Harmonic Coordinates

Tony DeRose, Mark Meyer


Generalizations of barycentric coordinates in two and higher dimensions have been shown to have a number of applications in recent years, including finite element analysis, the definition of S-patches (n-sided generalizations of Bezier surfaces), free-form deformations, mesh parametrization, and interpolation. In this paper we present a new form of d dimensional generalized barycentric coordinates. The new coordinates are defined as solutions to Laplace's equation subject to carefully chosen boundary conditions. Since solutions to Laplace's equation are called harmonic functions, we call the new construction harmonic coordinates. We show that harmonic coordinates possess several properties that make them more attractive than mean value coordinates when used to define two and three dimensional deformations.

Additional materials: [movie.mp4]

Available as Pixar Technical Memo #06-02

Other versions: