Mean value coordinates for closed triangular meshes

Constructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics. Such an interpolant has many uses in applications such as shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh P, we show that these coordinates are continuous everywhere and smooth on the interior of P. The coordinates are linear on the triangles of P and can reproduce linear functions on the interior of P. To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.

[1]  W. Fleming.  Functions of Several Variables , 1965 .

[2]  Thomas W. Sederberg,et al.  Free-form deformation of solid geometric models , 1986, SIGGRAPH.

[3]  Tony DeRose,et al.  A multisided generalization of Bézier surfaces , 1989, TOGS.

[4]  Sabine Coquillart,et al.  Extended free-form deformation: a sculpturing tool for 3D geometric modeling , 1990, SIGGRAPH.

[5]  Joe D. Warren,et al.  Barycentric coordinates for convex polytopes , 1996, Adv. Comput. Math..

[6]  Kenneth I. Joy,et al.  Free-form deformations with lattices of arbitrary topology , 1996, SIGGRAPH.

[7]  Michael S. Floater,et al.  Parametrization and smooth approximation of surface triangulations , 1997, Comput. Aided Geom. Des..

[8]  Michael S. Floater,et al.  Parametric Tilings and Scattered Data Approximation , 1998, Int. J. Shape Model..

[9]  K. Hormann,et al.  MIPS: An Efficient Global Parametrization Method , 2000 .

[10]  Mark Meyer,et al.  Intrinsic Parameterizations of Surface Meshes , 2002, Comput. Graph. Forum.

[11]  Mark Meyer,et al.  Generalized Barycentric Coordinates on Irregular Polygons , 2002, J. Graphics, GPU, & Game Tools.

[12]  Andrei Khodakovsky,et al.  Globally smooth parameterizations with low distortion , 2003, ACM Trans. Graph..

[13]  Katsutoshi Ootsubo,et al.  t-FFD: free-form deformation by using triangular mesh , 2003, SM '03.

[14]  Elisabeth Anna Malsch,et al.  Algebraic Construction of Smooth Interpolants on Polygonal Domains , 2003 .

[15]  M. Floater.  Mean value coordinates , 2003, Computer Aided Geometric Design.

[16]  John M. Schreiner,et al.  Inter-surface mapping , 2004, SIGGRAPH 2004.

[17]  Kai Hormann,et al.  Barycentric coordinates for arbitrary polygons in the plane , 2005 .

[18]  Kai Hormann,et al.  Surface Parameterization: a Tutorial and Survey , 2005, Advances in Multiresolution for Geometric Modelling.

[19]  Martin Reimers,et al.  Mean value coordinates in 3D , 2005, Comput. Aided Geom. Des..

[20]  Mathieu Desbrun,et al.  Barycentric coordinates for convex sets , 2007, Adv. Comput. Math..