mirror of https://github.com/CGAL/cgal
Fix some minor glitches in headings
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Philipp Möller
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967cc9b616
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6615bcee09
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@ -6,7 +6,6 @@ namespace CGAL {
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\anchor ChapterCombinatorialMap
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\author Guillaume Damiand
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\autotoc
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# Introduction # {#Combinatorial_mapIntroduction}
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@ -204,8 +203,7 @@ in a given building from rooms to rooms by traversing doors.
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\todo Solve the problem with the links to linear cell complex package that do not work.
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# Data Structure Presentation #
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\anchor sec_presentation
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# Data Structure Presentation # {#sec_presentation}
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In this section, we describe <I>d</I>D combinatorial maps in terms of data
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structure and operations. Mathematical definitions are provided in
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@ -213,8 +211,7 @@ Section \ref sec_definition "Mathematical Definitions", and a package
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description is given in
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Section \ref secsoftwaredesign "Software Design".
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## Combinatorial Map and Darts ##
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\anchor sseccombimapanddarts
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## Combinatorial Map and Darts ## {#sseccombimapanddarts}
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A <I>d</I>D combinatorial map is a set of darts <I>D</I>. A dart
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<I>d0</I> is an element that can be <I>linked</I> with <I>d</I>+1
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@ -263,8 +260,7 @@ with a little gray segment joining them (for example \f$
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segments (for example \f$ \beta_3\f$(1)=5).
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</b></center>
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## Cells as Sets of Darts ##
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\anchor sseccellsinmap
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## Cells as Sets of Darts ## {#sseccellsinmap}
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A cell in a <I>d</I>D combinatorial map is implicitly represented by a
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subset of darts. In this section, we will see how to retrieve all
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@ -425,9 +421,7 @@ obtain the set of darts {1,2,3,4}. Intuitively we "forget" \f$
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\beta_3\f$ and we obtain the set of darts of the facet containing dart
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1 restricted to the volume containing this dart.
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## How to Associate Information to Cells ##
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\anchor ssecassociateattributes
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## How to Associate Information to Cells ## {#ssecassociateattributes}
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Combinatorial maps only describe the cells of the subdvision, and all
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the incidence and adjacency relations between these cells. This is not
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@ -478,8 +472,7 @@ containing a color in RGB format. Only three 2-cells of the
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combinatorial map, among the ten, are associated to a 2-attribute.
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</b></center>
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## Combinatorial Map Properties ##
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\anchor sseccombimapvalidity
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## Combinatorial Map Properties ## {#sseccombimapvalidity}
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There are some conditions that a combinatorial map must satisfy to be
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valid. Some of them have already been given about the \f$ \beta\f$
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@ -587,8 +580,7 @@ example to modify locally a combinatorial map in a really fast way. In
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such a case, additional operations may be needed to restore these
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validity conditions.
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# Software Design # {#Combinatorial_mapSoftwarre}
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\anchor secsoftwaredesign
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# Software Design # {#secsoftwaredesign}
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The diagram in \ref figure9 "Figure 9" shows the different
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classes of the package. `Combinatorial_map` is the main class
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@ -606,8 +598,7 @@ accessed through <I>handles</I>. A handle is a model of the
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\anchor figure9
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\image html Diagramme_class.png "Figure 9: UML diagram of the main classes of the package. k is the number of non void attributes."
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## Combinatorial Maps ##
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\anchor sseccombinatorialmap
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## Combinatorial Maps ## {#sseccombinatorialmap}
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The class `Combinatorial_map<d,Items,Alloc>` is a model of the
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`CombinatorialMap` concept. It has three template parameters standing
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@ -653,8 +644,7 @@ for a handle to the <I>i</I>-attributes (and the const version
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\todo Add links to the corresponding types in CombinatorialMap concept asap the types were defined correctly in the concept.
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## Combinatorial Map Items ##
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\anchor ssecitem
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## Combinatorial Map Items ## {#ssecitem}
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The `CombinatorialMapItems` concept defines dart and attribute types
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of a combinatorial map. It contains one inner class named
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@ -688,8 +678,7 @@ The class `Combinatorial_map_min_items<d>` is a model of the
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behaviors. It defines `Dart<d,CMap>` as type of dart, and
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`Attributes` as empty tuple.
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## Darts ##
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\anchor ssecdarts
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## Darts ## {#ssecdarts}
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The class `Dart<d,CMap>`, a model of the `::Dart` concept, defines a
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<I>d</I>D dart. It has two template parameters standing for the
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@ -714,8 +703,7 @@ combinatorial map class. Users can provide their own model of the
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`::Dart` concept, and pass it to the combinatorial map with the help of
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a custom item class.
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## Cell Attributes ##
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\anchor ssecattributes
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## Cell Attributes ## {#ssecattributes}
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The class `Cell_attribute<CMap,Info_,Tag,OnMerge,OnSplit>`, a model of
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the `CellAttribute` concept, represents an attribute associated with a
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@ -757,8 +745,7 @@ What we said for the dart also holds for the cell attribute. The
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combinatorial map can be used with any user defined model of the
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`CellAttribute` concept.
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## Example of Combinatorial Map Definition ##
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\anchor ssecexampledef
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## Example of Combinatorial Map Definition ## {#ssecexampledef}
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Here comes an example of two combinatorial map definitions. The first
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case `Example_cmap4` defines a 4D combinatorial map which uses all the
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@ -806,8 +793,7 @@ following, we denote by
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`dh0`, `dh1`, `dh2` the dart handles for the darts `d0`, `d1`, `d2`,
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respectively. That is `d0 == *dh0`.
