From 8c7d87ad57a0df96067b9d31fc26bf67bbc5274c Mon Sep 17 00:00:00 2001 From: Guillaume Damiand Date: Mon, 4 Jul 2011 15:03:32 +0000 Subject: [PATCH] remove $$s --- .../Combinatorial_map/Combinatorial_map.tex | 477 +++++++++--------- 1 file changed, 240 insertions(+), 237 deletions(-) diff --git a/Combinatorial_map/doc_tex/Combinatorial_map/Combinatorial_map.tex b/Combinatorial_map/doc_tex/Combinatorial_map/Combinatorial_map.tex index 3d53c545272..a08d4617a7e 100644 --- a/Combinatorial_map/doc_tex/Combinatorial_map/Combinatorial_map.tex +++ b/Combinatorial_map/doc_tex/Combinatorial_map/Combinatorial_map.tex @@ -5,6 +5,10 @@ \def\betazero{\ccTexHtml{$\beta_0$}{β0}} \def\betaun{\ccTexHtml{$\beta_1$}{β1}} +\def\betadeux{\ccTexHtml{$\beta_2$}{β2}} +\def\betai{\ccTexHtml{$\beta_i$}{βi}} + +\def\pmun{\ccTexHtml{$p^{-1}$}{p-1}} % \newcommand{\orb}[1]{⟨#1⟩} \newcommand{\orb}[1]{\langle{}#1\rangle{}} @@ -27,14 +31,13 @@ dimension.\footnote{A 2D combinatorial map is equivalent to a halfedge We denote \cell{i} for an \emph{i}-dimensional cell (for example in 3D, 0-cells are \emph{vertices}, 1-cells are \emph{edges}, 2-cells are \emph{facets}, and 3-cells are \emph{volumes}). A \emph{boundary - relation} is defined on these cells, giving for each \cell{i} $c$ -the set of \cells{(i-1)} contained in the boundary of $c$. Two cells -$c_1$ and $c_2$ are \emph{incident} if there is a path of cells, + relation} is defined on these cells, giving for each \cell{i} \emph{c} +the set of \cells{(i-1)} contained in the boundary of \emph{c}. Two cells +\emph{c1} and \emph{c2} are \emph{incident} if there is a path of cells, starting from the cell of biggest dimension to the other cell, such that each cell of the path (except the first one) belongs to the -boundary of the previous cell in the path. Two \cells{i} $c$ and $c'$ -are \emph{adjacent} if there is an \cell{(i-1)} incident to both $c$ -and $c'$. You can see an example of a 2D object and a 3D object in +boundary of the previous cell in the path. Two \cells{i} \emph{c3} and \emph{c4} are \emph{adjacent} if there is an \cell{(i-1)} incident to both \emph{c3} +and \emph{c4}. You can see an example of a 2D object and a 3D object in Figure~\ref{fig-exemple-3Dmanifold} showing some cells of the subdivision and some adjacency and incidence relations. \begin{figure}[ht] @@ -55,21 +58,21 @@ subdivision and some adjacency and incidence relations. \end{ccHtmlOnly} \caption{Example of subdivided objects that can be described by combinatorial maps. \textbf{Left}: A 2D object composed of - three facets (2-cells), named $f_1$, $f_2$ and $f_3$, nine - edges (1-cells) and seven vertices (0-cells). $f_1$ and - $f_2$ are adjacent along edge $e_1$, thus $e_1$ is incident both - to $f_1$ and $f_2$. Vertex $v_1$ is incident to edge $e_1$, thus - $v_1$ is incident to $f_1$ and $f_2$ by transitivity. + three facets (2-cells), named \emph{f1}, \emph{f2} and \emph{f3}, nine + edges (1-cells) and seven vertices (0-cells). \emph{f1} and + \emph{f2} are adjacent along edge \emph{e1}, thus \emph{e1} is incident both + to \emph{f1} and \emph{f2}. Vertex \emph{v1} is incident to edge \emph{e1}, thus + \emph{v1} is incident to \emph{f1} and \emph{f2} by transitivity. \textbf{Right}: A 3D object (only partially represented for vertices and edges) - composed of three volumes (3-cells), named $vol_1$, $vol_2$ - and $vol_3$, twelve facets (2-cells) (there is one facet - $f_4$ between $vol_1$ and $vol_2$, and similarly between $vol_1$ - and $vol_3$ and $vol_2$ and $vol_3$), sixteen edges (1-cells), - and eight vertices (0-cells). $vol_1$ and $vol_2$ are adjacent - along facet $f_4$, thus $f_4$ is incident both to $vol_1$ and - $vol_2$. Edge $e_4$ is incident to the three facets between - $vol_1$ and $vol_2$, $vol_1$ and $vol_3$, and $vol_2$ and - $vol_3$. $e_4$ is also incident to the three volumes by + composed of three volumes (3-cells), named \emph{vol1}, \emph{vol2} + and \emph{vol3}, twelve facets (2-cells) (there is one facet + \emph{f4} between \emph{vol1} and \emph{vol2}, and similarly between \emph{vol1} + and \emph{vol3} and \emph{vol2} and \emph{vol3}), sixteen edges (1-cells), + and eight vertices (0-cells). \emph{vol1} and \emph{vol2} are adjacent + along facet \emph{f4}, thus \emph{f4} is incident both to \emph{vol1} and + \emph{vol2}. Edge \emph{e4} is incident to the three facets between + \emph{vol1} and \emph{vol2}, \emph{vol1} and \emph{vol3}, and \emph{vol2} and + \emph{vol3}. \emph{e4} is also incident to the three volumes by transitivity.