Created 星期六 01 四月 2017
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2D Conforming Triangulations and Meshes
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Conforming Triangulations
Definitions
A triangulation is a Delaunay triangulation if the circumscribing circle of any facet of the triangulation contains no vertex in its interior. A constrained Delaunay triangulation is a constrained triangulation which is as much Delaunay as possible. The circumscribing circle of any facet of a constrained Delaunay triangulation contains in its interior no data point visible from the facet.
An edge is said to be a Delaunay edge if it is inscribed in an empty circle (containing no data point in its interior). This edge is said to be a Gabriel edge if its diametrical circle is empty.
A constrained Delaunay triangulation is said to be a conforming Delaunay triangulation if every constrained edge is a Delaunay edge. Because any edge in a constrained Delaunay triangulation is either a Delaunay edge or a constrained edge, a conforming Delaunay triangulation is in fact a Delaunay triangulation. The only difference is that some of the edges are marked as constrained edges.
A constrained Delaunay triangulation is said to be a conforming Gabriel triangulation if every constrained edge is a Gabriel edge. The Gabriel property is stronger than the Delaunay property and each Gabriel edge is a Delaunay edge. Conforming Gabriel triangulations are thus also conforming Delaunay triangulations.
Any constrained Delaunay triangulation can be refined into a conforming Delaunay triangulation or into a conforming Gabriel triangulation by adding vertices, called Steiner vertices, on constrained edges until they are decomposed into subconstraints small enough to be Delaunay or Gabriel edges.
Delaunay三角网
Constrained Delaunay triangulations can be refined into conforming triangulations by the two following global functions:
template<class CDT> void make_conforming_Delaunay_2 (CDT& t)
template<class CDT> void make_conforming_Gabriel_2 (CDT& t)
CGAL::Constrained_Delaunay_triangulation_2<K> CDT;