A spectral characterization of the Delaunay triangulation
Computer Aided Geometric Design，2010，27（4）： | 2010年05月01日 | doi.org/10.1016/j.cagd.2010.02.002
The Delaunay triangulation of a planar point set is a fundamental construct in computational geometry. A simple algorithm to generate it is based on flips of diagonal edges in convex quads. We characterize the effect of a single edge flip in a triangulation on the geometric Laplacian of the triangulation, which leads to a simpler and shorter proof of a theorem of Rippa that the Dirichlet energy of any piecewise-linear scalar function on a triangulation obtains its minimum on the Delaunay triangulation. Using Rippa's theorem, we provide a spectral characterization of the Delaunay triangulation, namely that the spectrum of the geometric Laplacian is minimized on this triangulation. This spectral theorem then leads to a simpler proof of a theorem of Musin that the harmonic index also obtains its minimum on the Delaunay triangulation.