} \label{fig-exemple-3Dmanifold} \end{figure} @@ -79,8 +82,8 @@ cells and the incidence and adjacency relations, using only one basic element called \emph{dart}, and a set of \emph{pointers} between these darts. A dart can be thought as a part of an oriented edge (1-cell), together with a part of incident cells of dimensions 0, 2, 3,\ldots, -\emph{d}. When a dart $d_0$ describe a part of an \cell{i} $c$, we say that -$d_0$ \emph{belongs} to $c$, and that $c$ \emph{contains} $d_0$. Let +\emph{d}. When a dart \emph{d0} describe a part of an \cell{i} \emph{c}, we say that +\emph{d0} \emph{belongs} to \emph{c}, and that \emph{c} \emph{contains} \emph{d0}. Let us look at the example in Figure~\ref{fig-exemple-combi-maps} showing the 2D and 3D combinatorial maps describing the two objects given in Figure~\ref{fig-exemple-3Dmanifold}. @@ -110,7 +113,7 @@ Figure~\ref{fig-exemple-3Dmanifold}. \end{figure} First let us start in 2D (Figure~\ref{fig-exemple-combi-maps} (Left)). -Facet $f_1$ is described by four darts. These darts are linked +Facet \emph{f1} is described by four darts. These darts are linked together with pointers. Starting from a dart and following a \betaun{} %\mbeta{1} pointer, we get to a dart which belongs to the same facet but to the next edge (1-cell, which explains the index~1 of~$\beta_1$). @@ -118,7 +121,7 @@ Starting from any dart and following $\beta_1$ pointers, we can reach exactly all the darts describing the facet. Starting from a dart and following a $\beta_2$ pointer, we get to a dart which belongs to the same edge but to the neighboring facet (2-cell, which explains the -index~$2$ of~$\beta_2$). Starting from any dart and following +index~2 of~$\beta_2$). Starting from any dart and following $\beta_2$ pointers, we can reach exactly all the darts describing the edge (in 2D one or two darts). @@ -126,8 +129,8 @@ Things are slightly different for vertices. Indeed, each $\beta_i$ points to a dart belonging to a different \cell{i}, but also to a different 0-cell (vertex). This is so because two linked darts have opposite orientations. For this reason, starting from any dart -belonging to a vertex $v$, we have to follow $\beta_2$ then $\beta_1$ -to reach exactly the darts describing the vertex $v$. In fact, by +belonging to a vertex \emph{v}, we have to follow $\beta_2$ then $\beta_1$ +to reach exactly the darts describing the vertex \emph{v}. In fact, by composing two $\beta_i$s, we always obtain a dart belonging to the same vertex. @@ -138,7 +141,7 @@ example (Figure~\ref{fig-exemple-combi-maps} (Right)). In addition to $\beta_1$ and $\beta_2$ of the 2D case, there is a new pointer $\beta_3$. -If we take a closer look at the central edge $e_4$ shown in +If we take a closer look at the central edge \emph{e4} shown in Figure~\ref{fig-intuitive-exemple} (Left), we can see that it is described by six darts linked together. Starting from a dart and following a $\beta_3$ pointer, we get to a dart which belongs to the @@ -148,7 +151,7 @@ a dart and following a $\beta_2$ pointer, we get to a dart which belongs to the same edge, to the same volume, but to the neighboring facet (2-cell). Starting from any of these six darts and following $\beta_2$ and $\beta_3$ pointers, we can reach exactly the six darts -describing edge $e_4$. +describing edge \emph{e4}. % % Finally, following a $\beta_1$ pointer, we get to a % dart which belongs to the same facet, the same volume but on the @@ -175,9 +178,9 @@ describing edge $e_4$. \end{ccHtmlOnly} \caption{Two zooms on the 3D combinatorial map given in Figure~\ref{fig-exemple-combi-maps} (Right). - \textbf{Left}:~Zoom around the central edge $e_4$ which details + \textbf{Left}:~Zoom around the central edge \emph{e4} which details the six darts belonging to the edge. \textbf{Right}:~Zoom - around the facet between volumes $vol_2$ and $vol_3$ which + around the facet between volumes \emph{vol2} and \emph{vol3} which details the eight darts belonging to the facet.} \label{fig-intuitive-exemple} \end{figure} @@ -206,26 +209,26 @@ volume. % this reason, starting from any dart belonging to a vertex $v$, we have For vertices, we have to follow $\beta_2$ then $\beta_1$, and $\beta_3$ then $\beta_1$ to reach exactly the darts describing the -vertex $v$. Indeed, as in 2D, we have to compose two $\beta_i$s to +vertex \emph{v}. Indeed, as in 2D, we have to compose two $\beta_i$s to obtain a dart belonging to the same vertex. In some cases, the general rule that by following a $\beta_i$ we get a dart which belongs to the neighboring \cell{i} is not true, as for example for darts belonging to the boundary of the represented object. For example, in Figure~\ref{fig-exemple-3Dmanifold} (Left), any dart -$d_0$ that does not belong to edge $e_1$, $e_2$ and $e_3$ -belongs to a 2-cell, and there is no neighboring facet along the edge containing $d_0$. +\emph{d0} that does not belong to edge \emph{e1}, \emph{e2} and \emph{e3} +belongs to a 2-cell, and there is no neighboring facet along the edge containing \emph{d0}. % this is the % case for $\beta_2$ for all the darts that does not belong to edge % $e_1$, $e_2$ and $e_3$. % Indeed, these darts belong to a 2-cell, and % there is no neighboring facet along the edge containing these darts. Another example is in Figure~\ref{fig-exemple-3Dmanifold} (Right), for -any dart $d_0$ that belongs to facet $f_5$. +any dart \emph{d0} that belongs to facet \emph{f5}. % this is the case for % $\beta_3$ for example for all the darts belonging to facet % $f_5$. Indeed, these darts -$d_0$ belongs to volume $vol_2$, but there is no neighboring volume +\emph{d0} belongs to volume \emph{vol2}, but there is no neighboring volume along