CGAL 6.1 - 2D Periodic Hyperbolic Triangulations

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##

Iordan Iordanov and Monique Teillaud
This package enables building and handling triangulations of point sets on the two dimensional hyperbolic Bolza surface. Triangulations are built incrementally and can be modified by insertion or removal of vertices. Point location facilities are also offered. The package provides Delaunay triangulations and offers primitives to build the dual Voronoi diagrams.
Introduced in: CGAL 4.14
Depends on: 2D Hyperbolic Delaunay Triangulations
BibTeX: cgal:i-p4ht2-17-25b
License: GPL
Windows Demo: Periodic Hyperbolic Delaunay Triangulation

CGAL traditionally represents a triangulation via its faces and vertices. We represent a triangulation of the Bolza surface via the canonical representatives of its faces in the hyperbolic plane. This package provides the necessary objects and functions to work with Delaunay triangulations of the Bolza surface.

Each vertex gives access to one of its incident faces, and stores an input point. We additionally allow each vertex to store (temporarily) a hyperbolic translation to facilitate the insertion of new points and the removal of existing vertices.

Each face gives access to its three incident vertices and to its three adjacent faces. We enable a face to store additionally three hyperbolic translations. When applied to the three points stored in the vertices of the face, these translations produce the canonical representative of the face in the hyperbolic plane.

The three vertices of a face are indexed with 0, 1, and 2 in positive (counterclockwise) orientation. The orientation of faces on the Bolza surface is defined as the orientation of their canonical representatives in the hyperbolic plane.

## Classified Reference Pages

## Concepts

- `Periodic_4HyperbolicTriangulationTraits_2` adds supplementary requirements to the concept `HyperbolicDelaunayTriangulationTraits_2` that are necessary for periodic hyperbolic triangulations.
- `Periodic_4HyperbolicDelaunayTriangulationTraits_2` refines the concept `Periodic_4HyperbolicTriangulationTraits_2` by adding requirements necessary for the computation of periodic hyperbolic Delaunay triangulations.
- `Periodic_4HyperbolicTriangulationFaceBase_2` describes the requirements for faces to store hyperbolic translations.
- `Periodic_4HyperbolicTriangulationVertexBase_2` describes the requirements for vertices to store hyperbolic translations.

## Classes

- `CGAL::Periodic_4_hyperbolic_triangulation_2` gives access to the elements of the triangulation (vertices, edges, and faces), and functionality common to triangulations in general, such as point location. This class does not support insertion or removal of points.
- `CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_2` enables the construction and modification of Delaunay triangulations of the Bolza surface.
- `CGAL::Periodic_4_hyperbolic_Delaunay_triangulation_traits_2` is a model for the concept `Periodic_4HyperbolicDelaunayTriangulationTraits_2`.
- `CGAL::Periodic_4_hyperbolic_triangulation_face_base_2` is a model of the concept `Periodic_4HyperbolicTriangulationFaceBase_2` and represents a face of the triangulation.
- `CGAL::Periodic_4_hyperbolic_triangulation_vertex_base_2` is a model of the concept `Periodic_4HyperbolicTriangulationVertexBase_2` and represents a vertex of the triangulation.
- `CGAL::Hyperbolic_octagon_translation` is a class that represent hyperbolic translations of the fundamental group \(\mathcal G\) of the Bolza surface.

## Modules
| Concepts
| Main Classes
| Traits Classes
| Vertex and Face Base Classes
| Hyperbolic Translations Classes