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Changed structure of Reference Manual main page; taken Mael's comments into consideration, but followed the Ref page for P3M3
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Iordan Iordanov
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@ -40,7 +40,49 @@ triangulations and offers primitives to build the dual Voronoi diagrams.}
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\cgalPkgShortInfoEnd
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\cgalPkgDescriptionEnd
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\cgalModifBegin
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\cgal traditionally represents a triangulation via its faces and vertices. We represent a triangulation of the
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Bolza surface via the canonical representatives of its faces in the hyperbolic plane. This package provides the
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necessary objects and functions to work with Delaunay triangulations of the Bolza surface.
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Each vertex gives access to one of its incident faces, and stores an input point. We additionally allow each
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vertex to store (temporarily) a hyperbolic translation to facilitate the insertion of new points and the removal
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of existing vertices.
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Each face gives access to its three incident vertices and to its three adjacent faces. We enable a face
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to store additionally three hyperbolic translations. When applied to the three points stored in the vertices of
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the face, these translations produce the canonical representative of the face in the hyperbolic plane.
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The three vertices of a face are indexed with 0, 1, and 2 in positive (counter-clockwise) orientation. The
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orientation of faces on the Bolza surface is defined as the orientation of their canonical representatives in
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the hyperbolic plane.
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\cgalClassifedRefPages
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## Concepts ##
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The concepts in this package provide an interface for working with objects in the hyperbolic plane, including
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hyperbolic translations, which allow us to represent a triangulation of the Bolza surface. The concept
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`Periodic_4HyperbolicDelaunayTriangulationTraits_2` refines the concept `HyperbolicDelaunayTriangulationTraits_2`
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and adds the functionality needed to apply hyperbolic translations to geometric objects in the hyperbolic plane.
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- `Periodic_4HyperbolicDelaunayTriangulationTraits_2`
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The concepts `Periodic_4HyperbolicTriangulationDSFaceBase_2` and `Periodic_4HyperbolicTriangulationDSFaceBase_2`
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describe the requirements for the faces and the vertices of a triangulation of the Bolza surface. A face needs
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to be able to store hyperbolic translations as a canonical representative. A vertex needs to be able to store
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temporarily hyperbolic translations to facilitate insertion and removal procedures.
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- `Periodic_4HyperbolicTriangulationDSFaceBase_2`
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- `Periodic_4HyperbolicTriangulationDSVertexBase_2`
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Finally, the concept `HyperbolicOctagonTranslation` describes the requirements for an object that represents
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a hyperbolic translation.
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- `HyperbolicOctagonTranslation`
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## Classes ##
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The main class of the 2D Periodic Triangulation package is `CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2`.
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This class allows to construct and modify Delaunay triangulations of the Bolza surface. A secondary
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class made available is `CGAL::Periodic_4_hyperbolic_triangulation_2`, which gives access
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@ -48,55 +90,20 @@ to the elements of the triangulation (vertices, edges, and faces), and functiona
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triangulations in general, such as point location. The class `CGAL::Periodic_4_hyperbolic_triangulation_2`
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does <i>not</i> support insertion or removal of points. `CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2`
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inherits `CGAL::Periodic_4_hyperbolic_triangulation_2`.
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Both triangulation classes take a geometric traits and a triangulation data structure as template parameters.
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The geometric traits class must be a model of the concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2`
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to provide all predicates and constructions needed by the triangulation classes. The default triangulation data
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structure used is `CGAL::Triangulation_data_structure_2`.
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A triangulation is stored as a collection of vertices and faces that are linked together through incidence
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and adjacency relations. Each face gives access to its three incident vertices and to its three adjacent faces.
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Each vertex gives access to one of its incident faces. The three vertices of a face are indexed with 0, 1, and
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2 in positive (counter-clockwise) orientation. The orientation of faces on the Bolza surface is defined as
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the orientation of their canonical representatives in the underlying space, which is the hyperbolic plane
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\f$ \mathbb H^2\f$.
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To specify canonical representatives, which may contain vertices both inside and outside the original domain,
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we store additionally three hyperbolic translations, corresponding to each vertex. The hyperbolic translations
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are models of the concept `HyperbolicOctagonTranslation`.
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# Concepts #
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- `Periodic_4HyperbolicDelaunayTriangulationTraits_2`
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- `HyperbolicOctagonTranslation`
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- `Periodic_4HyperbolicTriangulationDSFaceBase_2`
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- `Periodic_4HyperbolicTriangulationDSVertexBase_2`
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- `HyperbolicOctagonTranslation`
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# Classes #
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## Main Classes ##
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- `CGAL::Periodic_4_hyperbolic_triangulation_2`
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- `CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2`
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## Traits Classes ##
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The package provides a model for the concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2`, which
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is the class `CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_traits_2`.
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- `CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_traits_2`
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## Vertex and Face Base Classes ##
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Models for the face and vertex concepts are provided as well.
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- `CGAL::Periodic_4_hyperbolic_triangulation_ds_face_base_2`
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- `CGAL::Periodic_4_hyperbolic_triangulation_ds_vertex_base_2`
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## Hyperbolic Translations Classes ##
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Finally, the class provides a model for the concept `HyperbolicOctagonTranslation`.
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- `CGAL::Hyperbolic_octagon_translation`
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## Enums ##
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- `CGAL::Periodic_4_hyperbolic_triangulation_2::Locate_type`
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\cgalModifEnd
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*/
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