%-------------------------------------------------------- %| CGAL Manual : tpm.tex %| %| Soon to be split into User and Reference Manuals %-------------------------------------------------------- %| Specification of topological planar map %| %| 31 Mar 2000 - Shai Hirsch, %| changes for the 31/3/2000 deadline %| %| Version 1.1 - Iddo Hanniel %| changes after lutz's comments in MPI %| Version 1.0 - Iddo Hanniel %| %-------------------------------------------------------- \def\Ipe#1{\def\IPEfile{#1}\input{#1}} \renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\normal}[1]{\eta_{#1}} \newenvironment{dfn}{{\vspace*{1ex} \noindent \bf Definition }}{\vspace*{1ex}} \newcommand{\bigdef}[2]{\index{#1}\begin{dfn} {\rm #2} \end{dfn}} \newenvironment{proof}{{\em Proof:}}{\hfill{\hfill\rule{2mm}{2mm}}} \newcommand{\comment}[1]{{\sf * #1 *}} \newcommand{\ncomment}[1]{\noindent {\sf * #1 * }} \newcommand{\intsupplanes}{P} \def\C{{\cal C}} \def\G{{\cal G}} \def\F{{\cal F}} \def\I{{\cal I}} \def\U{{\cal U}} \def\M{{\cal M}} \def\eps{{\varepsilon}} \def\bd{{\partial}} \def\dm{{\cal D}} \newcommand{\Section}[1]{Section~{\protect\ref{#1}}} \newcommand{\Chapter}[1]{Chapter~{\protect\ref{#1}}} % restores original settings for \parskip and \parindent \ccParDims \chapter{Topological Maps} \label{I1_ChapterTopologicalMap} % +=============================================================+ \section{Introduction} \label{TPM_sec:intro} The topological map (\ccc{Topological_map}) is a {\em combinatorial} structure with no geometric information. Therefore, it can also be used as a base class for deriving geometric subdivisions (e.g, 2D planar maps) with different geometries (e.g, on a sphere or torus). \section{Basic Terms and Software Design} The class is parametrized with the \ccc{Dcel} type which should model the \ccc{TopologicalMapDcel} concept. The \ccc{Dcel} (Doubly Connected Edge List) is the underlying combinatorial data structure (also know as the halfedge data structure). The \ccc{Planar_map_2} class (Chapter~\ref{I1_ChapterPlanarMap}) is derived from the \ccc{Topological_map} class and it describes an embedding of a topological map in the Euclidean plane. This chapter and Chapter \ref{I1_ChapterPlanarMap} describe the \ccc{Topological_map} class and the \ccc{Planar_map} class respectively. These classes supply the ability to maintain subdivisions of the plane induced by collections of curves. In this chapter we introduce the \ccc{topological map}. In this section we briefly review the concepts underlying the data structures described in the following sections as well as the functionality of \ccc{Topological_map} in a nutshell. \begin{figure} \begin{ccTexOnly} \centerline{ %\includegraphics{my_face.ps} \Ipe{my_face.ipe} } \end{ccTexOnly} \caption{A face, an edge, and a vertex \label{fig:face}} \begin{ccHtmlOnly}

\end{ccHtmlOnly} \end{figure} % \ccHtmlNoLinksFrom prevents Vertex from being linked to hds' vertex \ccHtmlNoLinksFrom{ \paragraph{Topological Map, Vertex, Edge, Face:} } A topological map is a graph that consists of vertices V, edges E, faces F and an incidence relation on them. %\ccSeeAlso{Polyhedron} Each edge is represented by two halfedges with opposite orientations. A {\em face} of the topological map is defined by the ordered circular sequences (inner and outer) of halfedges along its boundary. \paragraph{Incidence:} If a vertex $v$ is an endpoint of an edge $e$, then we say that $v$ and $e$ are {\em incident} to each other. Similarly, a face and an edge on its boundary are incident, and a face and a vertex on its boundary are incident (including edges and vertices that are not connected to the outer boundary --- see below). \ccHtmlNoLinksFrom{ \paragraph{Halfedge, Twin, Source, Target:} } We consider each edge $e$ to be two-sided, representing it by two directed {\em halfedges} \lcTex{$\vec{e}$}\lcHtml{$e$} and \lcTex{${\rm Twin}(\vec{e})$}\lcHtml{${\rm Twin}({e})$} (In other packages the twin halfedge is called $opposite$). A halfedge \lcTex{$\vec{e}$}\lcHtml{$e$} is an ordered pair $(u,v)$ of its endpoints, and it is directed from $u$, the {\em source}, to $v$, the {\em target} (there is no need to store both in each halfedge since \lcTex{${\rm Target}(\vec{e}) \equiv {\rm Source}({\rm Twin}(\vec{e}))$}% \lcHtml{${\rm Target}({e}) \equiv {\rm Source}({\rm Twin}({e}))$}). We consider each halfedge to lie on the boundary of a single face. %---the face lying to our left as we traverse the edge from source to target. %We consider each edge $e$ to be two-sided, representing it by two %directed {\em halfedges} \ccTexHtml{$\vec{e}$}{$e$} and %\ccTexHtml{${\rm Twin}(\vec{e})$}{${\rm Twin}(e)$} (in other %places the twin halfedge is called $Opposite$). %A halfedge \ccTexHtml{$\vec{e}$}{$e$} %is an ordered pair $(u,v)$ of its incident vertices, and %it is directed from $u$, the {\em source}, to $v$, the {\em target} (there %is no need to store both in each halfedge since %\ccTexHtml{${\rm