor {
+exch
+aload pop pop
+4 { 4 index mul 4 1 roll } repeat
+5 -1 roll pop
+setcmykcolor
+} bdef
+}{
+/setcustomcolor {
+exch
+[ exch /Separation exch dup 4 get exch /DeviceCMYK exch
+0 4 getinterval
+[ exch /dup load exch cvx {mul exch dup}
+/forall load /pop load dup] cvx
+] setcolorspace setcolor
+} bdef
+} ifelse
+} ifelse
+/ik 0 def
+/iy 0 def
+/im 0 def
+/ic 0 def
+/imagetint {
+ic .3 mul
+im .59 mul
+iy .11 mul
+ik add add add dup
+1 gt {pop 1} if
+} bdef
+/setcmykcolor where {
+pop
+}{
+/setcmykcolor {
+/ik xdef /iy xdef /im xdef /ic xdef
+imagetint
+1 exch sub setgray
+} bdef
+} ifelse
+/customcolorimage where {
+pop
+}{
+L1? {
+/customcolorimage{
+gsave
+colorexists {
+aload pop pop
+/ik xdef /iy xdef /im xdef /ic xdef
+currentcolortransfer
+{ik mul ik sub 1 add} addprocs
+4 1 roll {iy mul iy sub 1 add} addprocs
+4 1 roll {im mul im sub 1 add} addprocs
+4 1 roll {ic mul ic sub 1 add} addprocs
+4 1 roll setcolortransfer
+/magentabuf 0 string def
+/yellowbuf 0 string def
+/blackbuf 0 string def
+{
+colorimagebuffer dup length magentabuf length ne
+{
+dup length dup dup
+/magentabuf exch string def
+/yellowbuf exch string def
+/blackbuf exch string def
+} if
+dup magentabuf copy yellowbuf copy
+blackbuf copy pop
+} addprocs
+{magentabuf}{yellowbuf}{blackbuf} true 4 colorimage
+}{
+aload pop pop /ik xdef /iy xdef /im xdef /ic xdef /tint
+imagetint def
+currenttransfer
+{tint mul 1 tint sub add} addprocs settransfer image
+} ifelse
+grestore
+} bdef
+}{
+/customcolorimage {
+gsave
+[ exch /Separation exch dup 4 get exch /DeviceCMYK exch
+0 4 getinterval
+[ exch /dup load exch cvx {mul exch dup}
+/forall load /pop load dup] cvx
+] setcolorspace
+10 dict begin
+/ImageType 1 def
+/DataSource exch def
+/ImageMatrix exch def
+/BitsPerComponent exch def
+/Height exch def
+/Width exch def
+/Decode [1 0] def
+currentdict end
+image
+grestore
+} bdef
+} ifelse
+} ifelse
+/setseparationgray where {
+pop
+}{
+L1? {
+/setseparationgray { 1 exch sub dup dup dup setcmykcolor } bdef
+}{
+/setseparationgray {
+[/Separation /All /DeviceCMYK {dup dup dup}] setcolorspace
+1 exch sub setcolor
+} bdef
+} ifelse
+} ifelse
+/separationimage where { pop }
+{
+/separationimage {
+gsave
+1 1 1 1 (All)
+findcmykcustomcolor customcolorimage
+grestore
+} bdef
+} ifelse
+currentdict readonly pop end
+%%EndResource
+%%BeginResource: procset pdflev15044 6.0 1
+%%Copyright: Copyright 1987-2002 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: PDF operators, Level 1, with emulated separations (TN 5044)
+/_ColorSep5044? true dd
+/docinitialize {
+PDF begin
+/_defaulthalftone
+/currenthalftone where
+{ pop currenthalftone }
+{ 4 dict dup begin
+currentscreen
+/SpotFunction exch def
+/Angle exch def
+/Frequency exch def
+/HalftoneType 1 def
+end }
+ifelse
+dd
+/currentcolortransfer where
+{ pop /_defaultcolortransfer [ currentcolortransfer ] dd }
+{ /_defaultcolortransfer [currenttransfer dup dup dup] dd }
+ifelse
+end
+} bd
+/initialize {
+/overprintstack null dd
+sep_ops begin
+50 dict begin
+_defaulthalftone sethalftone
+} bd
+/terminate {
+end end
+} bd
+/currentcolortransfer where
+{ pop }
+{
+/setcolortransfer
+{
+settransfer pop pop pop
+} bd
+} ifelse
+/pl {
+transform
+0.25 sub round 0.25 add exch
+0.25 sub round 0.25 add exch
+itransform
+} bd
+/m { _sa? { pl } if moveto } bd
+/l { _sa? { pl } if lineto } bd
+/c
+{
+_sa? {3 {pl 6 2 roll} repeat} if
+curveto
+} bd
+/ri/pop ld
+/setSA { /_sa? xdd } bd
+/re
+{
+_sa?
+{
+8 dict begin
+/:h exch def
+/:w exch def
+/:y exch def
+/:x exch def
+:x :y pl
+/:ymin exch def /:xmin exch def
+:x :w add :y :h add pl
+/:ymax exch def /:xmax exch def
+:xmin :ymin moveto
+:xmax :ymin lineto
+:xmax :ymax lineto
+:xmin :ymax lineto
+closepath
+end
+}
+{
+4 2 roll moveto
+1 index 0 rlineto
+0 exch rlineto
+neg 0 rlineto
+closepath
+} ifelse
+} bd
+/q
+{
+gsave
+[currentoverprint overprintstack] /overprintstack xdd
+}
+[/overprintstack] bld
+/Q
+{
+overprintstack aload pop /overprintstack xdd setoverprint
+grestore
+}
+[/overprintstack] bld
+/AlmostFull?
+{ dup maxlength exch length sub 2 le
+} bd
+/Expand
+{ 1 index maxlength mul cvi dict
+dup begin exch { def } forall end
+} bd
+/xput
+{ 3 2 roll
+dup 3 index known not
+{ dup AlmostFull? { 1.5 Expand } if
+} if
+dup 4 2 roll put
+} bd
+/defineRes
+{ _categories 1 index known not
+{ /_categories _categories 2 index 10 dict xput store
+} if
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+} bd
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+} bd
+/findRes {
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+} bd
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+} bind dd
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+} bd
+/concatandsettransfer {
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+} bd
+/concatandsetcolortransfer {
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+6 -1 roll 4 -1 roll concattransferfuncs 5 1 roll
+4 -1 roll 3 -1 roll concattransferfuncs 3 1 roll
+concattransferfuncs
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+} if
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+} if
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+} if
+} bd
+/tn5044sethalftone
+{
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+colorplate get
+here not {
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+} if
+}
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+begin
+/TransferFunction here
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+} forall
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+}
+{
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+} ifelse
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+} bd
+/paintimage
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+{
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+{currentfile magentastr readstring pop}
+{currentfile yellowstr readstring pop}
+{currentfile blackstr readstring pop
+currentfile graystr readstring pop pop}
+}
+{ {currentfile cyanstr readhexstring pop}
+{currentfile magentastr readhexstring pop}
+{currentfile yellowstr readhexstring pop}
+{currentfile blackstr readhexstring pop
+currentfile graystr readhexstring pop pop}
+} ifelse
+true 4 colorimage
+}
+{
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+/binaryOK exch def
+[
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+dup
+/currentfile cvx
+[ /cyanstr /magentastr /yellowstr /blackstr /graystr ]
+3 -1 roll 1 sub get cvx
+binaryOK { /readstring } { /readhexstring } ifelse cvx
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+} for
+] cvx bind
+end
+[
+colorplate 6 eq {
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+negativecolorplate { 0 } { 1 } ifelse
+} if
+colorplate 4 le
+{
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+} if
+colorplate 6 ne
+{
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+aload pop
+} if
+] cvx
+gsave
+systemdict /settransfer get exec
+systemdict /image get exec
+grestore
+} ifelse
+} bd
+%%EndResource
+%%BeginResource: procset pdftext 6.0 1
+%%Copyright: Copyright 1987-2001 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: Text operators for PDF
+PDF /PDFText 78 dict dup begin put
+/docinitialize
+{
+/resourcestatus where {
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+/CIDParams /ProcSet resourcestatus {
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+false /CIDParams /ProcSet findresource /SetBuildCompatible get exec
+} if
+} if
+PDF begin
+PDFText /_pdfDefineIdentity-H known
+{ PDFText /_pdfDefineIdentity-H get exec}
+if
+end
+} bd
+/initialize {
+PDFText begin
+} bd
+/terminate { end } bd
+Level2?
+{
+/_safeput
+{
+3 -1 roll load 3 1 roll put
+}
+bd
+}
+{
+/_safeput
+{
+2 index load dup dup length exch maxlength ge
+{ dup length 5 add dict copy
+3 index xdd
+}
+{ pop }
+ifelse
+3 -1 roll load 3 1 roll put
+}
+bd
+}
+ifelse
+/pdf_has_composefont? systemdict /composefont known def
+/CopyFont {
+{
+1 index /FID ne 2 index /UniqueID ne and
+{ def } { pop pop } ifelse
+} forall
+} bd
+/Type0CopyFont
+{
+exch
+dup length dict
+begin
+CopyFont
+[
+exch
+FDepVector
+{
+dup /FontType get 0 eq
+{
+1 index Type0CopyFont
+/_pdfType0 exch definefont
+}
+{
+/_pdfBaseFont exch
+2 index exec
+}
+ifelse
+exch
+}
+forall
+pop
+]
+/FDepVector exch def
+currentdict
+end
+} bd
+Level2? {currentglobal true setglobal} if
+/cHexEncoding
+[/c00/c01/c02/c03/c04/c05/c06/c07/c08/c09/c0A/c0B/c0C/c0D/c0E/c0F/c10/c11/c12
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+/cF7/cF8/cF9/cFA/cFB/cFC/cFD/cFE/cFF] def
+Level2? {setglobal} if
+/modEnc {
+/_enc xdd
+/_icode 0 dd
+counttomark 1 sub -1 0
+{
+index
+dup type /nametype eq
+{
+_enc _icode 3 -1 roll put
+_icode 1 add
+}
+if
+/_icode xdd
+} for
+cleartomark
+_enc
+} bd
+/trEnc {
+/_enc xdd
+255 -1 0 {
+exch dup -1 eq
+{ pop /.notdef }
+{ Encoding exch get }
+ifelse
+_enc 3 1 roll put
+} for
+pop
+_enc
+} bd
+/TE {
+/_i xdd
+StandardEncoding 256 array copy modEnc
+_pdfEncodings exch _i exch put
+} bd
+/TZ
+{
+/_usePDFEncoding xdd
+findfont
+dup length 6 add dict
+begin
+{
+1 index /FID ne { def } { pop pop } ifelse
+} forall
+/pdf_origFontName FontName def
+/FontName exch def
+currentdict /PaintType known
+{ PaintType 2 eq {/PaintType 0 def} if }
+if
+_usePDFEncoding 0 ge
+{
+/Encoding _pdfEncodings _usePDFEncoding get def
+pop
+}
+{
+_usePDFEncoding -1 eq
+{
+counttomark 0 eq
+{ pop }
+{
+Encoding 256 array copy
+modEnc /Encoding exch def
+}
+ifelse
+}
+{
+256 array
+trEnc /Encoding exch def
+}
+ifelse
+}
+ifelse
+pdf_EuroProcSet pdf_origFontName known
+{
+pdf_origFontName pdf_AddEuroGlyphProc
+} if
+Level2?
+{
+currentdict /pdf_origFontName undef
+} if
+FontName currentdict
+end
+definefont pop
+}
+bd
+Level2?
+{
+/TZG
+{
+currentglobal true setglobal
+2 index _pdfFontStatus
+{
+2 index findfont
+false setglobal
+3 index findfont
+true setglobal
+ne
+{
+2 index findfont dup rcheck
+{
+dup length dict begin
+{
+1 index /FID ne { def } { pop pop } ifelse
+} forall
+currentdict end
+}
+if
+3 index exch definefont pop
+}
+if
+} if
+setglobal
+TZ
+} bd
+}
+{
+/TZG {TZ} bd
+} ifelse
+Level2?
+{
+currentglobal false setglobal
+userdict /pdftext_data 5 dict put
+pdftext_data
+begin
+/saveStacks
+{
+pdftext_data
+begin
+/vmmode currentglobal def
+false setglobal
+count array astore /os exch def
+end
+countdictstack array dictstack pdftext_data exch /ds exch put
+cleardictstack pdftext_data /dscount countdictstack put
+pdftext_data /vmmode get setglobal
+} bind def
+/restoreStacks
+{
+pdftext_data /vmmode currentglobal put false setglobal
+clear cleardictstack
+pdftext_data /ds get dup
+pdftext_data /dscount get 1 2 index length 1 sub
+{ get begin dup } for
+pop pop
+pdftext_data /os get aload pop
+pdftext_data /vmmode get setglobal
+} bind def
+/testForClonePrinterBug
+{
+currentglobal true setglobal
+/undefinedCategory /Generic /Category findresource
+dup length dict copy /Category defineresource pop
+setglobal
+pdftext_data /saveStacks get exec
+pdftext_data /vmmode currentglobal put false setglobal
+/undefined /undefinedCategory { resourcestatus } stopped
+pdftext_data exch /bugFound exch put
+pdftext_data /vmmode get setglobal
+pdftext_data /restoreStacks get exec
+pdftext_data /bugFound get
+} bind def
+end
+setglobal
+/pdf_resourcestatus
+pdftext_data /testForClonePrinterBug get exec
+{
+{
+pdftext_data /saveStacks get exec
+pdftext_data /os get dup dup length 1 sub
+dup 1 sub dup 0 lt { pop 0 } if
+exch 1 exch { get exch dup } for
+pop pop
+{ resourcestatus }
+stopped
+{
+clear cleardictstack pdftext_data /restoreStacks get exec
+{ pop pop } stopped pop false
+}
+{
+count array astore pdftext_data exch /results exch put
+pdftext_data /restoreStacks get exec pop pop
+pdftext_data /results get aload pop
+}
+ifelse
+}
+}
+{ { resourcestatus } }
+ifelse
+bd
+}
+if
+Level2?
+{
+/_pdfUndefineResource
+{
+currentglobal 3 1 roll
+_pdf_FontDirectory 2 index 2 copy known
+{undef}
+{pop pop}
+ifelse
+1 index (pdf) exch _pdfConcatNames 1 index
+1 index 1 _pdfConcatNames 1 index
+5 index 1 _pdfConcatNames 1 index
+4
+{
+2 copy pdf_resourcestatus
+{
+pop 2 lt
+{2 copy findresource gcheck setglobal undefineresource}
+{pop pop}
+ifelse
+}
+{ pop pop}
+ifelse
+} repeat
+setglobal
+} bd
+}
+{
+/_pdfUndefineResource { pop pop} bd
+}
+ifelse
+Level2?
+{
+/_pdfFontStatus
+{
+currentglobal exch
+/Font pdf_resourcestatus
+{pop pop true}
+{false}
+ifelse
+exch setglobal
+} bd
+}
+{
+/_pdfFontStatusString 50 string def
+_pdfFontStatusString 0 (fonts/) putinterval
+/_pdfFontStatus
+{
+FontDirectory 1 index known
+{ pop true }
+{
+_pdfFontStatusString 6 42 getinterval
+cvs length 6 add
+_pdfFontStatusString exch 0 exch getinterval
+{ status } stopped
+{pop false}
+{
+{ pop pop pop pop true}
+{ false }
+ifelse
+}
+ifelse
+}
+ifelse
+} bd
+}
+ifelse
+Level2?
+{
+/_pdfCIDFontStatus
+{
+/CIDFont /Category pdf_resourcestatus
+{
+pop pop
+/CIDFont pdf_resourcestatus
+{pop pop true}
+{false}
+ifelse
+}
+{ pop false }
+ifelse
+} bd
+}
+if
+/_pdfString100 100 string def
+/_pdfComposeFontName
+{
+dup length 1 eq
+{
+0 get
+1 index
+type /nametype eq
+{
+_pdfString100 cvs
+length dup dup _pdfString100 exch (-) putinterval
+_pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval
+2 index exch cvs length
+add 1 add _pdfString100 exch 0 exch getinterval
+exch pop
+true
+}
+{
+pop pop
+false
+}
+ifelse
+}
+{
+false
+}
+ifelse
+dup {exch cvn exch} if
+} bd
+/_pdfConcatNames
+{
+exch
+_pdfString100 cvs
+length dup dup _pdfString100 exch (-) putinterval
+_pdfString100 exch 1 add dup _pdfString100 length exch sub getinterval
+3 -1 roll exch cvs length
+add 1 add _pdfString100 exch 0 exch getinterval
+cvn
+} bind def
+/_pdfTextTempString 50 string def
+/_pdfRegOrderingArray [(Adobe-Japan1) (Adobe-CNS1) (Adobe-Korea1) (Adobe-GB1)] def
+/_pdf_CheckCIDSystemInfo
+{
+1 index _pdfTextTempString cvs
+(Identity) anchorsearch
+{
+pop pop pop pop true
+}
+{
+false
+_pdfRegOrderingArray
+{
+2 index exch
+anchorsearch
+{ pop pop pop true exit}
+{ pop }
+ifelse
+}
+forall
+exch pop
+exch /CIDFont findresource
+/CIDSystemInfo get
+3 -1 roll /CMap findresource
+/CIDSystemInfo get
+exch
+3 -1 roll
+{
+2 copy
+/Supplement get
+exch
+dup type /dicttype eq
+{/Supplement get}
+{pop 0 }
+ifelse
+ge
+}
+{ true }
+ifelse
+{
+dup /Registry get
+2 index /Registry get eq
+{
+/Ordering get
+exch /Ordering get
+dup type /arraytype eq
+{
+1 index type /arraytype eq
+{
+true
+1 index length 1 sub -1 0
+{
+dup 2 index exch get exch 3 index exch get ne
+{ pop false exit}
+if
+} for
+exch pop exch pop
+}
+{ pop pop false }
+ifelse
+}
+{
+eq
+}
+ifelse
+}
+{ pop pop false }
+ifelse
+}
+{ pop pop false }
+ifelse
+}
+ifelse
+} bind def
+pdf_has_composefont?
+{
+/_pdfComposeFont
+{
+2 copy _pdfComposeFontName not
+{
+2 index
+}
+if
+(pdf) exch _pdfConcatNames
+dup _pdfFontStatus
+{ dup findfont 5 2 roll pop pop pop true}
+{
+4 1 roll
+1 index /CMap pdf_resourcestatus
+{
+pop pop
+true
+}
+{false}
+ifelse
+1 index true exch
+{
+_pdfCIDFontStatus not
+{pop false exit}
+if
+}
+forall
+and
+{
+1 index 1 index 0 get _pdf_CheckCIDSystemInfo
+{
+3 -1 roll pop
+2 index 3 1 roll
+composefont true
+}
+{
+pop pop exch pop false
+}
+ifelse
+}
+{
+_pdfComposeFontName
+{
+dup _pdfFontStatus
+{
+exch pop
+1 index exch
+findfont definefont true
+}
+{
+pop exch pop
+false
+}
+ifelse
+}
+{
+exch pop
+false
+}
+ifelse
+}
+ifelse
+{ true }
+{
+dup _pdfFontStatus
+{ dup findfont true }
+{ pop false }
+ifelse
+}
+ifelse
+}
+ifelse
+} bd
+}
+{
+/_pdfComposeFont
+{
+_pdfComposeFontName not
+{
+dup
+}
+if
+dup
+_pdfFontStatus
+{exch pop dup findfont true}
+{
+1 index
+dup type /nametype eq
+{pop}
+{cvn}
+ifelse
+eq
+{pop false}
+{
+dup _pdfFontStatus
+{dup findfont true}
+{pop false}
+ifelse
+}
+ifelse
+}
+ifelse
+} bd
+}
+ifelse
+/_pdfStyleDicts 4 dict dup begin
+/Adobe-Japan1 4 dict dup begin
+Level2?
+{
+/Serif
+/HeiseiMin-W3-83pv-RKSJ-H _pdfFontStatus
+{/HeiseiMin-W3}
+{
+/HeiseiMin-W3 _pdfCIDFontStatus
+{/HeiseiMin-W3}
+{/Ryumin-Light}
+ifelse
+}
+ifelse
+def
+/SansSerif
+/HeiseiKakuGo-W5-83pv-RKSJ-H _pdfFontStatus
+{/HeiseiKakuGo-W5}
+{
+/HeiseiKakuGo-W5 _pdfCIDFontStatus
+{/HeiseiKakuGo-W5}
+{/GothicBBB-Medium}
+ifelse
+}
+ifelse
+def
+/HeiseiMaruGo-W4-83pv-RKSJ-H _pdfFontStatus
+{/HeiseiMaruGo-W4}
+{
+/HeiseiMaruGo-W4 _pdfCIDFontStatus
+{/HeiseiMaruGo-W4}
+{
+/Jun101-Light-RKSJ-H _pdfFontStatus
+{ /Jun101-Light }
+{ SansSerif }
+ifelse
+}
+ifelse
+}
+ifelse
+/RoundSansSerif exch def
+/Default Serif def
+}
+{
+/Serif /Ryumin-Light def
+/SansSerif /GothicBBB-Medium def
+{
+(fonts/Jun101-Light-83pv-RKSJ-H) status
+}stopped
+{pop}{
+{ pop pop pop pop /Jun101-Light }
+{ SansSerif }
+ifelse
+/RoundSansSerif exch def
+}ifelse
+/Default Serif def
+}
+ifelse
+end
+def
+/Adobe-Korea1 4 dict dup begin
+/Serif /HYSMyeongJo-Medium def
+/SansSerif /HYGoThic-Medium def
+/RoundSansSerif SansSerif def
+/Default Serif def
+end
+def
+/Adobe-GB1 4 dict dup begin
+/Serif /STSong-Light def
+/SansSerif /STHeiti-Regular def
+/RoundSansSerif SansSerif def
+/Default Serif def
+end
+def
+/Adobe-CNS1 4 dict dup begin
+/Serif /MKai-Medium def
+/SansSerif /MHei-Medium def
+/RoundSansSerif SansSerif def
+/Default Serif def
+end
+def
+end
+def
+/TZzero
+{
+/_wmode xdd
+/_styleArr xdd
+/_regOrdering xdd
+3 copy
+_pdfComposeFont
+{
+5 2 roll pop pop pop
+}
+{
+[
+0 1 _styleArr length 1 sub
+{
+_styleArr exch get
+_pdfStyleDicts _regOrdering 2 copy known
+{
+get
+exch 2 copy known not
+{ pop /Default }
+if
+get
+}
+{
+pop pop pop /Unknown
+}
+ifelse
+}
+for
+]
+exch pop
+2 index 3 1 roll
+_pdfComposeFont
+{3 -1 roll pop}
+{
+findfont dup /FontName get exch
+}
+ifelse
+}
+ifelse
+dup /WMode 2 copy known
+{ get _wmode ne }
+{ pop pop _wmode 1 eq}
+ifelse
+{
+exch _wmode _pdfConcatNames
+dup _pdfFontStatus
+{ exch pop dup findfont false}
+{ exch true }
+ifelse
+}
+{
+dup /FontType get 0 ne
+}
+ifelse
+{
+dup /FontType get 3 eq _wmode 1 eq and
+{
+_pdfVerticalRomanT3Font dup length 10 add dict copy
+begin
+/_basefont exch
+dup length 3 add dict
+begin
+{1 index /FID ne {def}{pop pop} ifelse }
+forall
+/Encoding Encoding dup length array copy
+dup 16#27 /quotesingle put
+dup 16#60 /grave put
+_regOrdering /Adobe-Japan1 eq
+{dup 16#5c /yen put dup 16#a5 /yen put dup 16#b4 /yen put}
+if
+def
+FontName
+currentdict
+end
+definefont
+def
+/Encoding _basefont /Encoding get def
+/_fauxfont true def
+}
+{
+dup length 3 add dict
+begin
+{1 index /FID ne {def}{pop pop} ifelse }
+forall
+FontType 0 ne
+{
+/Encoding Encoding dup length array copy
+dup 16#27 /quotesingle put
+dup 16#60 /grave put
+_regOrdering /Adobe-Japan1 eq
+{dup 16#5c /yen put}
+if
+def
+/_fauxfont true def
+} if
+} ifelse
+/WMode _wmode def
+dup dup /FontName exch def
+currentdict
+end
+definefont pop
+}
+{
+pop
+}
+ifelse
+/_pdf_FontDirectory 3 1 roll _safeput
+}
+bd
+Level2?
+{
+/Tf {
+_pdf_FontDirectory 2 index 2 copy known
+{get exch 3 -1 roll pop}
+{pop pop}
+ifelse
+selectfont
+} bd
+}
+{
+/Tf {
+_pdf_FontDirectory 2 index 2 copy known
+{get exch 3 -1 roll pop}
+{pop pop}
+ifelse
+exch findfont exch
+dup type /arraytype eq
+{makefont}
+{scalefont}
+ifelse
+setfont
+} bd
+}
+ifelse
+/cshow where
+{
+pop /pdf_cshow /cshow load dd
+/pdf_remove2 {pop pop} dd
+}
+{
+/pdf_cshow {exch forall} dd
+/pdf_remove2 {} dd
+} ifelse
+/pdf_xshow
+{
+/_pdf_na xdd
+/_pdf_i 0 dd
+currentpoint
+/_pdf_y xdd
+/_pdf_x xdd
+{
+pdf_remove2
+_pdf_str1 exch 0 exch put
+_pdf_str1 /_pdf_showproc load exec
+{_pdf_na _pdf_i get} stopped
+{ pop pop }
+{
+_pdf_x _pdf_y moveto
+0
+rmoveto
+}
+ifelse
+_pdf_i 1 add /_pdf_i xdd
+currentpoint
+/_pdf_y xdd
+/_pdf_x xdd
+}
+exch
+pdf_cshow
+} bd
+/pdf_yshow
+{
+/_pdf_na xdd
+/_pdf_i 0 dd
+currentpoint
+/_pdf_y xdd
+/_pdf_x xdd
+{
+pdf_remove2
+_pdf_str1 exch 0 exch put
+_pdf_str1 /_pdf_showproc load exec
+{_pdf_na _pdf_i get} stopped
+{ pop pop }
+{
+_pdf_x _pdf_y moveto
+0 exch
+rmoveto
+}
+ifelse
+_pdf_i 1 add /_pdf_i xdd
+currentpoint
+/_pdf_y xdd
+/_pdf_x xdd
+}
+exch
+pdf_cshow
+} bd
+/pdf_xyshow
+{
+/_pdf_na xdd
+/_pdf_i 0 dd
+currentpoint
+/_pdf_y xdd
+/_pdf_x xdd
+{
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+} def
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+PDFText begin
+[39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
+/Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute
+/egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde
+/oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex
+/udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
+/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash
+/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef
+/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash
+/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef
+/guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe
+/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide
+/.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright
+/fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
+/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex
+/Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex
+/Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla
+/hungarumlaut/ogonek/caron
+0 TE
+[1/dotlessi/caron 39/quotesingle 96/grave 
+127/bullet/Euro/bullet/quotesinglbase/florin/quotedblbase/ellipsis
+/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE
+/bullet/Zcaron/bullet/bullet/quoteleft/quoteright/quotedblleft
+/quotedblright/bullet/endash/emdash/tilde/trademark/scaron
+/guilsinglright/oe/bullet/zcaron/Ydieresis/space/exclamdown/cent/sterling
+/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine
+/guillemotleft/logicalnot/hyphen/registered/macron/degree/plusminus
+/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla
+/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters
+/questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla
+/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis
+/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash
+/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls/agrave
+/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute
+/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde
+/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute
+/ucircumflex/udieresis/yacute/thorn/ydieresis
+1 TE
+end
+%%BeginResource: procset pdfasc.prc 6.0 1
+%%Copyright: Copyright 1992-2003 Adobe Systems Incorporated. All Rights Reserved.
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+} if
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+}
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+} if
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+} if
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Examples/MeshGenToolbox/doc/testingApp.tex