this facet. The general rule is also not true for unbounded cells. For example if we remove a dart in Figure~\ref{fig-exemple-combi-maps} (Left), we obtain an unbounded @@ -233,7 +236,7 @@ facet having a dart without next dart for $\beta_1$, and if we remove a facet in Figure~\ref{fig-exemple-combi-maps} (Right), we obtain an unbounded volume having some darts without neighboring facet for $\beta_2$. In such a case, there is a particular value called -$\varnothing$ used to describe that a dart $d_0$ is not linked to +$\varnothing$ used to describe that a dart \emph{d0} is not linked to another dart in dimension~\emph{i}. % No it is false (case when c_i=c'_i) and we exchange also c_0... @@ -302,20 +305,20 @@ Section~\ref{sec_definition}, and a package description is given in Section~\ref{sec-software-design}. \subsection{Combinatorial Map and Darts}\label{ssec-combi-map-and-darts} -A \emph{d}D combinatorial map is a set of darts $D$. A dart $d_0$ is an -element that can be \emph{linked} with $d+1$ darts by pointers called -$\mb{i}$, with $0 \leq i \leq d$. Dart $d_0$ is said \emph{i-free} -when $\mb{i}(d_0)=\varnothing$. Each $\mb{i}$, for $2 \leq i \leq d$, -is its own inverse, i.e., if dart $d_0$ is not \emph{i}-free, then -$\mb{i}(\mb{i}(d_0))=d_0$. This is different for $\mb{0}$ and -$\mb{1}$: $\mb{0}$ is the inverse of $\mb{1}$, i.e., if darts $d_1$ -and $d_2$ are such that $\mb{1}(d_1)=d_2$, then -$\mb{0}(d_2)=d_1$. Given dart $d_1$, if there is no dart $d_2$ such -that $\mb{1}(d_2)=d_1$, then $\mb{0}(d_1)=\varnothing$. $\varnothing$ -is a constant, which does not belong to the set of darts $D$ of the +A \emph{d}D combinatorial map is a set of darts \emph{D}. A dart \emph{d0} is an +element that can be \emph{linked} with \emph{d+1} darts by pointers called +$\mb{i}$, with 0 $\leq$ \emph{i} $\leq$ \emph{d}. Dart \emph{d0} is said \emph{i-free} +when $\mb{i}$(\emph{d0})=$\varnothing$. Each $\mb{i}$, for 2 $\leq$ \emph{i} $\leq$ \emph{d}, +is its own inverse, i.e., if dart \emph{d0} is not \emph{i}-free, then +$\mb{i}$($\mb{i}$(\emph{d0}))=\emph{d0}. This is different for $\mb{0}$ and +$\mb{1}$: $\mb{0}$ is the inverse of $\mb{1}$, i.e., if darts \emph{d1} +and \emph{d2} are such that $\mb{1}$(\emph{d1})=\emph{d2}, then +$\mb{0}$(\emph{d2})=\emph{d1}. Given dart \emph{d1}, if there is no dart \emph{d2} such +that $\mb{1}$(\emph{d2})=\emph{d1}, then $\mb{0}$(\emph{d1})=$\varnothing$. $\varnothing$ +is a constant, which does not belong to the set of darts \emph{D} of the combinatorial map. However, by definition $\varnothing$ is linked with -itself for all $\beta_i$s: $\forall i$, $0 \leq i \leq d$, -$\mb{i}(\varnothing)=\varnothing$. +itself for all $\beta_i$s: $\forall$ \emph{i}, 0 $\leq$ \emph{i} $\leq$ \emph{d}, +$\mb{i}(\varnothing)$=$\varnothing$. % There is a particular value called % $\varnothing$ used to describe that a dart $d_0$ is not linked to @@ -333,16 +336,16 @@ $\mb{i}(\varnothing)=\varnothing$. A combinatorial map is \emph{without i-boundary} if there is no \emph{i}-free dart, and it is \emph{without boundary} if it is without -\emph{i}-boundary for all dimensions $1 \leq i \leq d$. +\emph{i}-boundary for all dimensions 1 $\leq$ \emph{i} $\leq$ \emph{d}. We show in Figure~\ref{fig-exemple-carte3d} a 3D object and the corresponding 3D combinatorial map. This map has 40 darts represented by arrows, some darts being numbered. In this combinatorial map, we -have for example $\beta_1(1)=2$, $\beta_2(1)=10$, and -$\beta_3(1)=5$. This combinatorial map is without 1-boundary and -2-boundary, but has some 3-boundary, because some darts are $3$-free, -for example $\beta_3(10)=\varnothing$ and $\beta_3(12)=\varnothing$. +have for example $\beta_1$(1)=2, $\beta_2$(1)=10, and +$\beta_3$(1)=5. This combinatorial map is without 1-boundary and +2-boundary, but has some 3-boundary, because some darts are 3-free, +for example $\beta_3$(10)=$\varnothing$ and $\beta_3$(12)=$\varnothing$. % \def\LargFig{.4\textwidth} \begin{figure} @@ -361,14 +364,14 @@ for example $\beta_3(10)=\varnothing$ and $\beta_3(12)=\varnothing$. \end{ccHtmlOnly} \caption{Example of a 3D combinatorial map. \textbf{Left}:~A 3D object - made of two volumes adjacent along facet $f_2$. \textbf{Right}:~The + made of two volumes adjacent along facet \emph{f2}. \textbf{Right}:~The corresponding 3D combinatorial map. Darts are drawn with arrows, sometimes numbered. Two darts linked by $\beta_1$ are - drawn consecutively (for example $\beta_1(10)=11$), and two + drawn consecutively (for example $\beta_1$(10)=11), and two darts linked by $\beta_2$ are drawn parallel, in reverse orientations, with a little gray segment joining them (for - example $\beta_2(1)=10$). $\beta_3$ pointers are represented by - blue segments (for example $\beta_3(1)=5$).} + example $\beta_2$(1)=10). $\beta_3$ pointers are represented by + blue segments (for example $\beta_3$(1)=5).