Target}(\vec{e}) \equiv {\rm Source}({\rm Twin}(\vec{e}))$}{${\rm Target}(e) \equiv {\rm Source}({\rm Twin}(e))$}). %We consider each halfedge to lie on the boundary of a single face. %%---the face lying to our left as we traverse the edge from source to target. \paragraph{Connected Component of the Boundary (CCB):} Each connected component of the boundary of a face is %represented defined by a circular list of halfedges. %The list of halfedges of the %outer boundary component of a face is oriented counterclockwise, and %the list for each inner boundary component is oriented clockwise, see %Figure \ref{fig:DCEL}. For a face $f$ of a topological map, we call each connected component of the boundary of $f$ a {\em CCB}. A {\em bounded face} has a unique CCB that is defined to be its outer CCB. An {\em unbounded\/} face does not have an outer boundary. In the topological map we have one unbounded face. %If $f$ is %bounded, we call its outer boundary component the outer CCB. Except for the outer CCB, any other connected component of the boundary of $f$ is called a hole (or inner CCB), every face can have none or several holes. We say that the holes are {\em contained\/} inside the face. \ccHtmlNoLinksFrom{ \paragraph{Edges around a Vertex :} } Every maximal set of halfedges that share the same target can be viewed as a circular list of halfedges ordered %clockwise around their target vertex. It should be noted that the orientation of the edges around a vertex is opposite to that of the halfedges around a face, i.e., if edge $e2$ succeeds edge $e1$ in the order given around vertex $v$, then $e1$ succeeds $e2$ in the order given around the incident face $f$. Unlike the convention we adopt for \ccc{Planar_map} in Chapter~\ref{I1_ChapterPlanarMap} where the halfedges are oriented counterclockwise around a face and clockwise around a vertex, in the topological map the users are free to choose any other convention. \begin{figure} \begin{ccTexOnly} \centerline{ \Ipe{dcel.ipe} } \end{ccTexOnly} \caption{Source and target vertices, and twin halfedges \label{fig:DCEL}} \begin{ccHtmlOnly}

\end{ccHtmlOnly} \end{figure} \paragraph{Doubly Connected Edge List (DCEL):} For a topological map, its {\em DCEL} representation consists of a connected list of halfedges for every CCB of every face in the subdivision, with additional incidence information that enables us to traverse the subdivision. %In particular, for For each halfedge the DCEL stores a pointer to its twin halfedge and to the next halfedge around its incident face (see Figure~\ref{fig:DCEL}). In addition, for each halfedge the DCEL stores a pointer to the incident face and the target vertex. For each face the DCEL stores a pointer to a halfedge representing iwion of the requirements for a DCEL in our implementation. \subsection*{Functionality} The class \ccc{Topological_map} supplies the ability to maintain a topological map. The user can insert edges in various ways and then split, merge or remove them as well as move holes from one face to another. The vertices, edges and faces can be traversed in a linear way or any other fashion mentioned above. For a full reference of the class (i.e its associated types, its operations, etc.) read the \ccc{Topological_map Reference Pages}\lcTex{ (\ccRefPage{Tpm_ref_intro})}. % +=============================================================+ \section{Example Programs} \label{TPM_sec:example} We conclude this chapter with two example programs. The first example demonstrates a simple construction of a \ccc{Topological_map}. The second example demonstrates the ease with which additional information can be added to the \ccc{Topological_map}. % +-------------------------------------------------------------+ \subsection{Simple Topological Map} The example shows a simple construction of a \ccStyle{Topological_map}. It uses the base classes for vertex, halfedge and face and demonstrates the use of the three insertion functions. The function \ccc{is_valid()} checks the validity of the topological map. \ccIncludeExampleCode{Topological_map/example1.C} The output of the program is: \begin{verbatim} inserting edge e1 in face interior ...map is valid. inserting edge e2 from target vertex of e1 ...map is valid. inserting edge e3 between target vertices of e2 and e1->twin() ...map is valid. \end{verbatim} %%%%%%%%%%%%%%%%%% \subsection{Topological Map with Additional Information} The example shows a construction of a \ccStyle{Topological_map} with additional information in the faces. It uses inheritance from the face base class to add the information. \ccIncludeExampleCode{Topological_map/example2.C} The output of the program is: \begin{verbatim} inserting e1 in face interior... inserting e2 from vertex... inserting e3 between vertices of e2 and e1->twin()... setting info of the new face to 10... unbounded face info = 0 new face info = 10 \end{verbatim} % +-------------------------------------------------------------+ % EOF