+\documentclass[12pt,titlepage]{article}
+
+\include{Preamble}
+
+\author{Wen Yu}
+
+\begin{document}
+
+\begin{titlepage}
+
+% Fonts for the title page
+\newcommand{\texttl}{\textsf} %font for the title (Emil's version is \textsf)
+\newcommand{\textau}{\textsf} %font for the authors, date, etc. (Emils version is \textsc)
+
+\begin{center}
+ 
+\HRule \\[0.4cm]
+{ \huge \bfseries \texttl{Summary of Validation Testing for a Parallel Mesh Generation Toolbox}}\\[0.4cm]
+ 
+\HRule \\[2cm]
+
+\textau{\LARGE Wen Yu}\\[2cm]
+ 
+\textau{\LARGE {September 2008}}
+ 
+\vfill
+ 
+% Bottom of the page
+%\includegraphics[width=0.27\textwidth]{logo.pdf}
+\hfill
+\textau{\LARGE
+\raisebox{2pt}{\begin{tabular}[b]{r}
+\raisebox{8pt}{Computing and Software} \\
+McMaster University \end{tabular}}
+}
+\end{center}
+
+\end{titlepage}
+
+\tableofcontents
+
+%to get cross-references correct
+\refstepcounter{funnum} \label{RefiningMesh}
+\refstepcounter{funnum} \label{CoarseningMesh}
+\refstepcounter{funnum} \label{RefiningOrCoarsening}
+\refstepcounter{funnum} \label{MeshType}
+\refstepcounter{funnum} \label{ElmShape}
+\refstepcounter{funnum} \label{DomainDimension}
+\refstepcounter{funnum} \label{Conformal}
+\refstepcounter{funnum} \label{InputDefinition}
+\refstepcounter{funnum} \label{RCInstruction}
+\refstepcounter{funnum} \label{OutputStorage}
+\refstepcounter{funnum} \label{VertexUniqueID}
+\refstepcounter{funnum} \label{ElmUniqueID}
+\refstepcounter{funnum} \label{ElmTopology}
+\refstepcounter{funnum} \label{OutElmOrder}
+\refstepcounter{funnum} \label{OutVertexOrder}
+\refstepcounter{funnum} \label{Help}
+\refstepcounter{nfnum} \label{Performance}
+\refstepcounter{nfnum} \label{Precision}
+\refstepcounter{nfnum} \label{Exception}
+\refstepcounter{nfnum} \label{Portability}
+\refstepcounter{nfnum} \label{LookAndFeel}
+\refstepcounter{nfnum} \label{Usability}
+\refstepcounter{nfnum} \label{Maintainability}
+
+\refstepcounter{mnum} \label{mMemory}
+\refstepcounter{mnum} \label{mFileRW}
+\refstepcounter{mnum} \label{mKeyboard}
+\refstepcounter{mnum} \label{mScreen}
+\refstepcounter{mnum} \label{mInput}
+\refstepcounter{mnum} \label{mOutput}
+\refstepcounter{mnum} \label{mService}
+\refstepcounter{mnum} \label{mVertex}
+\refstepcounter{mnum} \label{mEdge}
+\refstepcounter{mnum} \label{mCell}
+\refstepcounter{mnum} \label{mMesh}
+\refstepcounter{mnum} \label{mRefining}
+\refstepcounter{mnum} \label{mCoarsening}
+
+\newpage
+
+\section{Introduction \label{AtestSecIntro}}
+This section gives an overview of the Testing Summary for a Parallel Mesh Generation Toolbox (PMGT). First, the purpose of the document is provided. Second, the scope of the testing is identified. Third, the organization of the document is summarized.
+
+\subsection{Purpose of the Document}
+This document specifies validation tests for a PMGT. The results of the tests and analysis are also provided. The intended audience is testers who are going to test the system and developers who are going to maintain the software. Note that test document is dynamic in the sense that it should be updated when the development of the system proceeds. 
+
+\subsection{Scope of the Testing }
+In general, the purpose of testing is to help produce quality software. Due to limits on the time available for testing, the scope of the testing of PMGT is restricted to test the most important test factors. Like other scientific computing software, correctness and efficiency are considered to be the two most important test factors for PMGT. For efficiency testing, the focus is on execution time rather than on storage. 
+
+\subsection{Organization of the Document}
+Section \ref{AtestSecIntro} (this section) is an introduction to the report. Section \ref{AtestSecTest} shows what is going to be tested and the coverage of the testing, with respect to the software requirements and the software design. Section \ref{AtestSecResult} gives the result of the testing and the analysis. 
+
+\section{Testing PMGT \label{AtestSecTest}}
+Test cases are listed in Section \ref{AtestSecTestcases}. The detailed information for these test cases can be found in Section \ref{AtestSecResult}. The traceability matrix in Section \ref{AmatrixR} shows the association between test cases and the functional and nonfunctional requirements that are specified in the Software Requirements Specification (SRS) document. Similarly, a traceability matrix for test cases and the leaf modules as introduced in the Module Guide (MG) as shown in Section \ref{AmatrixM}. Tracking these relations is useful for developing and maintaining the software.
+
+\subsection{Test Cases \label{AtestSecTestcases}}
+The correctness validation test is designed for verifying the functional requirements RefiningMesh (\fref{RefiningMesh}), CoarseningMesh (\fref{CoarseningMesh}), ElmShape (\fref{ElmShape}), and Conformal (\fref{Conformal}). Other requirements for correctness are trivial and are satisfied obviously. For example, since the vertices are stored in an array, the OutVertexOrder (\fref{OutVertexOrder}) requirement is met by outputting the vertices in the order as the order of them in the array. The tests are against above requirements are automated. The automated validation tests requirements (ACVTRs) are listed in Section \ref{ACVTRs}. Since the output mesh also can also be displayed on screen, it can be checked manually. The visual correctness validation tests requirements (VCVTRs) are listed in Section \ref{VCVTRs}. The test cases are in Section \ref{TCs}.
+\subsubsection{Automated Correctness Validation Tests Requirements \label{ACVTRs}} 
+A list of ACVTRs follows. All test cases should pass these tests. Some test cases relate to data definitions defined in the SRS. In these cases the related data definition defined is shown as D$x$, where $x$ is the number of the associated data definition given in the SRS. 
+\begin{itemize}
+\item The area of each element is greater than zero (referring to \textbf{D5}). %hard coded reference
+\item The boundary of the mesh is closed. (referring to \textbf{D15}). %hard coded reference
+\item The mesh is conformal (referring to \textbf{D16}). %hard coded reference
+\item The intersection of any two elements is empty (referring to \textbf{D17}). %hard coded reference
+\item The input mesh and output mesh {\em CoveringUp} each other (referring to \textbf{D19}). %hard coded reference
+\item The length of each edge is greater than zero. (This is required by the definition of a mesh, which is defined in the SRS.)
+\item The vertices of each element are listed in a counterclockwise order. (The counterclockwise order of the vertices for each element is not necessary for implementing PMGT. However, it is adopted by most meshing and FEA software. PMGT uses this convention.) 
+\item The output mesh conforms to the Euler Equation. (This requirement is not documented in the SRS. However, any mesh should implicitly satisfy the equation $nc+nv-ne=1$, where $nc$ is the number of cells, $nv$ is the number of vertices, and $ne$ is the number of edges.)
+\end{itemize}
+\subsubsection{Visual Correctness Validation Tests Requirements\label{VCVTRs}}
+The output meshes should also be visually checked to ensure that the following VCVTRs are met.
+\begin{itemize}
+\item No vertex is outside of the input domain.
+\item No vertex is inside of a cell.
+\item No dangling points or edges are present.
+\item All cells are connected.
+\item The mesh is conformal.
+\end{itemize}
+Some of the VCVTRs overlap with the ACVTRs. This redundancy provides increased confidence in case one testing method fails to catch an error. 
+\subsubsection{Test Cases\label{TCs}}
+The test cases developed involve testing meshes against the above requirements. In each test case, except the last one, the input mesh is refined and then coarsened. Two algorithms for refining are used. One algorithm is called {\em Split}. It splits one cell into three by adding a point in the centroid of the triangle and connecting the added point to the three original vertices. The other algorithm is simply call {\em Refine}. It refines the original mesh by longest edge bisection. 
+
+The name of each test case includes three parts. For example, test case $AxxCBN$ means that the test uses $Axx$ algorithm for refining, where $Axx$ equals $Split$ or $Refine$. The letter $C$ indicates that coarsening is performed. If the $C$ is missing, the input mesh is not coarsened. $B$ is the number of refinements before coarsening. If $B$ is S, the mesh is refined once and then coarsen once. If the $B$ is M, the mesh is refined multiple time before coarsening. $N$ is a number. If the $N$ is omitted, it means only one of this kind of test performed. Otherwise the same test procedures is used several times on different input meshes. The reason for using the same procedure is that the topology of the output meshes may differ for different input meshes.
+\begin{itemize}
+\item Test Case {\em SplitCS} (\refstepcounter{casenum}\cthecasenum \label{ATCscs}): This test case tests the correctness of PMGT. The input mesh is shown in Figure \ref{input1}. The refining and coarsening criterion is that the cells intersected with the vertical line, $x=0.6$, are Split once, then the cells of the new mesh that intersect with the vertical line are coarsened once. When the splitting and coarsening is done, the vertical line is moved to the right one unit ($x=x+1.0$), and another Splitting and coarsening is performed. This procedure is repeated until no cells intersect with the vertical line. 
+
+\item Test Case {\em RefineCS1} (\refstepcounter{casenum}\cthecasenum \label{ATCrcs1}): This test case tests the correctness of PMGT. The input mesh is the same as \cref{ATCscs}, which is shown in Figure \ref{input1}. There is a vertical line at $x = 0.6$. The refining and coarsening criterion is that the cells that intersect with the vertical line are refined once, then the cells of the new mesh that intersect with the vertical line are coarsened once. When the refining and coarsening are done, the vertical line is moved to the right one unit, and another refining and coarsening is performed. This procedure is repeated until no cells intersect with the vertical line. 
+
+\item Test Case {\em RefineCS2} (\refstepcounter{casenum}\cthecasenum \label{ATCrcs2}): This test case tests the correctness of PMGT. The refining and coarsening criterion, vertical line function, and the test procedure are the same as test case \cref{ATCrcs1}. However, the input mesh is different. The input mesh is showed in Figure \ref{input2}. 
+
+\item Test Case {\em RefineCM} (\refstepcounter{casenum}\cthecasenum \label{ATCrcm}): This test case tests the correctness of PMGT. The input mesh is shown in Figure \ref{input3}. There is a vertical line at $x = 0.5$. The refining and coarsening criterion is the size of the cells. The size of the cell is measured by the length of the longest edge of the cell. The cells that intersect with the vertical line are refined until the criterion is met. When the refining is done, the vertical line is moved to the right $0.6$ unit ($x=x+0.6$), and another refinement is performed. After five refinements are done, the cells to be left of the vertical line by up to 2 units are coarsened, until the coarsening criterion is met. The refining and coarsening are stopped when the vertical line moves to a position outside of the domain. 
+
+\item Test Case {\em RefineM} (\refstepcounter{casenum}\cthecasenum \label{ATCrm}): This test case tests the correctness of PMGT. The input mesh is shown in Figure \ref{input4}. There is an arc with radius of 0.7 unit going through the mesh. Cells that intersect with the arc are refined until the required number of refinements has been reached. 
+
+\item Test Case {\em Split} (\refstepcounter{casenum}\cthecasenum \label{ATCss}): This test case tests both the correctness and speed of PMGT. The input mesh is shown in Figure \ref{input5}. This test simply splits all cells of the mesh 4 times. It is done in both the serial version and the parallel version with different number of processors. The execution time of setting the cells to be refined and splitting the cells is measured. 
+\end{itemize}
+
+\subsection{Traceability Matrix for SRS \label{AmatrixR}}
+In the traceability matrix for software requirements, if a test case tests the functionality of a software requirement, there will be a check mark on the cell for the corresponding test case. In each row of the traceability matrix for software requirements (Table \ref{AtestTMr}), if the requirement in that row defines the correctness or the speed of the software, one or more cells in this row are checked. Otherwise, all cells in the row are empty. Table \ref{AtestTMr} shows that the test cases developed in Section \ref{AtestSecTestcases} assist with validating the correctness and speed of the software. The detailed information for each functional and nonfunctional requirements can be found in the SRS document.
+The names of the requirements and their corresponding numbers are listed below for convenience.
+\begin{description}
+\item[\fref{RefiningMesh}:] RefiningMesh 
+\item[\fref{CoarseningMesh}:] CoarseningMesh
+\item[\fref{RefiningOrCoarsening}:] RefiningOrCoarsening
+\item[\fref{MeshType}:] MeshType
+\item[\fref{ElmShape}:] ElmShape
+\item[\fref{DomainDimension}:] DomainDimension
+\item[\fref{Conformal}:] Conformal
+\item[\fref{InputDefinition}:] InputDefinition
+\item[\fref{RCInstruction}:] RCInstruction
+\item[\fref{OutputStorage}:] OutputStorage
+\item[\fref{VertexUniqueID}:] VertexUniqueID
+\item[\fref{ElmUniqueID}:] ElmUniqueID
+\item[\fref{ElmTopology}:] ElmTopology
+\item[\fref{OutElmOrder}:] OutElmOrder
+\item[\fref{OutVertexOrder}:] OutVertexOrder
+\item[\fref{Help}:] Help
+\item[\nref{Performance}:] Performance
+\item[\nref{Precision}:] Precision
+\item[\nref{Exception}:] Exception
+\item[\nref{Portability}:] Portability
+\item[\nref{LookAndFeel}:] LookAndFeel
+\item[\nref{Usability}:] Usability
+\item[\nref{Maintainability}:] Maintainability
+\end{description}
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|}
+\hline
+ &\cref{ATCscs} & \cref{ATCrcs1} & \cref{ATCrcs2} & \cref{ATCrcm} & \cref{ATCrm}&\cref{ATCss}\\
+\hline\hline
+\fref{RefiningMesh}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark\\\hline%f2
+\fref{CoarseningMesh}& \checkmark & \checkmark & \checkmark & \checkmark & &   \\\hline%f3
+\fref{RefiningOrCoarsening}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline%f1
+\fref{MeshType}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark&  \\\hline%f4
+\fref{ElmShape}& \checkmark & \checkmark & \checkmark &  \checkmark  & \checkmark& \\\hline%f5
+\fref{DomainDimension}& \checkmark & \checkmark &  \checkmark  & \checkmark & \checkmark& \\\hline%f6
+\fref{Conformal}&  \checkmark &\checkmark & \checkmark & \checkmark & \checkmark& \\\hline%f7
+\fref{InputDefinition}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\fref{RCInstruction}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\fref{OutputStorage}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \\\hline
+\fref{VertexUniqueID}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \\\hline
+\fref{ElmUniqueID}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark &  \\\hline
+\fref{ElmTopology}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark &  \\\hline
+\fref{OutElmOrder}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \\\hline
+\fref{OutVertexOrder}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \\\hline
+\fref{Help} & &  &    &  & &  \\\hline
+\nref{Performance} &  &  &    &  & & \checkmark \\\hline
+\nref{Precision} &  &  &    &  & &  \\\hline
+\nref{Exception} & &  &    &  & &  \\\hline
+\nref{Portability} & &  &    &  & &  \\\hline
+\nref{LookAndFeel} &   &  &    &  & &  \\\hline
+\nref{Usability} &   &  &    &  & &  \\\hline
+\nref{Maintainability} &  &  &    &  & &  \\\hline
+\end{tabular}
+\caption{Traceability Matrix: Test Cases and Requirements}
+\label{AtestTMr}
+\end{table}
+
+\subsection{Traceability Matrix for MG \label{AmatrixM}}
+Similar to Section \ref{AmatrixR}, the traceability matrix for modules (Table \ref{AtestTMm}) shows that the test cases validate the modules that are associated with correctness and speed. The names of modules appear in Table \ref{AtestTMm} are listed below. The detailed information for each module can be found in the MG document.
+\begin{description}
+\item[\mref{mMemory}:] Virtual Memory Module
+\item[\mref{mFileRW}:] File Read/Write Module
+\item[\mref{mKeyboard}:] Keyboard Input Module
+\item[\mref{mScreen}:] Screen Output Module
+\item[\mref{mInput}:] Input Format Module
+\item[\mref{mOutput}:] Output Format Module
+\item[\mref{mService}:] Service Module
+\item[\mref{mVertex}:] Vertex Module
+\item[\mref{mEdge}:] Edge Module
+\item[\mref{mCell}:] Cell Module
+\item[\mref{mMesh}:] Mesh Module
+\item[\mref{mRefining}:] Refining Module
+\item[\mref{mCoarsening}:] Coarsening Module
+\end{description}
+
+\begin{table}[ht]
+\centering
+\begin{tabular}{|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|p{0.06\textwidth}|}
+\hline
+ &\cref{ATCscs} & \cref{ATCrcs1} & \cref{ATCrcs2} & \cref{ATCrcm} & \cref{ATCrm}&\cref{ATCss}\\
+\hline\hline
+\mref{mMemory}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mFileRW}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mKeyboard}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mScreen}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mInput}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mOutput}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mService}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark &  \\\hline
+\mref{mVertex}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mEdge}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mCell}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mMesh}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mRefining}& \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
+\mref{mCoarsening}& \checkmark & \checkmark & \checkmark & \checkmark &  &  \\\hline
+\end{tabular} 
+\caption{Traceability Matrix: Test Cases and Modules}
+\label{AtestTMm}
+\end{table}
+
+\section{Results and Analysis\label{AtestSecResult}}
+The results of the test cases defined in Section \ref{TCs} are listed in Section \ref{AtestSectestresult}. The analysis, including charts that compare the execution time of the parallel version to the serial version are provided in Section \ref{AtestSecAnalysis}. 
+\subsection {Testing Results \label{AtestSectestresult}}
+The following tables list the testing results of each test case. The field {\em Test Case Number} and {\em Test Case Name} list the number and the name of each test case. The {\em Input field} gives the number of the figure that is the input for that test case, or a description of the input mesh. The {\em Expected Output} describes the requirements of the output mesh. The {\em Actual Output} gives the result of the test. The {\em Selected Output Mesh} field should give the output meshes. However, there are too many intermediate mesh to display, and displaying only the final mesh is too simple to illustrate the feature of the test case. Selected intermediate meshes and final mesh are included in the {\em Actual Output} field. The {\em Result} field indicates whether the test is passed or failed.
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Test Case Number} & \cref{ATCscs} \\ 
+{\bf Test Case Name} & SplitCS\\ 
+\hline
+Input & Figure \ref{input1}\\
+Expected Output & ACVTRs and VCVTRs listed in Section \ref{AtestSecTest} are met\\
+Actual Output & Summary of the correctness test:\\
+& 15 tests are performed.\\
+& 15 tests succeed.\\
+& 0 tests fail.\\
+Selected Output Mesh &Figure \ref{output11}, \ref{output12}, \ref{output13}\\
+Result & Passed\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Test Case Number} & \cref{ATCrcs1} \\ 
+{\bf Test Case Name} & RefineCS1\\ 
+\hline
+Input & Figure \ref{input1}\\
+Expected Output & ACVTRs and VCVTRs listed in Section \ref{AtestSecTest} are met\\
+Actual Output & Summary of the correctness test:\\
+& 15 tests are performed.\\
+& 15 tests succeed.\\
+& 0 tests fail.\\
+Selected Output Mesh & Figure \ref{output21}, \ref{output22}, \ref{output23}\\
+Result & Passed\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Test Case Number} & \cref{ATCrcs2} \\ 
+{\bf Test Case Name} & RefineCS2\\ 
+\hline
+Input & Figure \ref{input2}\\
+Expected Output & ACVTRs and VCVTRs listed in Section \ref{AtestSecTest} are met\\
+Actual Output & Summary of the correctness test:\\
+& 15 tests are performed.\\
+& 15 tests succeed.\\
+& 0 tests fail.\\
+Selected Output Mesh & Figure \ref{output31}, \ref{output32}, \ref{output33}\\
+Result & Passed\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Test Case Number} & \cref{ATCrcm} \\ 
+{\bf Test Case Name} & RefineCM\\ 
+\hline
+Input & Figure \ref{input3}\\
+Expected Output & ACVTRs and VCVTRs listed in Section \ref{AtestSecTest} are met\\
+Actual Output & Summary of the correctness test:\\
+& 15 tests are performed.\\
+& 15 tests succeed.\\
+& 0 tests fail.\\
+Selected Output Mesh & Figure \ref{output41}, \ref{output42}, \ref{output43}, \ref{output44}\\
+Result & Passed\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Test Case Number} & \cref{ATCrm} \\ 
+{\bf Test Case Name} & RefineM\\ 
+\hline
+Input & Figure \ref{input4}\\
+Expected Output & ACVTRs and VCVTRs listed in Section \ref{AtestSecTest} are met\\
+Actual Output & Summary of the correctness test:\\
+& 15 tests are performed.\\
+& 15 tests succeed.\\
+& 0 tests fail.\\
+Selected Output Mesh & Figure \ref{output51}, \ref{output52}, \ref{output53}\\
+Result & Passed\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\vspace{1cm}
+\begin{minipage}{\textwidth}
+\begin{tabular}{p{\colAwidth}|p{\colBwidth}} 
+\hline \hline 
+{\bf Test Case Number} & \cref{ATCss} \\ 
+{\bf Test Case Name} & SplitM\\ 
+\hline
+Input & Figure \ref{input4}\\
+Expected Output & ACVTRs and VCVTRs listed in Section \ref{AtestSecTest} are met\\
+& Execution time increases as the number of cells increases. Execution time decreases as the number of processors increases.\\
+Actual Output & Execution time as indicated in Figure \ref{output61}\\
+Selected Output Mesh & The mesh is too dense to be shown.\\
+Result & Passed\\
+\hline \hline
+\end{tabular}
+\end{minipage}
+
+\begin{figure}[ht]
+\centering
+\epsfig{file=output61.eps, width= \textwidth}
+\caption{Output of \cref{ATCss}}
+\label{output61}
+\end{figure}
+
+\subsection {Analysis \label{AtestSecAnalysis}}
+All of the test cases conform to the ACVTRs and VCVTRs listed in Section \ref{AtestSecTest}. The test result of \cref{ATCss} show that when the number of cells increased, the execution time increased, and when the number of processors increased, the execution time decreased. That is, this test is passed. Figure \ref{speedup} show the speedup when using different numbers of processors. The speedup is defined as
+\[ Speedup(n) = \frac{T_{1}}{T_{n}}
+\]
+Where $T_{1}$ is the execution time of the serial version, and $T_{n}$ is the execution time of the parallel version with $n$ processors. In general, $Speedup(n) < n$. However, for PMGT, when the number of cells is greater than 2700, $Speedup(n) > n$, which is a super linear speedup. Since the algorithms used for the serial version and the parallel version are the same, the super linear speedup is probably due to the cache effect. That is, when the numbers of processors increases, the size of the accumulated caches from different processors also increases. With the larger accumulated cache size, more, or even all, core data set can fit into the caches and the memory access time reduces dramatically. This may explain the extra speedup in additional to the speedup due to parallel computation.
+
+\begin{figure}
+\centering
+\epsfig{file=speedup.eps, width= \textwidth}
+\caption{Speedup for Different Numbers of Processors}
+\label{speedup}
+\end{figure}
+
+
+\begin{figure}
+\centering
+\epsfig{file=input1.eps, width= 0.7\textwidth}
+\caption{Input 1}
+\label{input1}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=input2.eps, width= 0.7\textwidth}
+\caption{Input 2}
+\label{input2}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=input3.eps, width= 0.7\textwidth}
+\caption{Input 3}
+\label{input3}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=input4.eps, width= 0.7\textwidth}
+\caption{Input 4}
+\label{input4}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=input5.eps, width= 0.7\textwidth}
+\caption{Input 5}
+\label{input5}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output11.eps, width=0.7\textwidth}
+\caption{Output 1 of \cref{ATCscs}}
+\label{output11}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output12.eps, width=0.7\textwidth}
+\caption{Output 2 of \cref{ATCscs}}
+\label{output12}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output13.eps, width=0.7\textwidth}
+\caption{Output 3 of \cref{ATCscs}}
+\label{output13}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output21.eps, width=0.7\textwidth}
+\caption{Output 1 of \cref{ATCrcs1}}
+\label{output21}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output22.eps, width=0.7\textwidth}
+\caption{Output 2 of \cref{ATCrcs1}}
+\label{output22}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output23.eps, width=0.7\textwidth}
+\caption{Output 3 of \cref{ATCrcs1}}
+\label{output23}
+\end{figure}
+
+\clearpage
+
+\begin{figure}
+\centering
+\epsfig{file=output31.eps, width=0.7\textwidth}
+\caption{Output 1 of \cref{ATCrcs2}}
+\label{output31}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output32.eps, width=0.7\textwidth}
+\caption{Output 2 of \cref{ATCrcs2}}
+\label{output32}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output33.eps, width=0.7\textwidth}
+\caption{Output 3 of \cref{ATCrcs2}}
+\label{output33}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output41.eps, width=0.7\textwidth}
+\caption{Output 1 of \cref{ATCrcm}}
+\label{output41}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output42.eps, width=0.7\textwidth}
+\caption{Output 2 of \cref{ATCrcm}}
+\label{output42}
+\end{figure}
+
+\begin{figure} 
+\centering
+\epsfig{file=output43.eps, width=0.7\textwidth}
+\caption{Output 3 of \cref{ATCrcm}}
+\label{output43}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output44.eps, width=0.7\textwidth}
+\caption{Output 4 of \cref{ATCrcm}}
+\label{output44}
+\end{figure}
+
+
+\begin{figure}
+\centering
+\epsfig{file=output51.eps, width=0.7\textwidth}
+\caption{Output 1 of \cref{ATCrm}}
+\label{output51}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output52.eps, width=0.7\textwidth}
+\caption{Output 2 of \cref{ATCrm}}
+\label{output52}
+\end{figure}
+
+\begin{figure}
+\centering
+\epsfig{file=output53.eps, width=0.7\textwidth}
+\caption{Output 3 of \cref{ATCrm}}
+\label{output53}
+\end{figure}
+
+%\newpage
+
+%\bibliography{WenRef}
+%\bibliographystyle{plainnat}
+
+\end{document}