} \label{fig-exemple-carte3d} \end{figure} @@ -384,7 +387,7 @@ given dart, and how incidence and adjacency relations are defined in terms of darts. The first important property of a combinatorial map is that -each dart belongs to an \cell{i}, $\forall i$, $0 \leq i \leq d$. +each dart belongs to an \cell{i}, $\forall$ \emph{i}, 0 $\leq$ \emph{i} $\leq$ \emph{d}. For example in 3D, a dart belongs to a vertex, an edge, a facet, and a volume. This means that a 3D combinatorial map containing an isolated dart contains exactly one vertex, one edge, one facet and one volume. @@ -395,45 +398,45 @@ dart contains exactly one vertex, one edge, one facet and one volume. The second important property is that cells of a combinatorial map correspond to specific \emph{orbits}. Given a set -$S \subseteq \{\beta_1,\ldots,\beta_d\}$ and a dart -$d_0$, the \emph{orbit} $\orb{S}(d_0)$ is the set of darts that can be -reached from $d_0$ by following any combination of any $\beta_i$'s in $S$ +\emph{S} $\subseteq \{\beta_1,\ldots,\beta_d\}$ and a dart +\emph{d0}, the \emph{orbit} $\orb{S}$(\emph{d0}) is the set of darts that can be +reached from \emph{d0} by following any combination of any $\beta_i$'s in \emph{S} and their inverses (to simplify notations, we can use for example -$\orb{\beta_1,\beta_4}(d_0)$ to denote $\orb{S}(d_0)$ with -$S=\{\beta_1,\beta_4\}$). +$\orb{\beta_1,\beta_4}$(\emph{d0}) to denote $\orb{S}$(\emph{d0}) with +\emph{S}=$\{\beta_1,\beta_4\}$). -Given a dart $d_0$, in general, $\mb{i}(d_0)$ (with $1\leq i \leq d$) -belongs to the same cells as $d_0$, only the \cell{i} and 0-cell are -different. There are two exceptions: (1)~if $d_0$ is \emph{i}-free, then -$\mb{i}(d_0)=\varnothing$; (2)~if $\mb{i}(d_0)$ belongs to the same \cell{i} -as $d_0$ (case of multi-incidence). For example if an edge is an isolated -loop, it is incident twice to the same vertex, then given a dart $d_0$ -belonging to this edge, $\mb{1}(d_0)$ goes to the next edge, which is in +Given a dart \emph{d0}, in general, $\mb{i}$(\emph{d0}) (with 1$\leq$ \emph{i} $\leq$ d) +belongs to the same cells as \emph{d0}, only the \cell{i} and 0-cell are +different. There are two exceptions: (1)~if \emph{d0} is \emph{i}-free, then +$\mb{i}$(\emph{d0})=$\varnothing$; (2)~if $\mb{i}$(\emph{d0}) belongs to the same \cell{i} +as \emph{d0} (case of multi-incidence). For example if an edge is an isolated +loop, it is incident twice to the same vertex, then given a dart \emph{d0} +belonging to this edge, $\mb{1}$(\emph{d0}) goes to the next edge, which is in fact the same edge. -Since $\mb{i}(d_0)$ (with $1\leq i \leq d$) allows to change the -current \cell{i}, all the darts that can be reached from $d_0$ by -using any combination of $\mb{j}$'s, $\forall j$, $1 \leq j \leq d$ and -$j\neq i$ and their inverse are contained in the same \cell{i} as -$d_0$. The \cell{i} containing $d_0$ is defined in terms of orbit by -$\orb{\beta_1,\ldots,\beta_{i-1},\beta_{i+1},\ldots,\beta_d}(d_0)$. +Since $\mb{i}$(\emph{d0}) (with 1$\leq$ \emph{i} $\leq$ \emph{d}) allows to change the +current \cell{i}, all the darts that can be reached from \emph{d0} by +using any combination of $\mb{j}$'s, $\forall$ \emph{j}, 1 $\leq$ \emph{j} $\leq$ \emph{d} and +\emph{j}$\neq$ \emph{i} and their inverse are contained in the same \cell{i} as +\emph{d0}. The \cell{i} containing \emph{d0} is defined in terms of orbit by +$\orb{\beta_1,\ldots,\beta_{i-1},\beta_{i+1},\ldots,\beta_d}$(\emph{d0}). % since $\mb{0}$ is not a relation % (but only a notation for $\mb{1}^{-1}$). Thus, -There is a special case for vertices. Given a dart $d_0$, the set of -darts contained in the same vertex as $d_0$ are the darts that can be -reached from $d_0$ by using any combination of $\mb{i}\circ\mb{j}$, -$\forall i,j$, $1 \leq i< j \leq d$, and their inverse. The 0-cell -containing $d_0$ is defined in terms of orbit by -$\orb{\{\mb{i}\circ\mb{j}|\forall i,j: 1\leq i \end{ccHtmlOnly} - \caption{UML diagram of the main classes of the package. $k$ is the number of + \caption{UML diagram of the main classes of the package. \emph{k} is the number of non void attributes.} \label{fig-diagram_class} \end{figure} @@ -893,8 +896,8 @@ handle to the type of used darts (given in the items class). iterate over specific subsets of darts of the combinatorial map (see Section~\ref{ssec-range}). It also defines several methods to link and to unlink darts by $\beta_i$s (see -Section~\ref{ssec-link-darts}). We said that a dart $d_0$ is \emph{i}-free -if $\mb{i}(d_0)=\varnothing$. The $\varnothing$ constant is +Section~\ref{ssec-link-darts}). We said that a dart \emph{d0} is \emph{i}-free +if $\mb{i}$(\emph{d0})=$\varnothing$. The $\varnothing$ constant is represented in the class \ccc{Combinatorial_map} through a \ccc{static const Dart_handle} called \nulldart. Finally, some high levels @@ -941,8 +944,8 @@ the tuple must be either a model of the \ccc{CellAttribute} concept or \ccc{void}, \emph{(i-1)}-attributes are disabled: we say that \emph{(i-1)}-attributes are \emph{void}. Otherwise, \emph{(i-1)}-attributes are enabled and have the given type: we say \emph{(i-1)}-attributes are -\emph{non void}. If the size of the tuple is \emph{k}, with $k < -dimension+1$, $\forall i: k \leq i \leq dimension$, \emph{i}-attributes are +\emph{non void}. If the size