Delaunay Quadrangulation by Two-coloring Vertices

Abstract

We introduce a bichromatic Delaunay quadrangulation principle by assigning the vertices of a Delaunay triangulation one of two colors, then discarding edges between vertices of the same color. We present algorithms for generating quadrangulations using this principle and simple refinements. The global vertex coloring ensures that only local refinements are needed to get all quads. This is in contrast to triangle-pairing algorithms, which get stuck with isolated triangles that require global refinement. We present two new sphere-packing algorithms for generating the colored triangulation, and we may also take as input a Delaunay refinement mesh and color it arbitrarily. These mesh non-convex planar domains with provable quality: quad angles in [10o,174o] and edges in [0.1, 2]r. The algorithms extend to curved surfaces and graded meshes. The “random” algorithm generates points with blue noise. The “advancing-front” algorithm produces large patches of boundary-aligned square tilings.