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+		/ct_GetInterval
+			{
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+				begin
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+				if
+				1 index ct_CID_STR_SIZE idiv
+				/arrayIndex exch def
+				2 index arrayIndex  get
+				2 index
+				arrayIndex ct_CID_STR_SIZE mul
+				sub
+					{
+					dup 3 index add 2 index length le
+						{
+						2 index getinterval
+						dst_string  dst_index 2 index putinterval
+						length dst_index add /dst_index exch def
+						exit
+						}
+						{
+						1 index length 1 index sub
+						dup 4 1 roll
+						getinterval
+						dst_string  dst_index 2 index putinterval
+						pop dup dst_index add /dst_index exch def
+						sub
+						/arrayIndex arrayIndex 1 add def
+						2 index dup length arrayIndex gt
+							  { arrayIndex get }
+							  {
+							  pop
+							  exit
+							  }
+						ifelse
+						0
+						}
+					ifelse
+					}
+				loop
+				pop pop pop
+				dst_string 0 dst_index getinterval
+				end
+			} bind def
+		ct_Level2?
+			{
+			/ct_resourcestatus
+			currentglobal mark true setglobal
+				{ /unknowninstancename /Category resourcestatus }
+			stopped
+				{ cleartomark setglobal true }
+				{ cleartomark currentglobal not exch setglobal }
+			ifelse
+				{
+					{
+					mark 3 1 roll /Category findresource
+						begin
+						ct_Vars /vm currentglobal put
+						({ResourceStatus} stopped) 0 () /SubFileDecode filter cvx exec
+							{ cleartomark false }
+							{ { 3 2 roll pop true } { cleartomark false } ifelse }
+						ifelse
+						ct_Vars /vm get setglobal
+						end
+					}
+				}
+				{ { resourcestatus } }
+			ifelse bind def
+			/CIDFont /Category ct_resourcestatus
+				{ pop pop }
+				{
+				currentglobal  true setglobal
+				/Generic /Category findresource
+				dup length dict copy
+				dup /InstanceType /dicttype put
+				/CIDFont exch /Category defineresource pop
+				setglobal
+				}
+			ifelse
+			ct_UseNativeCapability?
+				{
+				/CIDInit /ProcSet findresource begin
+				12 dict begin
+				begincmap
+				/CIDSystemInfo 3 dict dup begin
+				  /Registry (Adobe) def
+				  /Ordering (Identity) def
+				  /Supplement 0 def
+				end def
+				/CMapName /Identity-H def
+				/CMapVersion 1.000 def
+				/CMapType 1 def
+				1 begincodespacerange
+				<0000> <FFFF>
+				endcodespacerange
+				1 begincidrange
+				<0000> <FFFF> 0
+				endcidrange
+				endcmap
+				CMapName currentdict /CMap defineresource pop
+				end
+				end
+				}
+			if
+			}
+			{
+			/ct_Category 2 dict begin
+			/CIDFont  10 dict def
+			/ProcSet	2 dict def
+			currentdict
+			end
+			def
+			/defineresource
+				{
+				ct_Category 1 index 2 copy known
+					{
+					get
+					dup dup maxlength exch length eq
+						{
+						dup length 10 add dict copy
+						ct_Category 2 index 2 index put
+						}
+					if
+					3 index 3 index put
+					pop exch pop
+					}
+					{ pop pop /defineresource /undefined ct_raise }
+				ifelse
+				} bind def
+			/findresource
+				{
+				ct_Category 1 index 2 copy known
+					{
+					get
+					2 index 2 copy known
+						{ get 3 1 roll pop pop}
+						{ pop pop /findresource /undefinedresource ct_raise }
+					ifelse
+					}
+					{ pop pop /findresource /undefined ct_raise }
+				ifelse
+				} bind def
+			/resourcestatus
+				{
+				ct_Category 1 index 2 copy known
+					{
+					get
+					2 index known
+					exch pop exch pop
+						{
+						0 -1 true
+						}
+						{
+						false
+						}
+					ifelse
+					}
+					{ pop pop /findresource /undefined ct_raise }
+				ifelse
+				} bind def
+			/ct_resourcestatus /resourcestatus load def
+			}
+		ifelse
+		/ct_CIDInit 2 dict
+			begin
+			/ct_cidfont_stream_init
+				{
+					{
+					dup (Binary) eq
+						{
+						pop
+						null
+						currentfile
+						ct_Level2?
+							{
+								{ cid_BYTE_COUNT () /SubFileDecode filter }
+							stopped
+								{ pop pop pop }
+							if
+							}
+						if
+						/readstring load
+						exit
+						}
+					if
+					dup (Hex) eq
+						{
+						pop
+						currentfile
+						ct_Level2?
+							{
+								{ null exch /ASCIIHexDecode filter /readstring }
+							stopped
+								{ pop exch pop (>) exch /readhexstring }
+							if
+							}
+							{ (>) exch /readhexstring }
+						ifelse
+						load
+						exit
+						}
+					if
+					/StartData /typecheck ct_raise
+					}
+				loop
+				cid_BYTE_COUNT ct_CID_STR_SIZE le
+					{
+					2 copy cid_BYTE_COUNT string exch exec
+					pop
+					1 array dup
+					3 -1 roll
+					0 exch put
+					}
+					{
+					cid_BYTE_COUNT ct_CID_STR_SIZE div ceiling cvi
+					dup array exch 2 sub 0 exch 1 exch
+						{
+						2 copy
+						5 index
+						ct_CID_STR_SIZE
+						string
+						6 index exec
+						pop
+						put
+						pop
+						}
+					for
+					2 index
+					cid_BYTE_COUNT ct_CID_STR_SIZE mod string
+					3 index exec
+					pop
+					1 index exch
+					1 index length 1 sub
+					exch put
+					}
+				ifelse
+				cid_CIDFONT exch /GlyphData exch put
+				2 index null eq
+					{
+					pop pop pop
+					}
+					{
+					pop /readstring load
+					1 string exch
+						{
+						3 copy exec
+						pop
+						dup length 0 eq
+							{
+							pop pop pop pop pop
+							true exit
+							}
+						if
+						4 index
+						eq
+							{
+							pop pop pop pop
+							false exit
+							}
+						if
+						}
+					loop
+					pop
+					}
+				ifelse
+				} bind def
+			/StartData
+				{
+				mark
+					{
+					currentdict
+					dup /FDArray get 0 get /FontMatrix get
+					0 get 0.001 eq
+						{
+						dup /CDevProc known not
+							{
+							/CDevProc 1183615869 internaldict /stdCDevProc 2 copy known
+								{ get }
+								{
+								pop pop
+								{ pop pop pop pop pop 0 -1000 7 index 2 div 880 }
+								}
+							ifelse
+							def
+							}
+						if
+						}
+						{
+						 /CDevProc
+							 {
+							 pop pop pop pop pop
+							 0
+							 1 cid_temp /cid_CIDFONT get
+							 /FDArray get 0 get
+							 /FontMatrix get 0 get div
+							 7 index 2 div
+							 1 index 0.88 mul
+							 } def
+						}
+					ifelse
+					/cid_temp 15 dict def
+					cid_temp
+						begin
+						/cid_CIDFONT exch def
+						3 copy pop
+						dup /cid_BYTE_COUNT exch def 0 gt
+							{
+							ct_cidfont_stream_init
+							FDArray
+								{
+								/Private get
+								dup /SubrMapOffset known
+									{
+									begin
+									/Subrs SubrCount array def
+									Subrs
+									SubrMapOffset
+									SubrCount
+									SDBytes
+									ct_Level2?
+										{
+										currentdict dup /SubrMapOffset undef
+										dup /SubrCount undef
+										/SDBytes undef
+										}
+									if
+									end
+									/cid_SD_BYTES exch def
+									/cid_SUBR_COUNT exch def
+									/cid_SUBR_MAP_OFFSET exch def
+									/cid_SUBRS exch def
+									cid_SUBR_COUNT 0 gt
+										{
+										GlyphData cid_SUBR_MAP_OFFSET cid_SD_BYTES ct_GetInterval
+										0 cid_SD_BYTES ct_cvnsi
+										0 1 cid_SUBR_COUNT 1 sub
+											{
+											exch 1 index
+											1 add
+											cid_SD_BYTES mul cid_SUBR_MAP_OFFSET add
+											GlyphData exch cid_SD_BYTES ct_GetInterval
+											0 cid_SD_BYTES ct_cvnsi
+											cid_SUBRS 4 2 roll
+											GlyphData exch
+											4 index
+											1 index
+											sub
+											ct_GetInterval
+											dup length string copy put
+											}
+										for
+										pop
+										}
+									if
+									}
+									{ pop }
+								ifelse
+								}
+							forall
+							}
+						if
+						cleartomark pop pop
+						end
+					CIDFontName currentdict /CIDFont defineresource pop
+					end end
+					}
+				stopped
+					{ cleartomark /StartData ct_reraise }
+				if
+				} bind def
+			currentdict
+			end def
+		/ct_saveCIDInit
+			{
+			/CIDInit /ProcSet ct_resourcestatus
+				{ true }
+				{ /CIDInitC /ProcSet ct_resourcestatus }
+			ifelse
+				{
+				pop pop
+				/CIDInit /ProcSet findresource
+				ct_UseNativeCapability?
+					{ pop null }
+					{ /CIDInit ct_CIDInit /ProcSet defineresource pop }
+				ifelse
+				}
+				{ /CIDInit ct_CIDInit /ProcSet defineresource pop null }
+			ifelse
+			ct_Vars exch /ct_oldCIDInit exch put
+			} bind def
+		/ct_restoreCIDInit
+			{
+			ct_Vars /ct_oldCIDInit get dup null ne
+				{ /CIDInit exch /ProcSet defineresource pop }
+				{ pop }
+			ifelse
+			} bind def
+		/ct_BuildCharSetUp
+			{
+			1 index
+				begin
+				CIDFont
+					begin
+					Adobe_CoolType_Utility /ct_BuildCharDict get
+						begin
+						/ct_dfCharCode exch def
+						/ct_dfDict exch def
+						CIDFirstByte ct_dfCharCode add
+						dup CIDCount ge
+							{ pop 0 }
+						if
+						/cid exch def
+							{
+							GlyphDirectory cid 2 copy known
+								{ get }
+								{ pop pop nullstring }
+							ifelse
+							dup length FDBytes sub 0 gt
+								{
+								dup
+								FDBytes 0 ne
+									{ 0 FDBytes ct_cvnsi }
+									{ pop 0 }
+								ifelse
+								/fdIndex exch def
+								dup length FDBytes sub FDBytes exch getinterval
+								/charstring exch def
+								exit
+								}
+								{
+								pop
+								cid 0 eq
+									{ /charstring nullstring def exit }
+								if
+								/cid 0 def
+								}
+							ifelse
+							}
+						loop
+			} def
+		/ct_SetCacheDevice
+			{
+			0 0 moveto
+			dup stringwidth
+			3 -1 roll
+			true charpath
+			pathbbox
+			0 -1000
+			7 index 2 div 880
+			setcachedevice2
+			0 0 moveto
+			} def
+		/ct_CloneSetCacheProc
+			{
+			1 eq
+				{
+				stringwidth
+				pop -2 div -880
+				0 -1000 setcharwidth
+				moveto
+				}
+				{
+				usewidths?
+					{
+					currentfont /Widths get cid
+					2 copy known
+						{ get exch pop aload pop }
+						{ pop pop stringwidth }
+					ifelse
+					}
+					{ stringwidth }
+				ifelse
+				setcharwidth
+				0 0 moveto
+				}
+			ifelse
+			} def
+		/ct_Type3ShowCharString
+			{
+			ct_FDDict fdIndex 2 copy known
+				{ get }
+				{
+				currentglobal 3 1 roll
+				1 index gcheck setglobal
+				ct_Type1FontTemplate dup maxlength dict copy
+					begin
+					FDArray fdIndex get
+					dup /FontMatrix 2 copy known
+						{ get }
+						{ pop pop ct_defaultFontMtx }
+					ifelse
+					/FontMatrix exch dup length array copy def
+					/Private get
+					/Private exch def
+					/Widths rootfont /Widths get def
+					/CharStrings 1 dict dup /.notdef
+						<d841272cf18f54fc13> dup length string copy put def
+					currentdict
+					end
+				/ct_Type1Font exch definefont
+				dup 5 1 roll put
+				setglobal
+				}
+			ifelse
+			dup /CharStrings get 1 index /Encoding get
+			ct_dfCharCode get charstring put
+			rootfont /WMode 2 copy known
+				{ get }
+				{ pop pop 0 }
+			ifelse
+			exch
+			1000 scalefont setfont
+			ct_str1 0 ct_dfCharCode put
+			ct_str1 exch ct_dfSetCacheProc
+			ct_SyntheticBold
+				{
+				currentpoint
+				ct_str1 show
+				newpath
+				moveto
+				ct_str1 true charpath
+				ct_StrokeWidth setlinewidth
+				stroke
+				}
+				{ ct_str1 show }
+			ifelse
+			} def
+		/ct_Type4ShowCharString
+			{
+			ct_dfDict ct_dfCharCode charstring
+			FDArray fdIndex get
+			dup /FontMatrix get dup ct_defaultFontMtx ct_matrixeq not
+				{ ct_1000Mtx matrix concatmatrix concat }
+				{ pop }
+			ifelse
+			/Private get
+			Adobe_CoolType_Utility /ct_Level2? get not
+				{
+				ct_dfDict /Private
+				3 -1 roll
+					{ put }
+				1183615869 internaldict /superexec get exec
+				}
+			if
+			1183615869 internaldict
+			Adobe_CoolType_Utility /ct_Level2? get
+				{ 1 index }
+				{ 3 index /Private get mark 6 1 roll }
+			ifelse
+			dup /RunInt known
+				{ /RunInt get }
+				{ pop /CCRun }
+			ifelse
+			get exec
+			Adobe_CoolType_Utility /ct_Level2? get not
+				{ cleartomark }
+			if
+			} bind def
+		/ct_BuildCharIncremental
+			{
+				{
+				Adobe_CoolType_Utility /ct_MakeOCF get begin
+				ct_BuildCharSetUp
+				ct_ShowCharString
+				}
+			stopped
+				{ stop }
+			if
+			end
+			end
+			end
+			end
+			} bind def
+		/BaseFontNameStr (BF00) def
+		/ct_Type1FontTemplate 14 dict
+			begin
+			/FontType 1 def
+			/FontMatrix  [0.001 0 0 0.001 0 0] def
+			/FontBBox  [-250 -250 1250 1250] def
+			/Encoding ct_cHexEncoding def
+			/PaintType 0 def
+			currentdict
+			end def
+		/BaseFontTemplate 11 dict
+			begin
+			/FontMatrix  [0.001 0 0 0.001 0 0] def
+			/FontBBox  [-250 -250 1250 1250] def
+			/Encoding ct_cHexEncoding def
+			/BuildChar /ct_BuildCharIncremental load def
+			ct_Clone?
+				{
+				/FontType 3 def
+				/ct_ShowCharString /ct_Type3ShowCharString load def
+				/ct_dfSetCacheProc /ct_CloneSetCacheProc load def
+				/ct_SyntheticBold false def
+				/ct_StrokeWidth 1 def
+				}
+				{
+				/FontType 4 def
+				/Private 1 dict dup /lenIV 4 put def
+				/CharStrings 1 dict dup /.notdef <d841272cf18f54fc13> put def
+				/PaintType 0 def
+				/ct_ShowCharString /ct_Type4ShowCharString load def
+				}
+			ifelse
+			/ct_str1 1 string def
+			currentdict
+			end def
+		/BaseFontDictSize BaseFontTemplate length 5 add def
+		/ct_matrixeq
+			{
+			true 0 1 5
+				{
+				dup 4 index exch get exch 3 index exch get eq and
+				dup not
+					{ exit }
+				if
+				}
+			for
+			exch pop exch pop
+			} bind def
+		/ct_makeocf
+			{
+			15 dict
+				begin
+				exch /WMode exch def
+				exch /FontName exch def
+				/FontType 0 def
+				/FMapType 2 def
+				/FontMatrix matrix def
+				/bfCount 1 index /CIDCount get 256 idiv 1 add
+					dup 256 gt { pop 256} if def
+				/Encoding
+					256 array 0 1 bfCount 1 sub { 2 copy dup put pop } for
+					bfCount 1 255 { 2 copy bfCount put pop } for
+					def
+				/FDepVector bfCount dup 256 lt { 1 add } if array def
+				BaseFontTemplate BaseFontDictSize dict copy
+					begin
+					/CIDFont exch def
+					CIDFont /FontBBox known
+						{ CIDFont /FontBBox get /FontBBox exch def }
+					if
+					CIDFont /CDevProc known
+						{ CIDFont /CDevProc get /CDevProc exch def }
+					if
