Ridges, Ravines and Singularities

Complexity of the delaunay triangulation of points on surfaces the smooth case

“…In the following, S is assumed to be a compact smooth oriented surface without boundary embedded in 3 . At a point p ∈ S, N (p) denotes the oriented unit vector normal to S at p. ρ1(p) and ρ2(p), ρ1(p) ≤ ρ2(p), denote respectively the minimum and the maximum principal (signed) curvatures at p.…”

“…In this section, we no longer assume that ρ2 is the maximal principal curvature, which may be either ρ1 or ρ2. Following [3], we choose coordinates in 3 such that p0 is at the origin, the (x, y)-plane is the tangent plane to the surface at p0, and the principal directions coincide with the x and y axis. We choose the orientation such that ρ2 ≥ 0.…”

“…We choose the orientation such that ρ2 ≥ 0. Using almost the same notations as in [3], the surface is then expressible as the graph of the function fS :…”

“…It is well known that the complexity of the Delaunay triangulation of N points in 3 , i.e. the number of its faces, can be as large as Ω(N 2 ).…”

“…[1,5,8,9]. The time complexity of those methods therefore crucially depends on the complexity of the triangulation of points scattered over a surface in 3 . Moreover, since output-sensitive algorithms are known for computing Delaunay triangulations [6,7], better bounds on the complexity of the Delaunay triangulation would immediately imply improved bounds on the time complexity of computing the Delaunay triangulation.…”