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## Iterating over Orbits, Cells, and Attributes ##
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\anchor ssecrange
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## Iterating over Orbits, Cells, and Attributes ## {#ssecrange}
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The combinatorial map offers iterators to traverse the darts of a
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specific orbit, to traverse all darts of one cell, or one dart per
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@ -889,8 +875,7 @@ For each range, there is an associated const range, a model of the
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`ConstRange` concept. You can find some examples of ranges in Section
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\ref ssecexample3DCM "A 3D Combinatorial Map".
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## Construction Operations ##
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\anchor ssecconstruction
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## Construction Operations ## {#ssecconstruction}
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Several global functions allow to create specific configurations of
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darts into a combinatorial map. Existing darts in the combinatorial
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@ -917,8 +902,7 @@ together by \f$ \beta_2\f$); dimension must be greater or equal than
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two.
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</UL>
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## Boolean Marks ##
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\anchor ssecadvmarks
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## Boolean Marks ## {#ssecadvmarks}
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\advanced It is often necessary to mark darts, for example to retrieve
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in <I>O(1)</I> if a given dart was already processed during a specific
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@ -960,7 +944,7 @@ to know which darts come from the first and second tetrahedron.
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\cgalexample{Combinatorial_map/map_3_marks.cpp}
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# Modification Operations #
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# Modification Operations # {#ssecmodoperations}
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Several operations allow to modify a given combinatorial map.
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There are two main categories of modification operations:
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@ -973,8 +957,7 @@ Section \ref sseclinkdarts "Sewing Orbits and Linking Darts");
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\ref ssecoperations "Removal and Insertion Operations").
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</UL>
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## Sewing Orbits and Linking Darts ##
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\anchor sseclinkdarts
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## Sewing Orbits and Linking Darts ## {#sseclinkdarts}
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The `CombinatorialMap` defines two groups of methods to modify the
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\f$ \beta\f$ pointers of existing darts.
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@ -1148,15 +1131,14 @@ attributes. In \ref figure10 "Figure 10" (Left), if we call
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no longer valid (for example dart 2 is 3-free and we should have \f$
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\beta_3\f$(2)=8).
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## Removal and Insertion Operations ##
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\anchor ssecoperations
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## Removal and Insertion Operations ## {#ssecoperations}
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The following high level operations are defined as global functions
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taking an instance `cm` of `CombinatorialMap` as first argument. All
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these methods ensure that given a valid combinatorial map and a
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possible operation, the modified combinatorial map is also valid.
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The first one is `remove_cell<CMap,i>(cm,dh0)` which modifies the
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The first one is `::remove_cell<CMap,i>(cm,dh0)` which modifies the
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combinatorial map to remove the <I>i</I>-cell containing dart `d0`,
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with 0\f$ \leq\f$<I>i</I>\f$ \leq\f$<I>d</I>. This operation is
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possible if <I>i</I>=<I>d</I> or if the given <I>i</I>-cell is
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@ -1248,8 +1230,7 @@ ssecexempleoperations "High Level Operations".
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# Examples # {#Combinatorial_mapExamples}
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## A 3D Combinatorial Map ##
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\anchor ssecexample3DCM
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## A 3D Combinatorial Map ## {#ssecexample3DCM}
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In this example, a 3-dimensional combinatorial map is constructed. Two
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combinatorial tetrahedra are created, then the numbers of cells of the
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@ -1381,9 +1362,7 @@ The output is:
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#Darts=24, #0-cells=4, #1-cells=6, #2-cells=4, #3-cells=1, #4-cells=2, #ccs=1, valid=1
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\endverbatim
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## Combinatorial Map With Attributes ##
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\anchor sseccombimapwithcolor
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## Combinatorial Map With Attributes ## {#sseccombimapwithcolor}
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In the following example, we illustrate how to specify the
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2-attributes in a 3D combinatorial map. For that, we define our own
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@ -1427,8 +1406,7 @@ contained in 2-attributes in an `int`). At the end, we obtain five
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2-attributes with 7 as value, five 2-attributes with 13 as value, and
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four 2-attributes having respectively 2, 2, 5 and 10 as values.
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# Mathematical Definitions # {#Combinatorial_mapMathematical}
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\anchor sec_definition
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# Mathematical Definitions # {#sec_definition}
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The initial definition of combinatorial map in any dimension is given
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in \cite cgal:l-tmbrc-91, \cite l-ndgcm-94. But it allows only to
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@ -86,7 +86,7 @@ UML diagram of the main classes of the package. Gray elements come from the \ref
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</b></center>
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\section sseclinearcellcomplex Linear Cell Complex
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# Linear Cell Complex # {#sseclinearcellcomplex}
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The `CGAL::Linear_cell_complex<d,d2,LCCTraits,Items,Alloc>` class
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is a model of the `CombinatorialMap` concept. It guarantees that
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@ -192,7 +192,7 @@ linear cell complex. As for a combinatorial map, it is also possible
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to use low level operations but additional operations may be needed to
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restore the validity conditions.
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\subsection sseclcclinkdarts Sewing and Unsewing
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## Sewing and Unsewing ## {#sseclcclinkdarts}
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As explained in the combinatorial map user manual,
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Section \ref sseclinkdarts, it is possible to glue two <I>i</I>-cells
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@ -3,7 +3,7 @@
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/*!
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\addtogroup PkgProperty_map
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\PkgDescriptionBegin{CGAL and Boost Property Maps,Pkg:Property_mapSummary}
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\PkgDescriptionBegin{CGAL and Boost Property Maps,PkgProperty_mapSummary}
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\PkgPicture{property_map.png}
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\PkgAuthors{Andreas Fabri and Laurent Saboret}
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\PkgDesc{This package provides a framework for interfacing \cgal data structures with algorithms expecting Boost Property Maps.}
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