+					currentdict
+					end
+				BaseFontNameStr 3 (0) putinterval
+				0 1 bfCount dup 256 eq { 1 sub } if
+					{
+					FDepVector exch
+					2 index BaseFontDictSize dict copy
+						begin
+						dup /CIDFirstByte exch 256 mul def
+						FontType 3 eq
+							{ /ct_FDDict 2 dict def }
+						if
+						currentdict
+						end
+					1 index  16
+					BaseFontNameStr  2 2 getinterval cvrs pop
+					BaseFontNameStr exch definefont
+					put
+					}
+				for
+				ct_Clone?
+					{ /Widths 1 index /CIDFont get /GlyphDirectory get length dict def }
+				if
+				FontName
+				currentdict
+				end
+			definefont
+			ct_Clone?
+				{
+				gsave
+				dup 1000 scalefont setfont
+				ct_BuildCharDict
+					begin
+					/usewidths? false def
+					currentfont /Widths get
+						begin
+						exch /CIDFont get /GlyphDirectory get
+							{
+							pop
+							dup charcode exch 1 index 0 2 index 256 idiv put
+							1 index exch 1 exch 256 mod put
+							stringwidth 2 array astore def
+							}
+						forall
+						end
+					/usewidths? true def
+					end
+				grestore
+				}
+				{ exch pop }
+			ifelse
+			} bind def
+		/ct_ComposeFont
+			{
+			ct_UseNativeCapability?
+				{
+				2 index /CMap ct_resourcestatus
+					{ pop pop exch pop }
+					{
+					/CIDInit /ProcSet findresource
+						begin
+						12 dict
+							begin
+							begincmap
+							/CMapName 3 index def
+							/CMapVersion 1.000 def
+							/CMapType 1 def
+							exch /WMode exch def
+							/CIDSystemInfo 3 dict dup
+								begin
+								/Registry (Adobe) def
+								/Ordering
+								CMapName ct_mkocfStr100 cvs
+								(Adobe-) search
+									{
+									pop pop
+									(-) search
+										{
+										dup length string copy
+										exch pop exch pop
+										}
+										{ pop (Identity)}
+									ifelse
+									}
+									{ pop  (Identity)  }
+								ifelse
+								def
+								/Supplement 0 def
+								end def
+							1 begincodespacerange
+							<0000> <FFFF>
+							endcodespacerange
+							1 begincidrange
+							<0000> <FFFF> 0
+							endcidrange
+							endcmap
+							CMapName currentdict /CMap defineresource pop
+							end
+						end
+					}
+				ifelse
+				composefont
+				}
+				{
+				3 2 roll pop
+				0 get /CIDFont findresource
+				ct_makeocf
+				}
+			ifelse
+			} bind def
+		/ct_MakeIdentity
+			{
+			ct_UseNativeCapability?
+				{
+				1 index /CMap ct_resourcestatus
+					{ pop pop }
+					{
+					/CIDInit /ProcSet findresource begin
+					12 dict begin
+					begincmap
+					/CMapName 2 index def
+					/CMapVersion 1.000 def
+					/CMapType 1 def
+					/CIDSystemInfo 3 dict dup
+						begin
+						/Registry (Adobe) def
+						/Ordering
+						CMapName ct_mkocfStr100 cvs
+						(Adobe-) search
+							{
+							pop pop
+							(-) search
+								{ dup length string copy exch pop exch pop }
+								{ pop (Identity) }
+							ifelse
+							}
+							{ pop (Identity) }
+						ifelse
+						def
+						/Supplement 0 def
+						end def
+					1 begincodespacerange
+					<0000> <FFFF>
+					endcodespacerange
+					1 begincidrange
+					<0000> <FFFF> 0
+					endcidrange
+					endcmap
+					CMapName currentdict /CMap defineresource pop
+					end
+					end
+					}
+				ifelse
+				composefont
+				}
+				{
+				exch pop
+				0 get /CIDFont findresource
+				ct_makeocf
+				}
+			ifelse
+			} bind def
+		currentdict readonly pop
+		end
+	end
+%%EndResource
+/currentpacking where{pop currentpacking true setpacking}if
+%%BeginResource: procset pdfvars 6.0 1
+%%Copyright: Copyright 1987-2002 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: definition of dictionary of variables used by PDF & PDFText procsets
+userdict /PDF 162 dict put
+userdict /PDFVars 89 dict dup begin put
+/docSetupDone false def
+/InitAll 0 def
+/TermAll 0 def
+/DocInitAll 0 def
+/DocTermAll 0 def
+/_pdfEncodings 2 array def
+/_pdf_str1 1 string def
+/_pdf_i 0 def
+/_pdf_na 0 def
+/_pdf_showproc 0 def
+/_italMtx [1 0 .212557 1 0 0] def
+/_italMtx_WMode1 [1 -.212557 0 1 0 0] def
+/_italMtxType0 [1 0 .1062785 1 0 0] def
+/_italMtx_WMode1Type0 [1 -.1062785 0 1 0 0] def
+/_basefont 0 def
+/_basefonto 0 def
+/_pdf_oldCIDInit null def
+/_pdf_FontDirectory 30 dict def
+/_categories 10 dict def
+/_sa? true def
+/_ColorSep5044? false def
+/nulldict 0 dict def
+/_processColors 0 def
+/overprintstack null def
+/_defaulttransfer currenttransfer def
+/_defaultflatness currentflat def
+/_defaulthalftone null def
+/_defaultcolortransfer null def
+/_defaultblackgeneration null def
+/_defaultundercolorremoval null def
+/_defaultcolortransfer null def
+PDF begin
+[/c/cs/cm/d/d0/f/h/i/j/J/l/m/M/n/q/Q/re/ri/S/sc/sh/Tf/w/W
+/applyInterpFunc/applystitchFunc/domainClip/encodeInput
+/initgs/int/limit/rangeClip
+/defineRes/undefineRes/findRes/setSA/pl
+/? /! /| /: /+ /GetGlyphDirectory
+/pdf_flushFilters /pdf_readstring /pdf_dictOp /pdf_image /pdf_maskedImage
+/pdf_shfill /pdf_sethalftone
+] {null def} bind forall
+end
+end
+%%EndResource
+PDFVars begin PDF begin
+%%BeginResource: procset pdfutil 6.0 1
+%%Copyright: Copyright 1993-2001 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: Basic utilities used by other PDF procsets
+/bd {bind def} bind def
+/ld {load def} bd
+/bld {
+dup length dict begin
+{ null def } forall
+bind
+end
+def
+} bd
+/dd { PDFVars 3 1 roll put } bd
+/xdd { exch dd } bd
+/Level2?
+systemdict /languagelevel known
+{ systemdict /languagelevel get 2 ge } { false } ifelse
+def
+/Level1? Level2? not def
+/Level3?
+systemdict /languagelevel known
+{systemdict /languagelevel get 3 eq } { false } ifelse
+def
+/getifknown {
+2 copy known { get true } { pop pop false } ifelse
+} bd
+/here {
+currentdict exch getifknown
+} bd
+/isdefined? { where { pop true } { false } ifelse } bd
+%%EndResource
+%%BeginResource: procset l2compat 6.0 1
+%%Copyright: Copyright 1987-2003 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: Level 1 emulation of level 2 operators
+/setcmykcolor isdefined? not
+{
+/setcmykcolor {
+1 sub 4 1 roll
+3 {
+3 index add neg dup 0 lt { pop 0 } if
+3 1 roll
+} repeat
+setrgbcolor
+pop
+} bd
+} if
+/rectclip isdefined? not
+{
+/rectclip { newpath re clip newpath } bd
+} if
+/rectfill isdefined? not
+{
+/rectfill { gsave newpath re fill grestore } bd
+} if
+/sethalftone isdefined? not
+{
+/sethalftone {
+begin
+HalftoneType 1 eq
+{ Frequency Angle /SpotFunction load setscreen }
+if
+end
+} bd
+} if
+Level1?
+{
+/pdf_show {show} bd
+/xshow
+{
+PDFVars /_pdf_showproc /pdf_show load put
+pdf_xshow
+} bd
+/yshow
+{
+PDFVars /_pdf_showproc /pdf_show load put
+pdf_yshow
+} bd
+/xyshow
+{
+PDFVars /_pdf_showproc /pdf_show load put
+pdf_xyshow
+} bd
+} if
+/getrampcolor {
+cvi
+/indx exch def
+[
+0 1 NumComp 1 sub {
+dup
+Samples exch get
+dup type /stringtype eq { indx get } if
+exch
+Scaling exch get aload pop
+3 1 roll
+mul add
+} for
+]
+L1setcolor
+} bd
+/sssetbackground { L1setcolor } bd
+%%EndResource
+%%BeginResource: procset pdf 6.0 1
+%%Copyright: Copyright 1998-2003 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: General operators for PDF, common to all Language Levels.
+/cm { matrix astore concat } bd
+/d /setdash ld
+/f /fill ld
+/h /closepath ld
+/i {dup 0 eq {pop _defaultflatness} if setflat} bd
+/j /setlinejoin ld
+/J /setlinecap ld
+/M /setmiterlimit ld
+/n /newpath ld
+/S /stroke ld
+/w /setlinewidth ld
+/W /clip ld
+/sg /setgray ld
+/initgs {
+0 setgray
+[] 0 d
+0 j
+0 J
+10 M
+1 w
+false setSA
+/_defaulttransfer load settransfer
+0 i
+/RelativeColorimetric ri
+newpath
+} bd
+/int {
+dup 2 index sub 3 index 5 index sub div 6 -2 roll sub mul
+exch pop add exch pop
+} bd
+/limit {
+dup 2 index le { exch } if pop
+dup 2 index ge { exch } if pop
+} bd
+/domainClip {
+Domain aload pop 3 2 roll
+limit
+} [/Domain] bld
+/applyInterpFunc {
+0 1 DimOut 1 sub
+{
+dup C0 exch get exch
+dup C1 exch get exch
+3 1 roll
+1 index sub
+3 index
+N exp mul add
+exch
+currentdict /Range_lo known
+{
+dup Range_lo exch get exch
+Range_hi exch get
+3 2 roll limit
+}
+{
+pop
+}
+ifelse
+exch
+} for
+pop
+} [/DimOut /C0 /C1 /N /Range_lo /Range_hi] bld
+/encodeInput {
+NumParts 1 sub
+0 1 2 index
+{
+dup Bounds exch get
+2 index gt
+{ exit }
+{ dup
+3 index eq
+{ exit }
+{ pop } ifelse
+} ifelse
+} for
+3 2 roll pop
+dup Bounds exch get exch
+dup 1 add Bounds exch get exch
+2 mul
+dup Encode exch get exch
+1 add Encode exch get
+int
+} [/NumParts /Bounds /Encode] bld
+/rangeClip {
+exch dup Range_lo exch get
+exch Range_hi exch get
+3 2 roll
+limit
+} [/Range_lo /Range_hi] bld
+/applyStitchFunc {
+Functions exch get exec
+currentdict /Range_lo known {
+0 1 DimOut 1 sub {
+DimOut 1 add -1 roll
+rangeClip
+} for
+} if
+} [/Functions /Range_lo /DimOut] bld
+/pdf_flushfilters
+{
+aload length
+{ dup status
+1 index currentfile ne and
+{ dup flushfile closefile }
+{ pop }
+ifelse
+} repeat
+} bd
+/pdf_readstring
+{
+1 index dup length 1 sub get
+exch readstring pop
+exch pdf_flushfilters
+} bind def
+/pdf_dictOp
+{
+3 2 roll
+10 dict copy
+begin
+_Filters dup length 1 sub get def
+currentdict exch exec
+_Filters pdf_flushfilters
+end
+} [/_Filters] bld
+/pdf_image {{image} /DataSource pdf_dictOp} bd
+/pdf_imagemask {{imagemask} /DataSource pdf_dictOp} bd
+/pdf_shfill {{sh} /DataSource pdf_dictOp} bd
+/pdf_sethalftone {{sethalftone} /Thresholds pdf_dictOp} bd
+/pdf_maskedImage
+{
+10 dict copy begin
+/miDict currentdict def
+/DataDict DataDict 10 dict copy def
+DataDict begin
+/DataSource
+_Filters dup length 1 sub get
+def
+miDict image
+_Filters pdf_flushfilters
+end
+end
+} [/miDict /DataDict /_Filters] bld
+/RadialShade {
+40 dict begin
+/background exch def
+/ext1 exch def
+/ext0 exch def
+/BBox exch def
+/r2 exch def
+/c2y exch def
+/c2x exch def
+/r1 exch def
+/c1y exch def
+/c1x exch def
+/rampdict exch def
+gsave
+BBox length 0 gt {
+newpath
+BBox 0 get BBox 1 get moveto
+BBox 2 get BBox 0 get sub 0 rlineto
+0 BBox 3 get BBox 1 get sub rlineto
+BBox 2 get BBox 0 get sub neg 0 rlineto
+closepath
+clip
+newpath
+} if
+c1x c2x eq
+{
+c1y c2y lt {/theta 90 def}{/theta 270 def} ifelse
+}
+{
+/slope c2y c1y sub c2x c1x sub div def
+/theta slope 1 atan def
+c2x c1x lt c2y c1y ge and { /theta theta 180 sub def} if
+c2x c1x lt c2y c1y lt and { /theta theta 180 add def} if
+}
+ifelse
+gsave
+clippath
+c1x c1y translate
+theta rotate
+-90 rotate
+{ pathbbox } stopped
+{ 0 0 0 0 } if
+/yMax exch def
+/xMax exch def
+/yMin exch def
+/xMin exch def
+grestore
+xMax xMin eq yMax yMin eq or
+{
+grestore
+end
+}
+{
+/max { 2 copy gt { pop } {exch pop} ifelse } bind def
+/min { 2 copy lt { pop } {exch pop} ifelse } bind def
+rampdict begin
+40 dict begin
+background length 0 gt { background sssetbackground gsave clippath fill grestore } if
+gsave
+c1x c1y translate
+theta rotate
+-90 rotate
+/c2y c1x c2x sub dup mul c1y c2y sub dup mul add sqrt def
+/c1y 0 def
+/c1x 0 def
+/c2x 0 def
+ext0 {
+0 getrampcolor
+c2y r2 add r1 sub 0.0001 lt
+{
+c1x c1y r1 360 0 arcn
+pathbbox
+/aymax exch def
+/axmax exch def
+/aymin exch def
+/axmin exch def
+/bxMin xMin axmin min def
+/byMin yMin aymin min def
+/bxMax xMax axmax max def
+/byMax yMax aymax max def
+bxMin byMin moveto
+bxMax byMin lineto
+bxMax byMax lineto
+bxMin byMax lineto
+bxMin byMin lineto
+eofill
+}
+{
+c2y r1 add r2 le
+{
+c1x c1y r1 0 360 arc
+fill
+}
+{
+c2x c2y r2 0 360 arc fill
+r1 r2 eq
+{
+/p1x r1 neg def
+/p1y c1y def
+/p2x r1 def
+/p2y c1y def
+p1x p1y moveto p2x p2y lineto p2x yMin lineto p1x yMin lineto
+fill
+}
+{
+/AA r2 r1 sub c2y div def
+AA -1 eq
+{ /theta 89.99 def}
+{ /theta AA 1 AA dup mul sub sqrt div 1 atan def}
+ifelse
+/SS1 90 theta add dup sin exch cos div def
+/p1x r1 SS1 SS1 mul SS1 SS1 mul 1 add div sqrt mul neg def
+/p1y p1x SS1 div neg def
+/SS2 90 theta sub dup sin exch cos div def
+/p2x r1 SS2 SS2 mul SS2 SS2 mul 1 add div sqrt mul def
+/p2y p2x SS2 div neg def
+r1 r2 gt
+{
+/L1maxX p1x yMin p1y sub SS1 div add def
+/L2maxX p2x yMin p2y sub SS2 div add def
+}
+{
+/L1maxX 0 def
+/L2maxX 0 def
+}ifelse
+p1x p1y moveto p2x p2y lineto L2maxX L2maxX p2x sub SS2 mul p2y add lineto
+L1maxX L1maxX p1x sub SS1 mul p1y add lineto
+fill
+}
+ifelse
+}
+ifelse
+} ifelse
+} if
+c1x c2x sub dup mul
+c1y c2y sub dup mul
+add 0.5 exp
+0 dtransform
+dup mul exch dup mul add 0.5 exp 72 div
+0 72 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+1 index 1 index lt { exch } if pop
+/hires exch def
+hires mul
+/numpix exch def
+/numsteps NumSamples def
+/rampIndxInc 1 def
+/subsampling false def
+numpix 0 ne
+{
+NumSamples numpix div 0.5 gt
+{
+/numsteps numpix 2 div round cvi dup 1 le { pop 2 } if def
+/rampIndxInc NumSamples 1 sub numsteps div def
+/subsampling true def
+} if
+} if
+/xInc c2x c1x sub numsteps div def
+/yInc c2y c1y sub numsteps div def
+/rInc r2 r1 sub numsteps div def
+/cx c1x def
+/cy c1y def
+/radius r1 def
+newpath
+xInc 0 eq yInc 0 eq rInc 0 eq and and
+{
+0 getrampcolor
+cx cy radius 0 360 arc
+stroke
+NumSamples 1 sub getrampcolor
+cx cy radius 72 hires div add 0 360 arc
+0 setlinewidth
+stroke
+}
+{
+0
+numsteps
+{
+dup
+subsampling { round } if
+getrampcolor
+cx cy radius 0 360 arc
+/cx cx xInc add def
+/cy cy yInc add def
+/radius radius rInc add def
+cx cy radius 360 0 arcn
+eofill
+rampIndxInc add
+}
+repeat
+pop
+} ifelse
+ext1 {
+c2y r2 add r1 lt
+{
+c2x c2y r2 0 360 arc
+fill
+}
+{
+c2y r1 add r2 sub 0.0001 le
+{
+c2x c2y r2 360 0 arcn
+pathbbox
+/aymax exch def
+/axmax exch def
+/aymin exch def
+/axmin exch def
+/bxMin xMin axmin min def
+/byMin yMin aymin min def
+/bxMax xMax axmax max def
+/byMax yMax aymax max def
+bxMin byMin moveto
+bxMax byMin lineto
+bxMax byMax lineto
+bxMin byMax lineto
+bxMin byMin lineto
+eofill
+}
+{
+c2x c2y r2 0 360 arc fill
+r1 r2 eq
+{
+/p1x r2 neg def
+/p1y c2y def
+/p2x r2 def
+/p2y c2y def
+p1x p1y moveto p2x p2y lineto p2x yMax lineto p1x yMax lineto
+fill
+}
+{
+/AA r2 r1 sub c2y div def
+AA -1 eq
+{ /theta 89.99 def}
+{ /theta AA 1 AA dup mul sub sqrt div 1 atan def}
