Minimal Delaunay triangulations of hyperbolic surfaces

Abstract

Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we will show that every hyperbolic surface of genus g has a simplicial Delaunay triangulation with O(g) vertices, where edges are given by distance paths. Then, we will construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we will show that the Ω( √ g) lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus g is tight for hyperbolic surfaces as well.

Delaunay triangulations of generalized Bolza surfaces

“…This can also be seen in the following way. It has been shown that every hyperbolic surface of genus g has a simplicial Delaunay triangulation with at most 151g vertices [22]. In particular, this upper bound does not depend on sys(M).…”

“…∠(p 1 0 p * 0 p 1 1 ) = π − 2∠(p 1 0 p * 0 p 0 0 ). (22) Equations ( 17) and (18) follow from the construction in the proof of Lemma 9 and from Theorem 2, respectively. Equation (19) holds because ∠(Op 0 0 v 1 ) = π 2 and the triangles [O, p 0 0 , p 0 1 ] and [v 1 , p 0 0 , p 0 1 ] are congruent.…”

“…Equation ( 21) is the sine rule [6,Chapter 7.12] in triangle [p 1 0 , p 0 0 , p * 0 ]. Finally, Equation (22) holds by symmetry. The circumradius R of an isosceles triangle with legs of length b and vertex angle α (see Figure 27 satisfies…”