Minimal roughness property of the Delaunay triangulation
Computer Aided Geometric Design
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Minimal roughness property of the Delaunay triangulation: a shorter approach
Computer Aided Geometric Design
Properties of the Delaunay triangulation
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
A Discrete Laplace–Beltrami Operator for Simplicial Surfaces
Discrete & Computational Geometry
A Monotonicity Property for Weighted Delaunay Triangulations
Discrete & Computational Geometry
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
Discrete Laplacians on general polygonal meshes
ACM SIGGRAPH 2011 papers
Extremum problems for eigenvalues of discrete Laplace operators
Computer Aided Geometric Design
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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.