+ifelse
+/SS1 90 theta add dup sin exch cos div def
+/p1x r2 SS1 SS1 mul SS1 SS1 mul 1 add div sqrt mul neg def
+/p1y c2y p1x SS1 div sub def
+/SS2 90 theta sub dup sin exch cos div def
+/p2x r2 SS2 SS2 mul SS2 SS2 mul 1 add div sqrt mul def
+/p2y c2y p2x SS2 div sub def
+r1 r2 lt
+{
+/L1maxX p1x yMax p1y sub SS1 div add def
+/L2maxX p2x yMax p2y sub SS2 div add def
+}
+{
+/L1maxX 0 def
+/L2maxX 0 def
+}ifelse
+p1x p1y moveto p2x p2y lineto L2maxX L2maxX p2x sub SS2 mul p2y add lineto
+L1maxX L1maxX p1x sub SS1 mul p1y add lineto
+fill
+}
+ifelse
+}
+ifelse
+} ifelse
+} if
+grestore
+grestore
+end
+end
+end
+} ifelse
+} bd
+/GenStrips {
+40 dict begin
+/background exch def
+/ext1 exch def
+/ext0 exch def
+/BBox exch def
+/y2 exch def
+/x2 exch def
+/y1 exch def
+/x1 exch def
+/rampdict exch def
+gsave
+BBox length 0 gt {
+newpath
+BBox 0 get BBox 1 get moveto
+BBox 2 get BBox 0 get sub 0 rlineto
+0 BBox 3 get BBox 1 get sub rlineto
+BBox 2 get BBox 0 get sub neg 0 rlineto
+closepath
+clip
+newpath
+} if
+x1 x2 eq
+{
+y1 y2 lt {/theta 90 def}{/theta 270 def} ifelse
+}
+{
+/slope y2 y1 sub x2 x1 sub div def
+/theta slope 1 atan def
+x2 x1 lt y2 y1 ge and { /theta theta 180 sub def} if
+x2 x1 lt y2 y1 lt and { /theta theta 180 add def} if
+}
+ifelse
+gsave
+clippath
+x1 y1 translate
+theta rotate
+{ pathbbox } stopped
+{ 0 0 0 0 } if
+/yMax exch def
+/xMax exch def
+/yMin exch def
+/xMin exch def
+grestore
+xMax xMin eq yMax yMin eq or
+{
+grestore
+end
+}
+{
+rampdict begin
+20 dict begin
+background length 0 gt { background sssetbackground gsave clippath fill grestore } if
+gsave
+x1 y1 translate
+theta rotate
+/xStart 0 def
+/xEnd x2 x1 sub dup mul y2 y1 sub dup mul add 0.5 exp def
+/ySpan yMax yMin sub def
+/numsteps NumSamples def
+/rampIndxInc 1 def
+/subsampling false def
+xStart 0 transform
+xEnd 0 transform
+3 -1 roll
+sub dup mul
+3 1 roll
+sub dup mul
+add 0.5 exp 72 div
+0 72 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt
+1 index 1 index lt { exch } if pop
+mul
+/numpix exch def
+numpix 0 ne
+{
+NumSamples numpix div 0.5 gt
+{
+/numsteps numpix 2 div round cvi dup 1 le { pop 2 } if def
+/rampIndxInc NumSamples 1 sub numsteps div def
+/subsampling true def
+} if
+} if
+ext0 {
+0 getrampcolor
+xMin xStart lt
+{ xMin yMin xMin neg ySpan rectfill } if
+} if
+/xInc xEnd xStart sub numsteps div def
+/x xStart def
+0
+numsteps
+{
+dup
+subsampling { round } if
+getrampcolor
+x yMin xInc ySpan rectfill
+/x x xInc add def
+rampIndxInc add
+}
+repeat
+pop
+ext1 {
+xMax xEnd gt
+{ xEnd yMin xMax xEnd sub ySpan rectfill } if
+} if
+grestore
+grestore
+end
+end
+end
+} ifelse
+} bd
+/currentdistillerparams where { pop currentdistillerparams /CoreDistVersion get 5000 lt}{true}ifelse
+{
+/PDFMark5 {cleartomark} bd
+}
+{
+/PDFMark5 {pdfmark} bd
+}ifelse
+/ReadByPDFMark5
+{
+2 dict begin
+/makerString exch def string /tmpString exch def
+{
+currentfile tmpString readline pop
+makerString anchorsearch
+{
+pop pop cleartomark exit
+}
+{
+3 copy /PUT PDFMark5 pop 2 copy (\n) /PUT PDFMark5
+} ifelse
+}loop
+end
+}bd
+%%EndResource
+%%BeginResource: procset sep_ops 6.0 1
+%%Copyright: Copyright 1987-2001 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: Support for Separations in Level 1, following the conventions of Tech Note 5044
+userdict /sep_ops 50 dict dup begin put
+/bdef {bind def} bind def
+/xdef {exch def} bdef
+/colorimagebuffer {
+0 1 2 index length 1 sub {
+dup 2 index exch get 255 exch sub 2 index 3 1 roll put
+} for
+} bdef
+/addprocs {
+[ 3 1 roll
+/exec load
+dup 3 1 roll
+] cvx
+} bdef
+/L1? {
+systemdict /languagelevel known {
+systemdict /languagelevel get 2 lt
+}{
+true
+} ifelse
+} bdef
+/colorexists {
+statusdict /processcolors known {
+statusdict /processcolors get exec
+}{
+/deviceinfo where {
+pop deviceinfo /Colors known {
+deviceinfo /Colors get
+statusdict /processcolors {
+deviceinfo /Colors known {
+deviceinfo /Colors get
+}{
+1
+} ifelse
+} put
+}{
+1
+} ifelse
+}{
+1
+} ifelse
+} ifelse
+1 gt
+} bdef
+/colorplate colorexists { 0 } { 5 } ifelse def
+/negativecolorplate false def
+/setcmykcolor where {
+pop
+gsave
+1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
+0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
+0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub
+0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub
+4 {4 copy} repeat
+grestore
+1 dict begin
+/foureq {
+4 index eq 8 1 roll
+4 index eq 8 1 roll
+4 index eq 8 1 roll
+4 index eq 8 1 roll
+pop pop pop pop and and and
+} def
+1 0 0 0 foureq {/colorplate 1 store} if
+0 1 0 0 foureq {/colorplate 2 store} if
+0 0 1 0 foureq {/colorplate 3 store} if
+0 0 0 1 foureq {/colorplate 4 store} if
+0 0 0 0 foureq {/colorplate 6 store} if
+end
+} if
+0 systemdict /currenttransfer get exec exec
+1 systemdict /currenttransfer get exec exec
+2 copy
+eq
+{
+/colorplate 6 store
+pop
+/negativecolorplate exch 0.5 lt store
+}
+{
+gt /negativecolorplate exch store
+}
+ifelse
+/findcmykcustomcolor where { pop }
+{
+/findcmykcustomcolor {
+[ 6 1 roll ] readonly
+} bdef
+} ifelse
+/setoverprint where {
+pop
+}{
+/setoverprint {
+pop
+} bdef
+} ifelse
+/currentoverprint where {
+pop
+}{
+/currentoverprint {
+false
+} bdef
+} ifelse
+/setcustomcolor where {
+pop
+}{
+L1? {
+/setcustomcolor {
+exch
+aload pop pop
+4 { 4 index mul 4 1 roll } repeat
+5 -1 roll pop
+setcmykcolor
+} bdef
+}{
+/setcustomcolor {
+exch
+[ exch /Separation exch dup 4 get exch /DeviceCMYK exch
+0 4 getinterval
+[ exch /dup load exch cvx {mul exch dup}
+/forall load /pop load dup] cvx
+] setcolorspace setcolor
+} bdef
+} ifelse
+} ifelse
+/ik 0 def
+/iy 0 def
+/im 0 def
+/ic 0 def
+/imagetint {
+ic .3 mul
+im .59 mul
+iy .11 mul
+ik add add add dup
+1 gt {pop 1} if
+} bdef
+/setcmykcolor where {
+pop
+}{
+/setcmykcolor {
+/ik xdef /iy xdef /im xdef /ic xdef
+imagetint
+1 exch sub setgray
+} bdef
+} ifelse
+/customcolorimage where {
+pop
+}{
+L1? {
+/customcolorimage{
+gsave
+colorexists {
+aload pop pop
+/ik xdef /iy xdef /im xdef /ic xdef
+currentcolortransfer
+{ik mul ik sub 1 add} addprocs
+4 1 roll {iy mul iy sub 1 add} addprocs
+4 1 roll {im mul im sub 1 add} addprocs
+4 1 roll {ic mul ic sub 1 add} addprocs
+4 1 roll setcolortransfer
+/magentabuf 0 string def
+/yellowbuf 0 string def
+/blackbuf 0 string def
+{
+colorimagebuffer dup length magentabuf length ne
+{
+dup length dup dup
+/magentabuf exch string def
+/yellowbuf exch string def
+/blackbuf exch string def
+} if
+dup magentabuf copy yellowbuf copy
+blackbuf copy pop
+} addprocs
+{magentabuf}{yellowbuf}{blackbuf} true 4 colorimage
+}{
+aload pop pop /ik xdef /iy xdef /im xdef /ic xdef /tint
+imagetint def
+currenttransfer
+{tint mul 1 tint sub add} addprocs settransfer image
+} ifelse
+grestore
+} bdef
+}{
+/customcolorimage {
+gsave
+[ exch /Separation exch dup 4 get exch /DeviceCMYK exch
+0 4 getinterval
+[ exch /dup load exch cvx {mul exch dup}
+/forall load /pop load dup] cvx
+] setcolorspace
+10 dict begin
+/ImageType 1 def
+/DataSource exch def
+/ImageMatrix exch def
+/BitsPerComponent exch def
+/Height exch def
+/Width exch def
+/Decode [1 0] def
+currentdict end
+image
+grestore
+} bdef
+} ifelse
+} ifelse
+/setseparationgray where {
+pop
+}{
+L1? {
+/setseparationgray { 1 exch sub dup dup dup setcmykcolor } bdef
+}{
+/setseparationgray {
+[/Separation /All /DeviceCMYK {dup dup dup}] setcolorspace
+1 exch sub setcolor
+} bdef
+} ifelse
+} ifelse
+/separationimage where { pop }
+{
+/separationimage {
+gsave
+1 1 1 1 (All)
+findcmykcustomcolor customcolorimage
+grestore
+} bdef
+} ifelse
+currentdict readonly pop end
+%%EndResource
+%%BeginResource: procset pdflev15044 6.0 1
+%%Copyright: Copyright 1987-2002 Adobe Systems Incorporated. All Rights Reserved.
+%%Title: PDF operators, Level 1, with emulated separations (TN 5044)
+/_ColorSep5044? true dd
+/docinitialize {
+PDF begin
+/_defaulthalftone
+/currenthalftone where
+{ pop currenthalftone }
+{ 4 dict dup begin
+currentscreen
+/SpotFunction exch def
+/Angle exch def
+/Frequency exch def
+/HalftoneType 1 def
+end }
+ifelse
+dd
+/currentcolortransfer where
+{ pop /_defaultcolortransfer [ currentcolortransfer ] dd }
+{ /_defaultcolortransfer [currenttransfer dup dup dup] dd }
+ifelse
+end
+} bd
+/initialize {
+/overprintstack null dd
+sep_ops begin
+50 dict begin
+_defaulthalftone sethalftone
+} bd
+/terminate {
+end end
+} bd
+/currentcolortransfer where
+{ pop }
+{
+/setcolortransfer
+{
+settransfer pop pop pop
+} bd
+} ifelse
+/pl {
+transform
+0.25 sub round 0.25 add exch
+0.25 sub round 0.25 add exch
+itransform
+} bd
+/m { _sa? { pl } if moveto } bd
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+} ifelse
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+} bind def
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+} bind def
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+} bind def
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+}
+{
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+}
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+}
+}
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+}
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+}
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+}
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+} bd
+}
+{
+/_pdfFontStatusString 50 string def
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+{
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+}
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+} bd
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+}
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+} bd
+}
+if
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+cvn
+} bind def
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+/_pdfRegOrderingArray [(Adobe-Japan1) (Adobe-CNS1) (Adobe-Korea1) (Adobe-GB1)] def
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+}
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+} bind def
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+}
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+}
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+} bd
+}
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+}
+ifelse
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+}
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+}
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+}
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+{
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+}stopped
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+}
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+}
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+}
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+}
+{
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+}
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+} if
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+}
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+} bd
+}
+{
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+} def
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+PDFText begin
+[39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis
+/Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute
+/egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde
+/oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex
+/udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls
+/registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash
+/.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef
+/.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash
+/questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef
+/guillemotleft/guillemotright/ellipsis/space/Agrave/Atilde/Otilde/OE/oe
+/endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide
+/.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright
+/fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand
+/Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex
+/Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex
+/Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla
+/hungarumlaut/ogonek/caron
+0 TE
+[1/dotlessi/caron 39/quotesingle 96/grave 
+127/bullet/Euro/bullet/quotesinglbase/florin/quotedblbase/ellipsis
+/dagger/daggerdbl/circumflex/perthousand/Scaron/guilsinglleft/OE
+/bullet/Zcaron/bullet/bullet/quoteleft/quoteright/quotedblleft
+/quotedblright/bullet/endash/emdash/tilde/trademark/scaron
+/guilsinglright/oe/bullet/zcaron/Ydieresis/space/exclamdown/cent/sterling
+/currency/yen/brokenbar/section/dieresis/copyright/ordfeminine
+/guillemotleft/logicalnot/hyphen/registered/macron/degree/plusminus
+/twosuperior/threesuperior/acute/mu/paragraph/periodcentered/cedilla
+/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters
+/questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla
+/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis
+/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply/Oslash
+/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls/agrave
+/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute
+/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde
+/ograve/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute
+/ucircumflex/udieresis/yacute/thorn/ydieresis
+1 TE
+end
+%%BeginResource: procset pdfasc.prc 6.0 1
+%%Copyright: Copyright 1992-2003 Adobe Systems Incorporated. All Rights Reserved.
+/ASR {
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+Dury 0 eq Durx 0 eq and Dlly 0 eq Dllx 0 eq and and
+{ shrink 0 gt { GClipBBox } { GPageBBox } ifelse }
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+angle 90 eq angle 270 eq or
+{
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+} if
+autorotate? PHt PW ne and DHt DW ne and
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+}
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+} if
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+%%PageTrailer
+%%Trailer
+%%DocumentNeededResources:
+%%+ font Arial-BoldMT
+%%EOF