of the tuple is \emph{k}, with \emph{k} < +dimension+1, $\forall$ \emph{i}: \emph{k} $\leq$ \emph{i} $\leq$ dimension, \emph{i}-attributes are void. The class \ccc{Combinatorial_map_min_items} is a model of the @@ -1114,7 +1117,7 @@ Sections~\ref{ssec-adv-marks}). In the following, we denote by The combinatorial map offers iterators to traverse the darts of a specific orbit, to traverse all darts of one cell, or -one dart per cell, and to traverse all $i$-attributes. +one dart per cell, and to traverse all \emph{i}-attributes. Instead of the \ccc{begin()/end()} member function pair as we know it from \stl\ containers, and from most \cgal\ data structures, the @@ -1125,16 +1128,16 @@ There are three different categories of dart range classes: \begin{itemize} \item \ccc{Dart_range}: range of all the darts of a combinatorial map; \item \ccc{Dart_of_orbit_range}: range of all the darts of - the orbit $\orb{Beta...}(d0)$ for a given $d0$. $Beta...$ is a - sequence of integers $i_1,\ldots,i_k$, each $i_j \in - \{0,\ldots,d\}$. These integers must satisfy: $i_1} for the orbit - $\orb{\mb{1},\mb{2}}(d0)$); + $\orb{\mb{1},\mb{2}}$(\emph{d0})); \item \ccc{Dart_of_cell_range}: range of all the darts of the \cell{i} containing a given dart. The \cell{i} is considered in - dimension \ccc{dim} (with $0 \leq dim \leq d$, $dim=d$ by default), - with $0\leq i \leq dim+1$. If $i=dim+1$, + dimension \ccc{dim} (with 0 $\leq$ \emph{dim} $\leq$ \emph{d}, \emph{dim}=\emph{d} by default), + with 0$\leq$ \emph{i} $\leq$ \emph{dim+1}. If \emph{i}=\emph{dim+1}, \ccc{Dart_of_cell_range} is the range of all the darts of the connected component containing a given dart. \end{itemize} @@ -1148,27 +1151,27 @@ There are three different categories of dart range classes: There are also two different classes of ranges containing one dart per \cell{i}. Note that in these classes, the dart of each \cell{i} can be any dart of the cell. Moreover, each \cell{i} (and \cell{j} in the -second case) is considered in dimension \ccc{dim} (with $0 \leq dim -\leq d$, $dim=d$ by default). +second case) is considered in dimension \ccc{dim} (with 0 $\leq$ \emph{dim} +$\leq$ \emph{d}, \emph{dim=d} by default). \begin{itemize} \item \ccc{One_dart_per_cell_range}: range containing one dart of - each \cell{i} of the combinatorial map, $0\leq i \leq dim+1$ (for + each \cell{i} of the combinatorial map, 0$\leq$ \emph{i} $\leq$ \emph{dim+1} (for example \ccc{One_dart_per_cell_range<2>} for the range of one dart per 2-cell of the combinatorial map); \item \ccc{One_dart_per_incident_cell_range}: range containing one dart of each \cell{i} incident to the \cell{j} - containing a given dart, with $0\leq i \leq dim+1$ and $0\leq j - \leq dim+1$ (for example + containing a given dart, with 0$\leq$ \emph{i} $\leq$ \emph{dim+1} and 0$\leq$ \emph{j} + $\leq$ \emph{dim+1} (for example \ccc{One_dart_per_incident_cell_range<0,3>} for the range of one dart per vertex of the volume incident to the starting dart). - If $i==j$, the range contains only the given dart. + If \emph{i}=\emph{j}, the range contains only the given dart. \end{itemize} The iterators of the \ccc{Dart_range} are bidirectional iterators, while the iterators of the other four ranges are forward iterators. The value type of all these iterators is \ccc{Dart}. -Additionally, there is a range over non void $i$-attributes: +Additionally, there is a range over non void \emph{i}-attributes: \ccc{Attribute_range::type}, having a bidirectional iterator with value type \ccc{Attribute_type::type}. @@ -1294,8 +1297,8 @@ $\beta$ pointers of existing darts. result of \ccc{sew<3>(1,5)} (or \ccc{sew<3>(2,8)}, or \ccc{sew<3>(3,7)}, or \ccc{sew<3>(4,6)}). Darts (1,5), (2,8), (3,7) and (4,6) are linked together by $\beta_3$. The two 2-cells - $c_1=\{1,2,3,4\}$ and $c_2=\{5,6,7,8\}$ are merged after the sew - into the 2-cell $\{1,2,3,4,5,6,7,8\}$. We are in the case where + \emph{c1}=\{1,2,3,4\} and \emph{c2}=\{5,6,7,8\} are merged after the sew + into the 2-cell \{1,2,3,4,5,6,7,8\}. We are in the case where the two attributes are non NULL, thus the first one is kept, and all the darts of $c_2$ are associated with the first attribute.} \label{fig-exemple-sew} @@ -1324,43 +1327,43 @@ The \ccc{link_beta} and \ccc{unlink_beta} methods only modify \end{ccAdvanced} \end{itemize} -Linking two darts $d_1$ and $d_2$ by $\beta_i$, with $2\leq i\leq d$ -and $d_1 \neq d_2$, consists in modifying two $\beta_i$ pointers such that -$\beta_i(d_1)=d_2$ and $\beta_i(d_2)=d_1$. For $i=1$, the modification -is $\beta_1(d_1)=d_2$ (and thus $\beta_0(d_2)=d_1$ by definition of -$\beta_0$); in this case we can have $d_1=d_2$ (a dart linked with +Linking two darts \emph{d1} and \emph{d2} by $\beta_i$, with 2$\leq$ \emph{i}$\leq$ \emph{d} +and \emph{d1} $\neq$ \emph{d2}, consists in modifying two $\beta_i$ pointers such that +$\beta_i$(\emph{d1})=\emph{d2} and $\beta_i$(\emph{d2})=\emph{d1}. For \emph{i}=1, the modification +is $\beta_1$(\emph{d1})=\emph{d2} (and thus $\beta_0$(\emph{d2})=\emph{d1} by definition of +$\beta_0$); in this case we can have \emph{d1=d2} (a