Mean value coordinates for arbitrary planar polygons

Barycentric coordinates for triangles are commonly used in computer graphics, geometric modeling, and other computational sciences because they provide a convenient way to linearly interpolate the data that is given at the corners of a triangle. The concept of barycentric coordinates can also be extended in several ways to convex polygons with more than three vertices, but most of these constructions break down when used in the nonconvex setting. Mean value coordinates offer a choice that is not limited to convex configurations, and we show that they are in fact well-defined for arbitrary planar polygons without self-intersections. Besides their many other important properties, these coordinate functions are smooth and allow an efficient and robust implementation. They are particularly useful for interpolating data that is given at the vertices of the polygons and we present several examples of their application to common problems in computer graphics and geometric modeling.

[1]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[2]  Frank Zeilfelder,et al.  Developments in bivariate spline interpolation , 2000 .

[3]  Kai Hormann,et al.  A quadrilateral rendering primitive , 2004, Graphics Hardware.

[4]  Martin D. Buhmann,et al.  Radial Basis Functions , 2021, Encyclopedia of Mathematical Geosciences.

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

[6]  Jianyun Chai,et al.  Contour interpolation and surface reconstruction of smooth terrain models , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[7]  M. Ehrgott,et al.  Geometric methods to solve max-ordering location problems , 1999 .

[8]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[9]  Kai Hormann,et al.  A general construction of barycentric coordinates over convex polygons , 2006, Adv. Comput. Math..

[10]  R. Sibson,et al.  A brief description of natural neighbor interpolation , 1981 .

[11]  Elisabeth Anna Malsch,et al.  Interpolations for temperature distributions: a method for all non-concave polygons , 2004 .

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

[13]  Tao Ju,et al.  Mean value coordinates for closed triangular meshes , 2005, ACM Trans. Graph..

[14]  R. Beatson,et al.  Fast evaluation of radial basis functions: I , 1992 .

[15]  K. Mardia,et al.  A review of image-warping methods , 1998 .

[16]  Mathieu Desbrun,et al.  A geometric construction of coordinates for convex polyhedra using polar duals , 2005, SGP '05.

[17]  Nira Dyn,et al.  Image Warping by Radial Basis Functions: Application to Facial Expressions , 1994, CVGIP Graph. Model. Image Process..

[18]  Sung Yong Shin,et al.  Scattered Data Interpolation with Multilevel B-Splines , 1997, IEEE Trans. Vis. Comput. Graph..

[19]  Heinrich Müller,et al.  Image warping with scattered data interpolation , 1995, IEEE Computer Graphics and Applications.

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

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

[22]  Gerald E. Farin,et al.  A Transfinite Form of Sibson's Interpolant , 1999, Discret. Appl. Math..

[23]  Richard K. Beatson,et al.  Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods , 2000, SIAM J. Sci. Comput..

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

[25]  Adam Finkelstein,et al.  A framework for geometric warps and deformations , 2002, TOGS.

[26]  Tony DeRose,et al.  Multiresolution analysis of arbitrary meshes , 1995, SIGGRAPH.

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

[28]  Gerald E. Farin,et al.  Surfaces over Dirichlet tessellations , 1990, Comput. Aided Geom. Des..

[29]  J. Warren,et al.  Mean value coordinates for closed triangular meshes , 2005, SIGGRAPH 2005.

[30]  John Lin,et al.  Smooth Two-Dimensional Interpolations: A Recipe for All Polygons , 2005, J. Graph. Tools.

[31]  Giovanni Ceva,et al.  De lineis rectis se invicem secantibus statica constructio , 1980 .

[32]  J. Warren.  On the Uniqueness of Barycentric Coordinates , 2003 .

[33]  Kokichi Sugihara,et al.  Voronoi-based interpolation with higher continuity , 2000, SCG '00.

[34]  Michael S. Floater,et al.  Mean value coordinates , 2003, Comput. Aided Geom. Des..

[35]  Elisabeth Anna Malsch,et al.  Shape functions for polygonal domains with interior nodes , 2004 .

[36]  HormannKai,et al.  Mean value coordinates for arbitrary planar polygons , 2006 .

[37]  N. Sukumar,et al.  Archives of Computational Methods in Engineering Recent Advances in the Construction of Polygonal Finite Element Interpolants , 2022 .

[38]  George Wolberg,et al.  Digital image warping , 1990 .

[39]  Sung Yong Shin,et al.  Image metamorphosis using snakes and free-form deformations , 1995, SIGGRAPH.

[40]  R. Sibson.  A vector identity for the Dirichlet tessellation , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

[41]  Ognyan Kounchev,et al.  Multivariate Polysplines: Applications to Numerical and Wavelet Analysis , 2001 .