Examples/MeshGenToolbox/mgt/Cell.h

+/*****************************************************************************/
+/*  PMGT                                                                     */
+/*                                                                           */
+/*  January, 2007                                                            */
+/*                                                                           */
+/*  Copyright 2007                                                           */
+/*  Wen Yu                                                                   */
+/*  Department of Computing and Software                                     */
+/*  McMaster University                                                      */
+/*  yuw4@mcmaster.ca                                                         */
+/*                                                                           */
+/*  This program may be freely redistributed under the condition that the    */
+/*    copyright notices are not removed. You may distribute modified versions*/
+/*    of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS  */
+/*    MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL     */
+/*    AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT  */
+/*    CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS.                */
+/*                                                                           */
+/*  This is the serial version of PMGT                                       */
+/*  This file includes classes for the cell module.			     */
+/*****************************************************************************/
+
+#ifndef CELL
+#define CELL
+#endif
+
+#ifndef VERTEX
+#include "Vertex.h"
+#endif
+
+#ifndef EDGE
+#include "Edge.h"
+#endif
+
+namespace PMGT{
+//namespace Cell{
+
+//using namespace Vertex;
+//using namespace Edge;
+
+class Cell{
+public:
+	Cell(int cId=-1, int LhId = -1, MFLAG mFlag=UNCHANGE, bool del=false){
+		cId_ = cId;
+		LhId_ = LhId;
+		mFlag_ = mFlag;
+		del_ = del;
+	}
+
+	inline int cId(){ return cId_;}
+	inline MFLAG mFlag(){ return mFlag_; }
+	inline int LhId(){ return LhId_; }
+	inline bool del(){ return del_; }
+
+
+	inline void setCId(int cId){ cId_ = cId; }
+	inline void setMflag(MFLAG flag){ mFlag_ = flag; }
+	inline void setLhId(int LhId){ LhId_ = LhId; }
+	inline void setDel(bool del) { del_ = del; }
+
+	inline void ccopy(Cell& other){
+		cId_ = other.cId_;
+		LhId_ = other.LhId_;
+		mFlag_ = other.mFlag_;
+		del_ = other.del_;
+	}
+
+private:
+	int		cId_;	//id
+	int		LhId_;	//longest halfedge
+	MFLAG	mFlag_; //flag for modification
+	bool 	del_;	// a flag to indicate if the cell
+					// is marked to be deleted
+}; // end of class Cell
+
+//} // end of namespace Cell
+
+
+} // end of namespace PMGT