dart linked with itself corresponds to an edge which is a loop). -Reciprocally, unlinking a given dart $d_0$ by $\beta_i$, with $2\leq -i\leq d$, consists in modifying two $\beta_i$ pointers -such that $\beta_i(\beta_i(d_0))=\varnothing$ and -$\beta_i(d_0)=\varnothing$. For $i=1$, the modification is -$\beta_1(d_0)=\varnothing$ (and thus -$\beta_0(\beta_1(d_0))=\varnothing$ by definition of $\beta_0$). Note +Reciprocally, unlinking a given dart \emph{d0} by $\beta_i$, with 2$\leq$ +\emph{i}$\leq$ \emph{d}, consists in modifying two $\beta_i$ pointers +such that $\beta_i(\beta_i$(\emph{d0}))=$\varnothing$ and +$\beta_i$(\emph{d0})=$\varnothing$. For \emph{i=1}, the modification is +$\beta_1$(\emph{d0})=$\varnothing$ (and thus +$\beta_0(\beta_1$(\emph{d0}))=$\varnothing$ by definition of $\beta_0$). Note that is it possible to unlink a given dart for $\beta_i$ only if it is -not $i$-free. +not \emph{i}-free. The \ccc{sew(dh1,dh2)} methods consist mainly to link two by two several darts by $\mb{i}$. This operation is possible only if there is -a bijection $f$ between all the darts of the orbit -$D_1=\orb{\mb{1},\ldots,\mb{i-2},\mb{i+2},\ldots,\mb{d}}(d_1)$ and -$D_2=\orb{\mb{1},\ldots,\mb{i-2},\mb{i+2},\ldots,\mb{d}}(d_2)$ -satisfying: $f(d_1)=d_2$, and for all $d'_1 \in D_1$, for all $j\in +a bijection \emph{f} between all the darts of the orbit +\emph{D1}=$\orb{\mb{1},\ldots,\mb{i-2},\mb{i+2},\ldots,\mb{d}}$(\emph{d1}) and +\emph{D2}=$\orb{\mb{1},\ldots,\mb{i-2},\mb{i+2},\ldots,\mb{d}}$(\emph{d2}) +satisfying: f(\emph{d1})=\emph{d2}, and for all \emph{e} $\in$ \emph{D1}, for all $j\in \{1,\ldots,i-2,i+2,\ldots,d\}$, -$f(\mb{j}(d'_1))=\mb{j}^{-1}(f(d'_1))$. Intuitively, this condition +\emph{f}($\mb{j}$(\emph{e}))=$\mb{j}^{-1}$(\emph{f}(\emph{e})). Intuitively, this condition ensures the validity of the combinatorial map by verifying that condition discussed in Section~\ref{ssec-combimap-validity} will be satisfied after the operation. This can be tested by using the method \ccc{is_sewable(dh1,dh2)}. For example, the function \ccc{is_sewable} would return \ccc{false} if we tried to sew a -triangular facet with a quad facet. Note that given two darts $d_1$ -and $d_2$, if there is such a bijection, it is uniquely defined. So giving +triangular facet with a quad facet. Note that given two darts \emph{d1} +and \emph{d2}, if there is such a bijection, it is uniquely defined. So giving the two darts as arguments of the \ccc{sew} is enough to retrieve all the pairs of darts to link. If such a bijection exists, the \ccc{sew(dh1,dh2)} operation -consists only in linking by $\beta_i$ each couple of darts $d_1$ and -$d_2$ such that $d_2=f(d_1)$. +consists only in linking by $\beta_i$ each couple of darts \emph{d1} and +\emph{d2} such that \emph{d2}=\emph{f}(\emph{d1}). % links by $\beta_i$ two by two all the darts of % the orbit $\orb{\mb{1},\ldots,\mb{i-2},\mb{i+2},\ldots,\mb{d}}(d1)$ @@ -1381,15 +1384,15 @@ In addition, the sew operations update the associations between darts and non void attributes in order to guarantee that all the darts belonging to a given cell are associated with the same attribute (which is a condition of combinatorial map validity). For each couple -of \cells{j} $c_1$ and $c_2$ that are merged into one \cell{j} during -the sew, we have to update the two associated attributes $attr_1$ and -$attr_2$. If both are NULL, there is nothing to do. If one is NULL +of \cells{j} \emph{c1} and \emph{c2} that are merged into one \cell{j} during +the sew, we have to update the two associated attributes \emph{attr1} and +\emph{attr2}. If both are NULL, there is nothing to do. If one is NULL and the other not, we only associate the non NULL attribute to all the darts of the resulting cell. When the two attributes are non NULL, we -first apply functor \ccc{On_merge} on the two attributes $attr_1$ and -$attr_2$ (see Section~\ref{ssec-attributes}). Then, we associate the -attribute $attr_1$ to all darts of the resulting \cell{j}. Finally, -attribute $attr_2$ is removed from the combinatorial map. +first apply functor \ccc{On_merge} on the two attributes \emph{attr1} and +\emph{attr2} (see Section~\ref{ssec-attributes}). Then, we associate the +attribute \emph{attr1} to all darts of the resulting \cell{j}. Finally, +attribute \emph{attr2} is removed from the combinatorial map. Note that when the two attributes are non NULL, the first one is kept. But user can customize this behavior in order to update the @@ -1402,34 +1405,34 @@ second attribute is kept. For example, in Figure~\ref{fig-exemple-sew}, we want to 3-sew the two initial volumes. \ccc{sew<3>(1,5)} links by $\mb{3}$ the pairs of -darts $(1,5)$, $(2,8)$, $(3,7)$ and $(4,6)$, thus the combinatorial map +darts (1,5), (2,8), (3,7) and (4,6), thus the combinatorial map obtained is valid. 