Reference Manual

Vincent Despré, Loïc Dubois, Camille Lanuel, Marc Pouget and Monique Teillaud

This package enables building and handling triangulations or Delaunay triangulations of closed orientable hyperbolic surfaces. It offers functions for the generation of the triangulation from a convex fundamental domain, the Delaunay flip algorithm and the construction of a portion of the lift of the triangulation in the Poincaré disk. A method is offered that generates such domains in genus two.

Introduced in: CGAL 6.1
Depends on: Combinatorial Maps
BibTeX: cgal:ddpt-thss-26b
License: GPL
Windows Demo: 2D Triangulations on Hyperbolic Surfaces

Classified Reference Pages

Note that the data structure for triangulations in this package is a CGAL::Combinatorial_map instead of the usual CGAL::Triangulation_data_structure_2 used for instance in the package 2D Periodic Hyperbolic Triangulations. In particular, this implies that vertices, edges and faces are encoded by darts of the combinatorial map so that most of the functions of the package are working with darts.

Concepts

• HyperbolicSurfaceTraits_2 is the concept for the template parameter of most classes of the package.
• HyperbolicSurfaceDelaunayTraits_2 is the concept for the template parameter of the class CGAL::Delaunay_triangulation_on_hyperbolic_surface_2.
• ComplexNumber describes a complex number type.

Classes

• CGAL::Triangulation_on_hyperbolic_surface_2 represents a triangulation of a hyperbolic surface. It offers functions for the generation of the triangulation from a convex fundamental domain, the Delaunay flip algorithm and the construction of a portion of the lift of the triangulation in the Poincaré disk.
• CGAL::Delaunay_triangulation_on_hyperbolic_surface_2 represents a Delaunay triangulation of a hyperbolic surface. It offers functions for the generation of the Delaunay triangulation from a convex fundamental domain, the insertion of a point in the Delaunay triangulation, two point location algorithms and the \( \varepsilon \)-net algorithm.
• CGAL::Hyperbolic_fundamental_domain_2 represents a fundamental domain of a hyperbolic surface.
• CGAL::Hyperbolic_fundamental_domain_factory_2 is a factory class, whose purpose is to generate some convex domains of surfaces of genus two.
• CGAL::Hyperbolic_isometry_2 represents an isometry in the Poincaré disk model. Facilities are offered to compose isometries, and apply an isometry to a point.

Models for HyperbolicSurfaceTraits_2, HyperbolicSurfaceDelaunayTraits_2 and ComplexNumber are provided: CGAL::Hyperbolic_surface_traits_2, CGAL::Hyperbolic_Surface_Delaunay_Traits_2 and CGAL::Complex_number.

Input/Output Functions

• operator<< and operator>> are overloaded for several classes of the package.

Dirichlet Domain Function

• CGAL::Dirichlet_vertices computes the vertices of a Dirichlet domain from a fundamental domain whose vertices represent a single point on the corresponding hyperbolic surface.

Topics
	Concepts
	Main Classes
	Traits Classes
	Input/Output Functions
	Dirichlet Domain Function