Terminal-edges Delaunay (small-angle based) algorithm for the quality triangulation problem

Abstract

The terminal-edge Delaunay algorithm, initially called Lepp–Delaunay algorithm, deals with the construction of size-optimal (adapted to the geometry) quality triangulation of complex objects. In two dimensions, the algorithm can be formulated in terms of the Delaunay insertion of both midpoints of terminal edges (the common longest-edge of a pair of Delaunay triangles) and midpoints of boundary related edges in the current mesh. For the processing of a small angled triangle in the current mesh, the terminal-edge is found as the final longest-edge of the finite chain of triangles that neighbor on a longest edge — the longest edge propagating path of the small angled triangle. Three boundary-related point insertion operations, which prevent nonconvergence behavior, are discussed in detail. The triangle improvement properties of the point insertion operations are used to prove that optimal-size triangulations, with smallest-angle greater than or equal to 30° are always produced.