2-attributes are updated so that all the darts belonging to the 2-cell containing dart 1 become associated to the same 2-attribute after the operation. % Similarly, \ccc{unsew} operations unlink $\mb{i}$ for all the darts -in the orbit $\orb{\mb{1},\ldots,\mb{i-2},\mb{i+2},\ldots,\mb{d}}(d_0)$, +in the orbit $\orb{\mb{1},\ldots,\mb{i-2},\mb{i+2},\ldots,\mb{d}}$(\emph{d0}), and thus guarantee to obtain a valid combinatorial map. This -operation is possible for any non $i$-free dart. +operation is possible for any non \emph{i}-free dart. As for the sew operations, attributes are updated to guarantee that two darts belonging to two different \cells{j} are -associated to two different $j$-attributes. If the unsew operation -splits a \cell{j} $c$ in two \cells{j} $c_1$ and $c_2$, and if $c$ is -associated to a $j$-attribute $attr_1$, then this attribute is duplicated -into $attr_2$, and all the darts belonging to $c_2$ are associated +associated to two different \emph{j}-attributes. If the unsew operation +splits a \cell{j} \emph{c} in two \cells{j} \emph{c1} and \emph{c2}, and if \emph{c} is +associated to a \emph{j}-attribute \emph{attr1}, then this attribute is duplicated +into \emph{attr2}, and all the darts belonging to \emph{c2} are associated with this new attribute. Finally, we call the functor \ccc{On_split} -on the two attributes $attr_1$ and $attr_2$ (see +on the two attributes \emph{attr1} and \emph{attr2} (see Section~\ref{ssec-attributes}). Let us consider the combinatorial map given in Figure~\ref{fig-exemple-sew} (Right). If we call \ccc{unsew<3>(2)}, we obtain the combinatorial map in Figure~\ref{fig-exemple-sew} (Left) (except for the color of the attribute associated to the -2-cell $\{5,6,7,8\}$ which would be \texttt{\#00ff00}). The \ccc{unsew<3>} +2-cell \{5,6,7,8\} which would be \texttt{\#00ff00}). The \ccc{unsew<3>} operation has duplicated the 2-attribute associated to the 2-cell -$\{1,2,3,4,5,6,7,8\}$ since this 2-cell is split in two after the +\{1,2,3,4,5,6,7,8\} since this 2-cell is split in two after the unsew operation. \begin{ccAdvanced} @@ -1467,9 +1470,9 @@ unsew operation. In Figure~\ref{fig-exemple-sew} (Left), if we call \ccc{link_beta<3>(1,5)}, in the resulting combinatorial map we have - now $\mb{3}(1)=5$ and $\mb{3}(5)=1$. This combinatorial map is no + now $\mb{3}$(1)=5 and $\mb{3}$(5)=1. This combinatorial map is no longer valid (for example dart 2 is 3-free and we should have - $\mb{3}(2)=8$). + $\mb{3}$(2)=8). \end{ccAdvanced} @@ -1499,7 +1502,7 @@ valid. The first one is \ccc{remove_cell(cm,dh0)} which modifies the combinatorial map to remove the \cell{i} containing dart \ccc{d0}, -with $0 \leq i \leq d$. This operation is possible if $i==d$ or if the given +with 0 $\leq$ \emph{i} $\leq$ \emph{d}. This operation is possible if \emph{i}=\emph{d} or if the given \cell{i} is incident to at most two \cells{(i+1)} which can be tested thanks to \ccc{is_removable(cm,dh0)}. If the removed \cell{i} was incident to two different \cells{(i+1)}, these two cells are @@ -1577,9 +1580,9 @@ example on Figure~\ref{fig-insert-vertex}). \ccc{remove_cell<1>} operations. \textbf{Left}:~Initial combinatorial map. \textbf{Right}:~After the insertion of two 1-cells: a first one between the two 0-cells containing darts - \ccc{d1} and \ccc{d2}, and a second one between the two 0-cells - containing darts \ccc{d3} and \ccc{d4}. Now if we remove the two - 1-cells containing darts \ccc{d5} and \ccc{d6}, we obtain the + \ccc{d2} and \ccc{d3}, and a second incident to the 0-cell + containing dart \ccc{d1}. Now if we remove the two + 1-cells containing darts \ccc{d4} and \ccc{d5}, we obtain the initial combinatorial map.} \label{fig-insert-edge} \end{figure} @@ -1587,12 +1590,12 @@ example on Figure~\ref{fig-insert-vertex}). \ccc{insert_cell_1_in_cell_2(cm,dh1,dh2)} adds a 1-cell in the 2-cell containing darts \ccc{d1} and \ccc{d2}, between the two 0-cells containing darts \ccc{d1} and \ccc{d2}. The 2-cell is split -in two. This operation is possible if $d1 \in \orb{\beta_1}(d2)$ +in two. This operation is possible if \emph{d1} $\in \orb{\beta_1}(d2)$ which can be tested thanks to \ccc{is_insertable_cell_1_in_cell_2(cm,dh1,dh2)}. In the example on Figure~\ref{fig-insert-edge}, it is possible to insert an edge -between darts $d1$ and $d2$, and between $d3$ and $d4$, but it is -not possible to insert an edge between $d1$ and $d3$. +between darts \emph{d1} and \emph{d2}, and between \emph{d3} and \emph{d4}, but it is +not possible to insert an edge between \emph{d1} and \emph{d3}. %\item \ccc{insert_dangling_cell_1_in_cell_2(cm,dh0)} adds a 1-cell in @@ -1754,7 +1757,7 @@ combinatorial map contains only one 3-cell. Indeed, the \ccc{sew<4>} operation has identified by pairs all the darts of the two 3-cells by definition of the sew operation (see Section~\ref{ssec-link-darts}) which, in 4D, links by $\beta_3$ all the darts in -$\orb{\mb{1},\mb{2}}(d_1)$ and in $\orb{\mb{1},\mb{2}}(d_2)$. The +$\orb{\mb{1},\mb{2}}$(\emph{d1}) and in $\orb{\mb{1},\mb{2}}$(\emph{d2}). The situation is similar (but in higher dimension) to the configuration where we have two triangles in a 3D combinatorial map, and you use \ccc{sew<3>} between these two triangles. The two triangles @@ -1836,47 +1839,47 @@ objects without boundaries. This definition was extended objects with boundaries, based on the notions of partial permutations