CGAL 5.6.3 - 2D Hyperbolic Delaunay Triangulations

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2D Hyperbolic Delaunay Triangulations Reference

##

Mikhail Bogdanov, Iordan Iordanov, and Monique Teillaud
This package enables building and handling Delaunay triangulations of point sets in the Poincaré disk model of the hyperbolic plane. Triangulations are built incrementally and can be modified by insertion and removal of vertices; point location facilities are also offered, as well as primitives to build the dual Voronoi diagrams.
Introduced in: CGAL 4.14
Depends on: 2D Triangulations
BibTeX: cgal:bt-ht2-17-25b
License: GPL
Windows Demo: Hyperbolic Delaunay Triangulation

The Delaunay triangulation of a set of points \(P\) in the hyperbolic plane \(\mathbb H^2\) is a two-dimensional connected simplicial complex with vertex set defined by the points \(P\).

## Classified Reference Pages

## Concepts

- `HyperbolicDelaunayTriangulationTraits_2` describes the requirements for an interface for geometric objects, constructions, and predicates in the hyperbolic plane.
- `HyperbolicTriangulationFaceBase_2` describes the requirements for faces of the hyperbolic Delaunay triangulation.
- `HyperbolicFaceData` describes the requirements for an hyperbolic marker of faces of the hyperbolic Delaunay triangulation, used to filter faces of the Euclidean Delaunay triangulation.

## Classes

- `CGAL::Hyperbolic_Delaunay_triangulation_2` is the main class of the 2D Hyperbolic Triangulation package. It enables the constructions of Delaunay triangulations in the hyperbolic plane. It offers all the functionalities provided by `CGAL::Delaunay_triangulation_2`, such as point location, insertion, and removal. Construction of the dual Voronoi diagram is also provided.
- `CGAL::Hyperbolic_triangulation_face_base_2`

Two models for the concept `HyperbolicDelaunayTriangulationTraits_2` are provided:

- `CGAL::Hyperbolic_Delaunay_triangulation_traits_2` is by default based upon `CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt` and guarantees exact constructions of Delaunay triangulations and dual objects for input points with algebraic coordinates.
- `CGAL::Hyperbolic_Delaunay_triangulation_CK_traits_2` is based upon `CGAL::Circular_kernel_2` and guarantees exact constructions of Delaunay triangulations and dual objects when the input points have rational coordinates.

The model `CGAL::Hyperbolic_Delaunay_triangulation_CK_traits_2` is faster than `CGAL::Hyperbolic_Delaunay_triangulation_traits_2` for points with rational coordinates.

## Modules
| Concepts
| Main Classes
| Traits Classes
| Face Classes