Nico Kruithof

This package allows to build and handle triangulations of point sets in the two dimensional flat torus. Triangulations are built incrementally and can be modified by insertion or removal of vertices. They offer point location facilities. The package provides Delaunay triangulations and offers nearest neighbor queries and primitives to build the dual Voronoi diagrams.

Introduced in: CGAL 4.3
Depends on: 2D Triangulations
BibTeX: cgal:k-pt2-13-24b
License: GPL
Windows Demo: Periodic Delaunay Triangulation

The main classes of the 2D Periodic Triangulation package are CGAL::Periodic_2_triangulation_2 and CGAL::Periodic_2_Delaunay_triangulation_2. They contain functionality to access triangulations and to run queries on them. CGAL::Periodic_2_Delaunay_triangulation_2 can construct and modify Delaunay triangulations. It takes the geometric traits as well as the triangulation data structure as template parameters.

The geometric traits class must be a model of the concept Periodic_2DelaunayTriangulationTraits_2. It contains all predicates and constructions that are needed by the functions in the triangulation classes.

The package uses CGAL::Triangulation_data_structure_2 to represent the triangulation. The faces and vertices need to be models of the concepts Periodic_2TriangulationFaceBase_2 and TriangulationVertexBase_2, respectively. A triangulation is stored as a collection of vertices and faces that are linked together through incidence and adjacency relations. Each face gives access to its three incident vertices and to its three adjacent cells. Each vertex gives access to one of its incident faces.

The three vertices of a cell are indexed with 0, 1, and 2 in positive orientation, the positive orientation being defined by the orientation of the underlying space \( \mathbb T_c^3\). The neighbors of a face are also indexed with 0, 1, 2 in such a way that the neighbor indexed by \( i\) is opposite to the vertex with the same index. See Figure 42.2.

In order to be able to specify the triangle that contain vertices both inside and outside the original domain we store additional offset information for each vertex of a face. These offsets are models of the concept Periodic_2Offset_2.