and partial involutions. -Intuitively, a \emph{partial permutation} on a finite set $E$ is a -mapping from $E\cup\{\varnothing\}$ to $E\cup\{\varnothing\}$ which is +Intuitively, a \emph{partial permutation} on a finite set \emph{E} is a +mapping from \emph{E}$\cup$\{$\varnothing$\} to \emph{E}$\cup$\{$\varnothing$\} which is injective on the subset of the domain that does not map to -$\varnothing$. More precisely, a mapping $p:E \cup \{\varnothing\} -\rightarrow E \cup \{\varnothing\}$ is a \emph{partial permutation} -defined on $E$ if: +$\varnothing$. More precisely, a mapping p:E $\cup$ \{$\varnothing$\} +$\rightarrow$ \emph{E} $\cup$ \{$\varnothing$\} is a \emph{partial permutation} +defined on \emph{E} if: \begin{enumerate} -\item $p(\varnothing)=\varnothing$; -\item $\forall e_1 \in E$, $\forall e_2 \in E$, - $p(e_1)=p(e_2)\neq \varnothing \Rightarrow e_1=e_2$. +\item \emph{p}($\varnothing$)=$\varnothing$; +\item $\forall$ \emph{e1} $\in$ \emph{E}, $\forall$ \emph{e2} $\in$ \emph{E}, + p(\emph{e1})=p(\emph{e2})$\neq \varnothing \Rightarrow$ \emph{e1}=\emph{e2}. \end{enumerate} -The inverse $p^{-1}$ of this partial permutation is also a partial +The inverse \pmun{} of this partial permutation is also a partial permutation and is defined by: \begin{enumerate} -\item $p^{-1}(\varnothing)=\varnothing$; -\item $\forall e \in E$, if it exists $a\in E$ such that $p(a)=e$, - then $p^{-1}(e)=a$, otherwise $p^{-1}(e)=\varnothing$. +\item \pmun{}($\varnothing$)=$\varnothing$; +\item $\forall$ \emph{e} $\in$ \emph{E}, if it exists \emph{a}$\in$ \emph{E} such that \emph{p}(\emph{a})=\emph{e}, + then \pmun{}(\emph{e})=a, otherwise \pmun{}(\emph{e})=$\varnothing$. \end{enumerate} -Let $E$ be a set, and $p$ a partial permutation on $E$. An element -$e$ is called a \emph{fixed point} for $p$ if $p(e)=e$. $p$ is a -\emph{partial involution} if $\forall e \in E: p(e)\neq \varnothing -\Rightarrow p(p(e))=e$. +Let \emph{E} be a set, and \emph{p} a partial permutation on \emph{E}. An element +\emph{e} is called a \emph{fixed point} for \emph{p} if \emph{p(e)=e}. \emph{p} is a +\emph{partial involution} if $\forall$ \emph{e} $\in$ \emph{E}: \emph{p}(\emph{e})$\neq \varnothing +\Rightarrow$ \emph{p}(\emph{p}(\emph{e}))=\emph{e}. Now we can give the definition of a combinatorial map in any dimension. -Let $d \geq 0$. A \emph{d}-dimensional combinatorial map (or -\emph{d}-map) is a (d+1)-tuple $M=(D,\beta_1,\ldots,\beta_d)$ +Let \emph{d} $\geq$ 0. A \emph{d}-dimensional combinatorial map (or +\emph{d}-map) is a (d+1)-tuple \emph{M}=(\emph{D},$\beta_1,\ldots,\beta_d)$ where: \begin{enumerate} -\item $D$ is a finite set of darts; -\item $\beta_1$ is a partial permutation on $D$; -\item $\forall i, \, 2 \leq i \leq d$, $\beta_i$ is a - partial involution on $D$ without fixed point; -\item\label{cond-composition} $\forall i:\, 0 \leq i \leq d-2$, - $\forall j:\, 3 \leq j \leq d$, $i+2 \leq j$, +\item \emph{D} is a finite set of darts; +\item $\beta_1$ is a partial permutation on \emph{D}; +\item $\forall$ \emph{i}, 2 $\leq$ \emph{i} $\leq$ \emph{d}, $\beta_i$ is a + partial involution on \emph{D} without fixed point; +\item\label{cond-composition} $\forall$ \emph{i}: 0 $\leq$ \emph{i} $\leq$ \emph{d-2}, + $\forall$ \emph{j}: 3 $\leq$ \emph{j} $\leq$ \emph{d}, \emph{i+2} $\leq$ \emph{j}, $\beta_{i}\circ\beta_{j}$ is a partial involution. \end{enumerate} -A $d$-dimensional combinatorial map represents a subdivision of an -orientable $d$-dimensional quasi-manifold. A dart is an abstract element +A \emph{d}-dimensional combinatorial map represents a subdivision of an +orientable \emph{d}-dimensional quasi-manifold. A dart is an abstract element which is only required to define partial permutations. The last line of the definition fixes constraints which guarantee the topological validity of the represented object, i.e., the fact that it is a @@ -1884,18 +1887,18 @@ quasi-manifold. This definition allows us to verify the validity of a given combinatorial map by checking if each item of the definition is satisfied. -Given a set of partial permutations $S=\{f_1,\ldots,f_k\}$, we denote +Given a set of partial permutations \emph{S}=\{$f_1,\ldots,f_k$\}, we denote by $\orb{S}$ the \emph{permutation group} generated by -$\{f_1,\ldots,f_k\}$ and whose group operation is the composition of -partial permutations. The orbit $\orb{f_1,\ldots,f_k}(a)$ is the set -of darts which can be obtained from $a$ by elements of $\orb{S}$: -$\orb{f_1,\ldots,f_k}(a)=\{\phi(a)|\phi \in \orb{S}\}\setminus\{\varnothing\}$. +\{$f_1,\ldots,f_k$\} and whose group operation is the composition of +partial permutations. The orbit $\orb{f_1,\ldots,f_k}$(\emph{a}) is the set +of darts which can be obtained from \emph{a} by elements of $\orb{S}$: +$\orb{f_1,\ldots,f_k}$(\emph{a})=\{$\phi$(\emph{a})|$\phi \in \orb{S}$\}$\setminus$\{$\varnothing$\}. -Let $d_0 \in D$ be a dart. Given $i$, $1\leq i \leq d$, -the \cell{i} containing $d_0$ is -$\orb{\beta_1,\ldots,\beta_{i-1},\beta_{i+1},\ldots,\beta_d}(d_0)$. -The $0$-cell containing $d_0$ is -$\orb{\{\mb{i}\circ\mb{j}|\forall i,j: 1\leq i