Examples/MeshGenToolbox/mgt/Coarsening.cpp

+/*****************************************************************************/
+/*  PMGT                                                                     */
+/*                                                                           */
+/*  January, 2007                                                            */
+/*                                                                           */
+/*  Copyright 2007                                                           */
+/*  Wen Yu                                                                   */
+/*  Department of Computing and Software                                     */
+/*  McMaster University                                                      */
+/*  yuw4@mcmaster.ca                                                         */
+/*                                                                           */
+/*  This program may be freely redistributed under the condition that the    */
+/*    copyright notices are not removed. You may distribute modified versions*/
+/*    of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS  */
+/*    MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL     */
+/*    AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT  */
+/*    CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS.                */
+/*                                                                           */
+/*  This is the serial version of PMGT                                       */
+/*****************************************************************************/
+#ifndef COARSENING
+#include "Coarsening.h"
+#endif
+
+namespace PMGT{
+
+	// coarsen the mesh by edge collapse
+	void Coarsening::edge_collapse(Mesh& mesh){
+		int i, j, k,
+			cId, // temp cell Id
+			eId, // temp edge Id
+			hId, // temp halfedge Id
+			vId, // temp vertex Id
+
+		 	acId, // adjacent cell
+		 	first_acId, // the first adjacent cell
+		 	rvId, // the vertex to be remained
+		 	dvId;  // the vertex to be deleted
+
+		 vector<int>	del_cells,	// cells to be deleted
+		 				mutual_halfedges, // halfedges of the mutual edge
+		 				del_halfedges, // halfedges to be deleted/merged
+		 			 	remain_halfedges, // halfedges to be remained
+		 			 	update_halfedges, 	// halfedges whose target vertices
+		 			 						// to be updated
+						update_cells; // cells whose edge length are changed
+
+		//int nc = mesh.n_of_c();
+		for (i=0; i<mesh.n_of_c(); i++){
+			bool legal = false;
+			if (mesh.cell(i).mFlag() == COARSEN && !mesh.cell(i).del()){
+				// check if it is legal to delete i
+
+				// check if one of the adjacent cells is to be coarsened
+				acId = mesh.cc_first(i);
+
+				for (int k=0; k<3; k++){ // try 3 neighbour of cell i
+//cout << "coarsen i=" << i << " acId=" << acId << " k=" << k <<endl;
+					if (acId == -1){
+						break;
+					}
+					// check if the i-th neighbour is to be coarsened
+					if (mesh.cell(acId).mFlag() != COARSEN){
+						acId = mesh.cc_next(i, acId);
+						continue;
+					}
+					/* check if the mutual edge is legal to be removed */
+
+					// find one halfedge of the mutual edge
+					hId = mesh.ch_first(i);
+
+					if (mesh.IcId_from_hId(mesh.ops_hId(hId)) != acId){
+						if (mesh.IcId_from_hId(mesh.ops_hId
+						(mesh.prev_hId(hId))) == acId){
+							hId = mesh.prev_hId(hId);
+						}
+						else if(mesh.IcId_from_hId(mesh.ops_hId
+						(mesh.next_hId(hId))) == acId){
+							hId = mesh.next_hId(hId);
+						}
+						else assert(false);
+					}
+
+					// find the vertex to be delated
+					int illegal_vId = -1; // the vertex that can not be deleted
+					bool both_illegal = false; // both vertices are illegal to be deleted
+					dvId = mesh.tvId_from_hId(hId);
+					rvId = mesh.svId_from_hId(hId);
+					int old_hId = hId;
+					for (int x=0; x<2; x++){ // try two vertices of hId
+						if (mesh.is_b_v(dvId)){
+							if (illegal_vId == -1){
+								illegal_vId = dvId;
+								dvId = mesh.svId_from_hId(old_hId);
+								rvId = mesh.tvId_from_hId(old_hId);
+								hId = mesh.ops_hId(old_hId);
+								continue;
+							}
+							else{
+								both_illegal = true;
+								break;
+							}
+						}
+//cout << "mutual_h:"<<hId<<",";
+						// reserve the entities that will be modified
+						del_cells.clear();
+						mutual_halfedges.clear();
+						del_halfedges.clear();
+						remain_halfedges.clear();
+						update_halfedges.clear();
+						update_cells.clear();
+						// set variables
+						mutual_halfedges.push_back(hId);
+						del_halfedges.push_back(mesh.next_hId(hId));
+						remain_halfedges.push_back(mesh.prev_hId(hId));
+						del_cells.push_back(mesh.IcId_from_hId(hId));
+
+						hId = mesh.ops_hId(hId);
+//cout << hId<< endl;
+
+						mutual_halfedges.push_back(hId);
+						del_halfedges.push_back(mesh.prev_hId(hId));
+						remain_halfedges.push_back(mesh.next_hId(hId));
+						del_cells.push_back(mesh.IcId_from_hId(hId));
+//cout << "del_cells "<< del_cells[0]<<","<<del_cells[1]<<endl;
+						assert(del_cells[0]==i || del_cells[0]==acId);
+						assert(del_cells[1]==i || del_cells[1]==acId);
+						assert(del_cells[0] != del_cells[1]);
+						assert(!mesh.is_b_v(dvId));
+
+						// find the halfedges whose target vertices are dvId
+						hId = mesh.vih_first(dvId);
+						do{
+							if (hId == -1){
+								break;
+							}
+							if (hId != mutual_halfedges[0] && hId != del_halfedges[1]
+									&& hId != mesh.ops_hId(del_halfedges[0])){
+								update_halfedges.push_back(hId);
+								//update_cells.push_back(mesh.IcId_from_hId(hId));
+							}
+//cout << "dvId="<<dvId<<"vih id=" << hId << endl;
+
+							hId = mesh.vih_next(dvId, hId);
+						}while (hId != mesh.vih_first(dvId));
+
+						// find the cells whose whose one vertices are dvId
+						cId = mesh.vc_first(dvId);
+						do{
+							if (cId ==-1){
+								break;
+							}
+							if (cId != del_cells[0] && cId != del_cells[1]){
+								update_cells.push_back(cId);
+								//update_cells.push_back(mesh.IcId_from_hId(hId));
+							}
+
+							cId = mesh.vc_next(dvId, cId);
+
+						}while (cId != mesh.vc_first(dvId));
+
+
+						for (j=0; j<update_cells.size(); j++){
+							vector<int> vIds;
+							vIds.clear();
+							vIds = mesh.vIds_from_cId(update_cells[j]);
+							for (int jj=0; jj<3; jj++){
+								if (vIds[jj] == dvId){
+									vIds[jj] = rvId;
+								}
+							}
+							if ( !mesh.is_ccw(vIds)){
+								break;// for j
+							}
+
+						}
+						if (j==update_cells.size()){
+							break; // from (for x)
+						}
+						// dvId is not legal to be deleted
+						if (illegal_vId == -1){
+							illegal_vId = dvId;
+							dvId = mesh.svId_from_hId(old_hId);
+							rvId = mesh.tvId_from_hId(old_hId);
+							hId = mesh.ops_hId(old_hId);
+							continue;
+						}
+						else {// both vertices are illegal to be deleted
+							both_illegal = true;
+							break; // from (for x)
+						}
+					}// for x
+
+					if (both_illegal){
+						acId = mesh.cc_next(i, acId);
+						if (acId == -1){
+							break;
+						}
+						continue;
+					}
+					else {
+						legal = true;
+						break; // from (for k) since dvId is found
+					}
+				} // for k
+				if (legal){
+//cout << "legal coarsen i=" << i << " acId=" << acId << endl;
+
+					// reset Mflag
+					for (j=0; j<update_halfedges.size(); j++){
+						mesh.halfedge(update_halfedges[j]).setTvId(rvId);
+
+					}
+					for (j=0; j<2; j++){
+						hId = mesh.ops_hId(del_halfedges[j]);
+						cId = mesh.IcId_from_hId(hId);
+						if (mesh.LhId_from_cId(cId) == hId){
+							int LhId = remain_halfedges[j];
+							int next_hId = mesh.next_hId(hId);
+							int prev_hId = mesh.prev_hId(hId);
+							if (mesh.dist(LhId) < mesh.dist(next_hId)){
+								LhId = next_hId;
+							}
+							if (mesh.dist(LhId) < mesh.dist(prev_hId)){
+								LhId = prev_hId;
+							}
+							mesh.cell(cId).setLhId(LhId);
+//cout << " ____cId = " << cId << " LhId = " << LhId <<endl;
+						}
+					}
+
+
+					for (j=0; j<update_cells.size(); j++){
+						mesh.cell(update_cells[j]).setMflag(UNCHANGE);
+					}
+//cout << "---update_cells:";
+//for (int ii=0; ii<update_cells.size(); ii++){
+//	cout <<" " << update_cells[ii]<< "("<<mesh.cell(update_cells[ii]).mFlag()<<")";
+//}
+
+//cout << endl;
+
+					// reset OhId
+					int svId = mesh.svId_from_hId(del_halfedges[1]);
+					if (mesh.OhId_from_vId(svId) == del_halfedges[1]){
+						//mesh.vertex(svId).setOhId(mesh.ccw_hId(del_halfedges[1]));
+						//assert(mesh.svId_from_hId(del_halfedges[1])==
+						//mesh.svId_from_hId(remain_halfedges[1]));
+						mesh.vertex(svId).setOhId(mesh.ops_hId(remain_halfedges[1]));
+//cout << "reset OhId of vertex " << svId << " to " << mesh.vertex(svId).OhId()  << endl;
+					}
+
+					svId = mesh.svId_from_hId(mesh.ops_hId(del_halfedges[0]));
+					if (mesh.OhId_from_vId(svId) == mesh.ops_hId(del_halfedges[0])){
+						//mesh.vertex(svId).setOhId(mesh.ccw_hId(mutual_halfedges[0]));
+						//assert(mesh.svId_from_hId(del_halfedges[0])==
+						//mesh.svId_from_hId(remain_halfedges[0]));
+						mesh.vertex(svId).setOhId(remain_halfedges[0]);
+
+//cout << "reset OhId of vertex " << svId << " to " << mesh.vertex(svId).OhId() <<  endl;
+					}
+					svId = mesh.svId_from_hId(mutual_halfedges[0]);
+					if (mesh.OhId_from_vId(svId) == mutual_halfedges[0]){
+						//mesh.vertex(svId).setOhId(mesh.ccw_hId(mutual_halfedges[0]));
+						//assert(mesh.svId_from_hId(mutual_halfedges[0])==
+						//mesh.svId_from_hId(remain_halfedges[0]));
+
+						mesh.vertex(svId).setOhId(remain_halfedges[1]);
+//cout << "reset OhId of vertex " << svId << " to " << mesh.vertex(svId).OhId()  << endl;
+					}
+
+
+					for (j=0; j<2; j++){
+						hId = mesh.ops_hId(del_halfedges[j]);
+
+
+						int phId = mesh.prev_hId(hId);
+						int nhId = mesh.next_hId(hId);
+						cId = mesh.IcId_from_hId(hId);
+
+						mesh.halfedge(remain_halfedges[j]).setIcId(cId);
+						mesh.halfedge(remain_halfedges[j]).setPhId(phId);
+						mesh.halfedge(remain_halfedges[j]).setNhId(nhId);
+						mesh.halfedge(nhId).setPhId(remain_halfedges[j]);
+						mesh.halfedge(phId).setNhId(remain_halfedges[j]);
+
+
+						mesh.halfedge(del_halfedges[j]).setIcId(-1);
+
+					}
+
+
+					// mark edges to be deleted
+					mesh.edge(mesh.eId_from_hId(mutual_halfedges[0])).setDel(true);
+					mesh.edge(mesh.eId_from_hId(del_halfedges[0])).setDel(true);
+					mesh.edge(mesh.eId_from_hId(del_halfedges[1])).setDel(true);
+
+//cout << "deleted edges:"<< mesh.eId_from_hId(mutual_halfedges[0])
+//<< " "<< mesh.eId_from_hId(del_halfedges[0])
+//<< " "<< mesh.eId_from_hId(del_halfedges[1])<< endl;
+
+					// mark cells to be deleted
+					mesh.cell(del_cells[0]).setDel(true);
+					mesh.cell(del_cells[1]).setDel(true);
+
+
+					// delete dvId
+					mesh.vertex(dvId).set_isolated_v();
+					mesh.vertex(dvId).setDel(true);
+				}// if (legal)
+			}// if COARSEN
+		} // for i
+
+		// delete entities
+		mesh.delete_cells();
+		mesh.delete_edges();//mesh.display_info();
+
+		mesh.delete_vertices();
+
+		// reset mFlags
+	}// edge_collapse
+
+}

Examples/MeshGenToolbox/mgt/Coarsening.h

+/*****************************************************************************/
+/*  PMGT                                                                     */
+/*                                                                           */
+/*  January, 2007                                                            */
+/*                                                                           */
+/*  Copyright 2007                                                           */
+/*  Wen Yu                                                                   */
+/*  Department of Computing and Software                                     */
+/*  McMaster University                                                      */
+/*  yuw4@mcmaster.ca                                                         */
+/*                                                                           */
+/*  This program may be freely redistributed under the condition that the    */
+/*    copyright notices are not removed. You may distribute modified versions*/
+/*    of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS  */
+/*    MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL     */
+/*    AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT  */
+/*    CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS.                */
+/*                                                                           */
+/*  This is the serial version of PMGT                                       */
+/*  This file contains the class for corsening module.                       */
+/*****************************************************************************/
+
+#ifndef COARSENING
+#define COARSENING
+#endif
+
+#ifndef MESH
+#include "Mesh.h"
+#endif
+
+namespace PMGT{
+class Coarsening{
+public:
+
+	// coarsen the mesh by edge collapse
+	void edge_collapse(Mesh& mesh);
+};
+}

Examples/MeshGenToolbox/mgt/Edge.h

+/*****************************************************************************/
+/*  PMGT                                                                     */
+/*                                                                           */
+/*  January, 2007                                                            */
+/*                                                                           */
+/*  Copyright 2007                                                           */
+/*  Wen Yu                                                                   */
+/*  Department of Computing and Software                                     */
+/*  McMaster University                                                      */
+/*  yuw4@mcmaster.ca                                                         */
+/*                                                                           */
+/*  This program may be freely redistributed under the condition that the    */
+/*    copyright notices are not removed. You may distribute modified versions*/
+/*    of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS  */
+/*    MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL     */
+/*    AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT  */
+/*    CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS.                */
+/*                                                                           */
+/*  This is the serial version of PMGT                                       */
+/*  This file includes classes for the edge module.							 */
+/*****************************************************************************/
+
+#ifndef EDGE
+#define EDGE
+#endif
+
+#ifndef VERTEX
+#include "Vertex.h"
+#endif
+
+namespace PMGT{
+//namespace Edge{
+
+//using namespace Vertex;
+
+
+class Halfedge{
+public:
+	Halfedge(int hId=-1, int TvId=-1, int IcId=-1, int PhId=-1, int NhId=-1){
+	hId_ = hId;
+	TvId_ = TvId;
+	IcId_ = IcId;
+	PhId_ = PhId;
+	NhId_ = NhId;
+	}
+
+	inline int hId(){ return hId_; }
+	inline int IcId(){ return IcId_; }
+	inline int TvId(){ return TvId_; }
+	inline int NhId(){ return NhId_; }
+	inline int PhId(){ return PhId_; }
+
+	inline void setHId(int hId){ hId_ = hId; }
+	inline void setTvId(int TvId){ TvId_ = TvId; }
+	inline void setIcId(int IcId){ IcId_ = IcId; }
+	inline void setNhId(int NhId){ NhId_ = NhId; }
+	inline void setPhId(int PhId){ PhId_ = PhId; }
+
+	inline bool is_boundary(){ return IcId_ == -1; }
+	//opposite halfedge
+	inline int ops_hId(){ return ((hId_ & 1) ? hId_-1: hId_+1); }
+	inline void hcopy(Halfedge& other){
+		hId_ = other.hId_;
+		TvId_ = other.TvId_;
+		IcId_ = other.IcId_;
+		PhId_ = other.PhId_;
+		NhId_ = other.NhId_;
+	}
+
+private:
+	int	hId_;	//id
+	int TvId_; 	//target vertex
+	int	IcId_;	//its cel
+	int	PhId_;	//previous halfedge
+	int NhId_;	//next halfedge
+
+}; // end of class Halfedge
+
+
+class Edge{
+public:
+	Edge(int svId=-2, int tvId=-1, int eId=-1, bool del=false){
+		assert(svId != tvId);
+		eId_ = eId;
+		pair_[0].setHId(eId << 1);
+		pair_[0].setTvId(tvId);
+		pair_[1].setHId((eId << 1)+1);
+		pair_[1].setTvId(svId);
+		del_ = del;
+	}
+
+
+	inline int eId(){ return eId_;}
+	inline Halfedge* pair(){ return pair_; }
+	inline bool del(){ return del_; }
+
+	inline void setEId(int eId){ eId_ = eId; }
+	inline void setPair(Halfedge& h1, Halfedge& h2){
+		pair_[0].hcopy(h1);
+		pair_[1].hcopy(h2);
+	}
+	inline void setDel(bool del) { del_ = del; }
+
+
+	inline bool is_isolated_e(){
+		return pair_[0].IcId() == -1
+			&& pair_[1].IcId() == -1;
+	}
+	inline void set_isolated_e(){
+		pair_[0].setIcId(-1);
+		pair_[1].setIcId(-1);
+	}
+
+	inline bool same (int svId, int tvId){
+		int v1Id = pair_[0].TvId();
+		int v2Id = pair_[1].TvId();
+		return ((v1Id==svId && v2Id==tvId) || (v1Id==tvId && v2Id==svId));
+	}
+
+	inline void ecopy(Edge& other){
+		eId_ = other.eId_;
+		pair_[0].hcopy(other.pair_[0]);
+		pair_[1].hcopy(other.pair_[1]);
+		del_ = other.del_;
+	}
+
+private:
+	int			eId_;		// id
+	Halfedge 	pair_[2]; 	//a pair of halfedges
+	bool 		del_;	// a flag to indicate if the edge
+						// is marked to be deleted
+}; // end of class Edge
+
+
+//} // end of namespace Edge
+
+
+} // end of namespace PMGT

Examples/MeshGenToolbox/mgt/Input.cpp

+/*****************************************************************************/
+/*  PMGT                                                                     */
+/*                                                                           */
+/*  January, 2007                                                            */
+/*                                                                           */
+/*  Copyright 2007                                                           */
+/*  Wen Yu                                                                   */
+/*  Department of Computing and Software                                     */
+/*  McMaster University                                                      */
+/*  yuw4@mcmaster.ca                                                         */
+/*                                                                           */
+/*  This program may be freely redistributed under the condition that the    */
+/*    copyright notices are not removed. You may distribute modified versions*/
+/*    of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS  */
+/*    MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL     */
+/*    AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT  */
+/*    CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS.                */
+/*                                                                           */
+/*  This is the serial version of PMGT                                       */
+/*****************************************************************************/
+
+#include "Input.h"
+
+namespace PMGT{
+
+
+	// Read the mesh from two files
+	// One is the coordinates of points
+	// The other is the vertex list of cells
+	void Input::read_file(Mesh& mesh, char* vertexFile, char* cellFile){
+/**/
+		FILE *v;
+		FILE *c;
+		Point p;
+		Ids ids;
+		vector<Point> points;
+		vector<Ids> cells;
+
+		char *s, *t;
+		char line[100];
+		bool error = false;
+
+		// Open the file.  If NULL is returned there was an error
+		if((v = fopen(vertexFile, "r")) == NULL) {
+			cout << "Error opening file " << vertexFile << "." << endl;
+			exit(1);
+		}
+
+		if((c = fopen(cellFile, "r")) == NULL) {
+			cout << "Error opening file " << cellFile << "." << endl;
+			exit(1);
+		}
+
+		//assert(vertices_.size() == 0);
+		//assert(edges_.size() == 0);
+		//assert(cells_.size() == 0);
+
+		// get vertices
+		points.clear();
+		int count = 0;
+		while( fgets(line, sizeof(line), v) != NULL ) {
+				// Get each line from the infile
+			s = line;
+			p.clear();
+
+			p.push_back(strtod(s, &t));
+			while  ( *s != ' ' && *s !='\n'){
+				s++;
+			}
+			if (*s == '\n'){
+				error = true;
+				break;
+			}
+			p.push_back(strtod(s, &t));
+			points.push_back(p);
+			count++;
+		}
+		fclose(v);  // Close the file
+		if (error){ // error message for input file
+			cout <<"Something wrong in your input file " << vertexFile <<
+			" line " << count+1 << endl;
+			exit(1);
+		}
+
+		// get cells
+		int n=points.size();
+		count = 0;
+		cells.clear();
+		while( fgets(line, sizeof(line), c) != NULL ) {
+				// Get each line from the infile
+			s = line;
+			ids.clear();
+
+			ids.push_back(strtol(s, &t, 10));
+
+			if (ids[ids.size()-1] <0 || ids[ids.size()-1]>=n){
+				error = true;
+				break;
+			}
+			while  ( *s != ' ' && *s !='\n'){
+				s++;
+			}
+			while(*s == ' ') s++;
+			if (*s == '\n'){
+				error = true;
+				break;
+			}
+
+			ids.push_back(strtol(s, &t, 10));
+			while(*s == ' ') s++;
+			if (ids[ids.size()-1] <0 || ids[ids.size()-1]>=n){
+				error = true;
+				break;
+			}
+
+			while  ( *s != ' ' && *s !='\n'){
+				s++;
+			}
+			while(*s == ' ') s++;
+			if (*s == '\n'){
+				error = true;
+				break;
+			}
+
+			ids.push_back(strtol(s, &t, 10));
+			if (ids[ids.size()-1] <0 || ids[ids.size()-1]>=n){
+				error = true;
+				break;
+			}
+			cells.push_back(ids);
+
+			count++;
+		}
+		fclose(c);
+		if (error){ // error message for input file
+			cout <<"Something wrong in your input file " << cellFile <<
+			" line " << count+1 <<endl;
+			exit(1);
+		}
+		mesh.create(points, cells);
+	}
+}
+

Examples/MeshGenToolbox/mgt/Input.h

+/*****************************************************************************/
+/*  PMGT                                                                     */
+/*                                                                           */
+/*  January, 2007                                                            */
+/*                                                                           */
+/*  Copyright 2007                                                           */
+/*  Wen Yu                                                                   */
+/*  Department of Computing and Software                                     */
+/*  McMaster University                                                      */
+/*  yuw4@mcmaster.ca                                                         */
+/*                                                                           */
+/*  This program may be freely redistributed under the condition that the    */
+/*    copyright notices are not removed. You may distribute modified versions*/
+/*    of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS  */
+/*    MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL     */
+/*    AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT  */
+/*    CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS.                */
+/*                                                                           */
+/*  This is the serial version of PMGT                                       */
+/*  This file includes classes for the input module.                         */
+/*****************************************************************************/
+
+#ifndef INPUT
+#define INPUT
+#endif
+
+#ifndef MESH
+#include "Mesh.h"
+#endif
+
+namespace PMGT{
+class Input{
+public:
+
+	// Read the mesh from two files
+	// One is the coordinates of points
+	// The other is the vertex list of cells
+	void read_file(Mesh& mesh, char* vertexFile, char* cellFile);
+};
+}
+
+

