2D Triangulations on Hyperbolic Surfaces Reference

Vincent Despré, Loïc Dubois, Marc Pouget and Monique Teillaud

This package enables building and handling 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:y-t2-25a
License: GPL
Windows Demo: 2D Triangulations on Hyperbolic Surfaces

Classified Reference Pages

Concepts

• HyperbolicSurfaceTraits_2 is the concept for the template parameter of most classes of the package.
• ComplexNumber describes a complex number type.

Classes

• CGAL::Triangulation_on_hyperbolic_surface_2 represents a triangulation of a closed orientable 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::Hyperbolic_fundamental_domain_2 represents a fundamental domain of a closed orientable 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 and ComplexNumber are provided: CGAL::Hyperbolic_surface_traits_2 and CGAL::Complex_number.

Input/Output Functions

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

Modules
	Concepts
	Main Classes
	Traits Classes
	Input/Output Functions