Examples/MeshGenToolbox/mgt/Mesh.cpp

+/*****************************************************************************/
+/*  PMGT                                                                     */
+/*                                                                           */
+/*  January, 2007                                                            */
+/*                                                                           */
+/*  Copyright 2007                                                           */
+/*  Wen Yu                                                                   */
+/*  Department of Computing and Software                                     */
+/*  McMaster University                                                      */
+/*  yuw4@mcmaster.ca                                                         */
+/*                                                                           */
+/*  This program may be freely redistributed under the condition that the    */
+/*    copyright notices are not removed. You may distribute modified versions*/
+/*    of this code UNDER THE CONDITION THAT THIS CODE AND ANY MODIFICATIONS  */
+/*    MADE TO IT IN THE SAME FILE REMAIN UNDER COPYRIGHT OF THE ORIGINAL     */
+/*    AUTHOR, BOTH SOURCE AND OBJECT CODE ARE MADE FREELY AVAILABLE WITHOUT  */
+/*    CHARGE, AND CLEAR NOTICE IS GIVEN OF THE MODIFICATIONS.                */
+/*                                                                           */
+/*  This is the serial version of PMGT                                       */
+/*****************************************************************************/
+
+#include "Mesh.h"
+
+namespace PMGT{
+//namespace Mesh{
+
+	// Returns the id of the vertex that has given coordinate.
+	// Returns -1 if the vertex does not exist.
+	int Mesh::find_vertex(Point point){
+		for (int i=0; i<vertices_.size(); i++){
+			if (vertices_[i].point() == point)
+				return i;
+		}
+		return -1;
+	}
+
+	// Returns the id of the edge that has given points.
+	// Returns -1 if the edge does not exist.
+	int  Mesh::find_edge(int svId, int tvId){
+		assert(is_valid_vId(svId));
+		assert(is_valid_vId(tvId));
+		for (int i=0; i<edges_.size(); i++){
+			if (edges_[i].pair()[0].TvId() == tvId){
+				if (edges_[i].pair()[1].TvId() == svId)
+					return i;
+			}
+			if (edges_[i].pair()[0].TvId() == svId){
+				if (edges_[i].pair()[1].TvId() == tvId)
+					return i;
+			}
+		}
+		return -1;
+	}
+
+	// Returns the id of the halfedge that has given start and targate points.
+	// Returns -1 if the halfedge does not exist.
+	int  Mesh::find_halfedge(int svId, int tvId){
+		assert(is_valid_vId(svId));
+		assert(is_valid_vId(tvId));
+		for (int i=0; i<edges_.size(); i++){
+			if (edges_[i].pair()[0].TvId() == tvId){
+				if (edges_[i].pair()[1].TvId() == svId)
+					return (i<<1);
+			}
+			if (edges_[i].pair()[0].TvId() == svId){
+				if (edges_[i].pair()[1].TvId() == tvId)
+					return (i<<1)+1;
+			}
+		}
+		return -1;
+	}
+
+	// Returns the id of the cell that has given vertices.
+	// Returns -1 if the cell does not exist.
+	// Make sure that vIds and corresponding halfedges exist.
+	int Mesh::find_cell(vector<int> vIds){
+		assert(vIds.size() == 3);
+		for (int i=0; i<3; i++){
+			assert(is_valid_vId(vIds[i]));
+		}
+		vector<int> hId(3);
+		for (int i=0; i<3; i++){
+			hId[i] = find_halfedge(vIds[i], vIds[(i+1)%3]);
+			if(hId[i] == -1) return -1;
+		}
+
+		for (int i=0; i<cells_.size(); i++){
+			for (int j=0; j<3; j++){
+				if (cells_[i].LhId()==hId[j]){
+					int k;
+					for (k=0; k<3; k++){
+						if (vIds[k]=tvId_from_hId(hId[j])){
+
+							break;
+						}
+					}
+					if (k<3 && vIds[(k+1)%3]==tvId_from_hId(hId[(j+1)%3])
+					&& vIds[(k+2)%3]==tvId_from_hId(hId[(j+2)%3])){
+						// three vertices is target vertices of three halfedges
+						return i;
+					}
+				}
+			}
+		}
+		return -1;
+	}
+
+
+	// Create a vertex and add it to the end of the vertex list.
+	// Returns false if the vertex already exists.
+	bool Mesh::add_vertex(Point point){
+		int vId = find_vertex(point);
+		if (vId != -1) return false;
+		Vertex v(point, vertices_.size());
+		vertices_.push_back(v);
+		return true;
+	}
+
+
+	// Create an edge and add it to the end of the edge list.
+	// Returns false if the edge already exists.
+	bool Mesh::add_edge(int svId, int tvId){
+		assert(is_valid_vId(svId));
+		assert(is_valid_vId(tvId));
+		int eId = find_edge(svId, tvId);
+		if (eId != -1){
+			cout << "add edge " << svId << " " << tvId << " failed" << endl;
+			return false;
+		}
+		Edge e(svId, tvId, edges_.size());
+		e.pair()[0].setIcId(-1);
+		e.pair()[1].setIcId(-1);
+		edges_.push_back(e);
+		edge(edges_.size()-1).pair()[0].setIcId(-1);
+		edge(edges_.size()-1).pair()[1].setIcId(-1);
+		return true;
+	}
+
+
+	// Create a cell and add it to the end of the vertex list.
+	// Returns false if the cell already exists.
+	bool Mesh::add_cell(vector<int> vIds){
+		assert(vIds.size() == 3);
+		for (int i=0; i<3; i++){
+			assert(is_valid_vId(vIds[i]));
+		}
+		int cId = find_cell(vIds);
+		if (cId != -1) return false;
+	assert(cId == -1);
+		vector<int> hId(3);
+		for (int i=0; i<3; i++){
+
+			hId[i] = find_halfedge(vIds[i], vIds[(i+1)%3]);
+			assert(hId[i] != -1);
+		}
+		int LhId = hId[0];
+		if (dist(hId[1]) > dist(LhId)) LhId = hId[1];
+		if (dist(hId[2]) > dist(LhId)) LhId = hId[2];
+
+		//Cell c(cells_.size(), LhId);
+		Cell c;
+		cells_.push_back(c);
+		cId = cells_.size()-1;
+		cell(cId).setCId(cId);
+		cell(cId).setLhId(LhId);
+		// resolve connectivity of edges and vertices
+		for (int i=0; i<3; i++){
+			halfedge(hId[i]).setIcId(cId);
+			halfedge(hId[i]).setNhId(hId[(i+1)%3]);
+			halfedge(hId[i]).setPhId(hId[(i+2)%3]);
+
+		}
+		for (int i=0; i<3; i++){
+			int OhId = vertex(vIds[i]).OhId();
+			if (OhId ==-1 || (!is_b_h(ops_hId(OhId)))){
+				vertex(vIds[i]).setOhId(hId[i]);
+			}
+			else if(svId_from_hId(OhId) != vIds[i]){
+				vertex(vIds[i]).setOhId(hId[i]);
+			}
+		}
+		return true;
+
+	}
+
+	// update the cell using given vertex.
+	// Returns false if the cell does not exist
+	// or the new cell is already exists.
+	bool Mesh::update_cell(vector<int> vIds, int cId){
+		if (!is_valid_cId(cId)) {
+			cout << "update cell " << cId << " failed. " <<
+			cId << " is not valid." << endl;
+			return false;
+		}
+		int newId = find_cell(vIds);
+		if (newId != -1){
+			cout << "update cell " << cId << " failed. The cell (" <<
+			 vIds[0]<<","<<vIds[1]<<","<<vIds[2]<<") exists."<<endl;
+			return false;
+		}
+
+		vector<int> hId(3);
+		for (int i=0; i<3; i++){
+			hId[i] = find_halfedge(vIds[i], vIds[(i+1)%3]);
+			assert(hId[i] != -1);
+		}
+		int LhId = hId[0];
+		if (dist(hId[1]) > dist(LhId)) LhId = hId[1];
+		if (dist(hId[2]) > dist(LhId)) LhId = hId[2];
+		Cell& c = cell(cId);
+		c.setLhId(LhId);
+		c.setMflag(UNCHANGE);
+
+		// resolve connectivity of edges and vertices
+		for (int i=0; i<3; i++){
+			halfedge(hId[i]).setIcId(c.cId());
+			halfedge(hId[i]).setNhId(hId[(i+1)%3]);
+			halfedge(hId[i]).setPhId(hId[(i+5)%3]);
+
+			int OhId = vertex(vIds[i]).OhId();
+			if (OhId ==-1 || (!is_b_h(ops_hId(OhId)))){
+				vertex(vIds[i]).setOhId(hId[i]);
+			}
+			else if(svId_from_hId(OhId) != vIds[i]){
+				vertex(vIds[i]).setOhId(hId[i]);
+			}
+		}
+		return true;
+
+	}
+
+
+	// refine two cells whose longest edges are oppsite to each other
+	bool Mesh::longest_ref2(int hId, int o_hId, int n_hId){
+
+		assert(o_hId == ops_hId(hId));
+		int cId	= IcId_from_hId(hId);
+		int ops_cId = IcId_from_hId(o_hId);
+		vector<int> vIds;
+		// reserve vertices
+		if (cId != -1){
+			//vIds = vIds_from_cId(cId);
+			vIds.push_back(tvId_from_hId(hId));
+			vIds.push_back(tvId_from_hId(next_hId(hId)));
+			vIds.push_back(svId_from_hId(hId));
+		}
+		vector<int> ops_vIds;// = vIds_from_cId(ops_cId);
+		ops_vIds.push_back(tvId_from_hId(o_hId));
+		ops_vIds.push_back(tvId_from_hId(next_hId(o_hId)));
+		ops_vIds.push_back(svId_from_hId(o_hId));
+
+		if (cId !=-1){
+			assert(ops_vIds[0] == vIds[2]);
+			assert(ops_vIds[2] == vIds[0]);
+		}
+
+		// reset mFlag
+		cell(ops_cId).setMflag(UNCHANGE);
+		if (cId != -1){
+			cell(cId).setMflag(UNCHANGE);
+		}
+
+		// add new vertex
+		Point p = mid_point(hId);
+		assert(add_vertex(p));
+		int new_vId = n_of_v()-1;
+
+		// add/update edges
+		assert(add_edge(ops_vIds[1], new_vId));
+		if (cId != -1){
+			assert(add_edge(vIds[1], new_vId));
+		}
+
+		halfedge(hId).setTvId(new_vId);
+		assert(add_edge(ops_vIds[2], new_vId));
+
+		// add/update cells
+		int next_svId=0;
+		if (n_hId != -1){
+			for(int k=0; k<2; k++){
+				if(ops_vIds[k]==svId_from_hId(n_hId)){
+					next_svId = k;
+					break;
+				}
+			}
+		}
+		vector<int> temp_vIds;
+		temp_vIds.clear();
+		temp_vIds.push_back(new_vId);
+		temp_vIds.push_back(ops_vIds[next_svId]);
+		temp_vIds.push_back(ops_vIds[next_svId+1]);
+		assert(update_cell(temp_vIds, ops_cId));
+
+		if (next_svId == 0){
+			temp_vIds[1] = ops_vIds[1];
+			temp_vIds[2] = ops_vIds[2];
+		}
+		else {
+			temp_vIds[1] = ops_vIds[0];
+			temp_vIds[2] = ops_vIds[1];
+		}
+
+		assert(add_cell(temp_vIds));
+		if (cId != -1){
+			temp_vIds.clear();
+			temp_vIds.push_back(new_vId);
+			temp_vIds.push_back(vIds[0]);
+			temp_vIds.push_back(vIds[1]);
+			assert(update_cell(temp_vIds, cId));
+
+			temp_vIds[1] = vIds[1];
+			temp_vIds[2] = vIds[2];
+			assert(add_cell(temp_vIds));
+		}
+	}
+
+
+
+	// ****** delete an entity *****
+
+	// Delete vertices that marked to be deleted.
+	bool Mesh::delete_vertices(){
+		for (int i=vertices_.size()-1; i>=0; i--){
+			if (vertices_[i].del() == true){
+
+				// the vertex to be deleted is not the last one
+				if (vertices_.size()-1 != i){
+					int vnId = vertices_.size()-1;
+					int hId = vih_first(vnId);
+
+					do{
+						int old_hId = hId;
+						hId = vih_next(vnId, hId);
+						halfedge(old_hId).setTvId(i);
+						if (hId == -1){
+							break;
+						}
+					}while (hId != vih_first(vnId));
+
+					vertices_[i].vcopy(vertices_[vnId]);
+					vertices_[i].setVId(i);
+
+				}
+				vertices_.pop_back();
+			}
+		}
+		return true;
+	}
+
+
+
+// Delete edges that marked to be deleted from the list.
+	bool Mesh::delete_edges(){
+		vector<int> vIds(2);
+		int j;
+		for (int i=edges_.size()-1; i>=0; i--){
+			if (edges_[i].del() == true){
+				//assert(IcId_from_hId(hId_from_eId(i,0)) == -1);
+				//assert(IcId_from_hId(hId_from_eId(i,1)) == -1);
+				if (i!=edges_.size()-1){
+
+
+					int enId = edges_.size()-1;
+					// relink last edge in the edge list
+					// this edge will be moved to to position i
+					vIds = vIds_from_eId(enId);
+					for (j=0; j<2; j++){
+						int hnj = hId_from_eId(enId,j);
+						int hij = hId_from_eId(i,j);
+						// resolve OhIds of endpoints
+						if (OhId_from_vId(vIds[j]) == hnj){
+							vertex(vIds[j]).setOhId(hij);
+						}
+
+
+						// resolve PhIds and NhIds
+						if (!is_b_h(hnj)){
+							halfedge(prev_hId(hnj)).setNhId(hij);
+							halfedge(next_hId(hnj)).setPhId(hij);
+						}
+
+						// resolve LhId of the incident cell
+						if (IcId_from_hId(hnj) != -1 &&
+						LhId_from_cId(IcId_from_hId(hnj))==hnj){
+							cell(IcId_from_hId(hnj)).setLhId(hij);
+						}
+					}// for j
+			// copy last edge
+					edges_[i].ecopy(edges_[enId]);
+					edges_[i].setEId(i);
+					edges_[i].pair()[0].setHId(i<<1);
+					edges_[i].pair()[1].setHId((i<<1)+1);
+				}// if i
+				// delete last edge
+				edges_.pop_back();
+
+
+			}
+		} // for i
+			return true;
+	}
+
+
+	// Delete cells that marked to be deleted from the list.
+	bool Mesh::delete_cells(){
+
+		//vector<int> hIds(3);
+		for (int i=cells_.size()-1; i>=0; i--){
+			if (cells_[i].del() == true){
+				// the cell to be deleted is not the last one
+				if (i!=cells_.size()-1){
+
+					int cnId = cells_.size()-1;
+					// relink last cell in the cell list
+					// this cell will be moved to cId
+					vector<int> hIds;
+					hIds.push_back(LhId_from_cId(cnId));
+					hIds.push_back(next_hId(LhId_from_cId(cnId)));
+					hIds.push_back(prev_hId(LhId_from_cId(cnId)));
+
+					for(int j=0; j<3; j++){
+						halfedge(hIds[j]).setIcId(i);
+					}
+					// copy last cell
+					cells_[i].ccopy(cells_[cnId]);
+					cells_[i].setCId(i);
+
+				}
+				// delete last cell
+				cells_.pop_back();
+			}
+		}
+			return true;
+	}
+
+
+	// create a mesh using two input variables.
+	// One is the coordinates of points
+	// The other is the vertex list of cells
+	bool Mesh::create(vector<Point> points, vector<Ids> cells){
+		int i,j;
+		int nv = points.size();
+		int nc = cells.size();
+		int eId[3];
+
+		vector<int> vIds;
+
+		for (i=0; i<nv; i++){
+			// point duplication check
+			if (!add_vertex(points[i])){
+				cout << "Duplicate point is detected " <<
+					"during mesh creatation."<< endl;
+				return false;
+			}
+		}
+
+		for (i=0; i<nc; i++){
+			vIds.clear();
+
+			for (j=0; j<3; j++){
+				// validation check
+				if (!is_valid_vId(cells[i][j])){
+
+					cout << "Invalid id is detected " <<
+					"during mesh creatation."<< endl;
+					return false;
+				}
+
+				// find vertices
+				vIds.push_back(cells[i][j]);
+
+				// boundary check
+				if (!is_b_v(vIds[j])){
+					cout << "Complex vertex is detected " <<
+					"during mesh creatation."<< endl;
+					return false;
+				}
+			}
+			// cell duplication check
+			if (find_cell(vIds) != -1){
+				cout << "Duplicate cell is detected " <<
+					"during mesh creatation."<< endl;
+				return false;
+			}
+
+			// add edges if necessary
+			for (j=0; j<3; j++){
+				eId[j] = find_edge(vIds[j], vIds[(j+1)%3]);
+				if (eId[j] == -1){
+					add_edge(vIds[j], vIds[(j+1)%3]);
+				}
+
+			}
+
+			// add the cell
+			add_cell(vIds);
+		}
+
+	}
+
+
+	// display the internal information of a mesh on the screen
+	void Mesh::display_info(){
+		vector<int> ids;
+
+		cout << endl <<"vertices:" << endl;
+		for (int i=0; i< vertices_.size(); i++){
+			cout << "id=" << vertex(i).vId() << " x=" << vertex(i).x() << " y=" << vertex(i).y() <<
+			" OhId=" << vertex(i).OhId() << endl;
+		}
+
+		cout <<endl << "edges:" << endl;
+		for (int i=0; i<edges_.size(); i++){
+			cout << "id=" << edge(i).eId() << " svId=" << vIds_from_eId(i)[0] <<
+			" tvId=" << vIds_from_eId(i)[1] << endl;
+			cout << "hid=" << edge(i).pair()[0].hId() << " prev=" <<  edge(i).pair()[0].PhId() <<
+			" next=" << edge(i).pair()[0].NhId() << " IcId=" << edge(i).pair()[0].IcId() << endl;
+			cout << "hid=" << edge(i).pair()[1].hId() << " prev=" <<  edge(i).pair()[1].PhId() <<
+			" next=" << edge(i).pair()[1].NhId() << " IcId=" << edge(i).pair()[1].IcId() << endl;
+		}
+
+		cout <<endl << "cells:" << endl;
+		for (int i=0; i< cells_.size(); i++){
+				cout << "id=" << cell(i).cId() << " LhId=" << cell(i).LhId() <<
+				" vIds=" << vIds_from_cId(i)[0] << ","
+						<< vIds_from_cId(i)[1] << ","
+						<< vIds_from_cId(i)[2] <<
+				" mFlag=" << cell(i).mFlag() << endl;
+		}
+	}
+
+
+//}
+}