Abstract
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.
Supplemental Material
- Beyer, W. H. 1987. CRC Standard Mathematical Tables (28th Edition). CRC Press.Google Scholar
- Coquillart, S. 1990. Extended free-form deformation: a sculpturing tool for 3d geometric modeling. In SIGGRAPH '90: Proceedings of the 17th annual conference on Computer graphics and interactive techniques, ACM Press, 187--196. Google ScholarDigital Library
- Desbrun, M., Meyer, M., and Alliez, P. 2002. Intrinsic Parameterizations of Surface Meshes. Computer Graphics Forum 21, 3, 209--218.Google ScholarCross Ref
- Fleming, W., Ed. 1977. Functions of Several Variables. Second edition. Springer-Verlag.Google Scholar
- Floater, M. S., and Hormann, K. 2005. Surface parameterization: a tutorial and survey. In Advances in Multiresolution for Geometric Modelling, N. A. Dodgson, M. S. Floater, and M. A. Sabin, Eds., Mathematics and Visualization. Springer, Berlin, Heidelberg, 157--186.Google Scholar
- Floater. M. S., Kos., G., and Reimers, M. 2005. Mean value coordinates in 3d. To appear in CAGD. Google ScholarDigital Library
- Floater, M. 1997. Parametrization and smooth approximation of surface triangulations. CAGD 14, 3, 231--250. Google ScholarDigital Library
- Floater, M. 1998. Parametric Tilings and Scattered Data Approximation. International Journal of Shape Modeling 4, 165--182.Google ScholarCross Ref
- Floater, M. S. 2003. Mean value coordinates. Comput. Aided Geom. Des. 20, 1, 19--27. Google ScholarDigital Library
- Hormann, K., and Greiner, G. 2000. MIPS - An Efficient Global Parametrization Method. In Curves and Surfaces Proceedings (Saint Malo, France), 152--163.Google Scholar
- Hormann, K. 2004. Barycentric coordinates for arbitrary polygons in the plane. Tech. rep., Clausthal University of Technology, September. http://www.in.tuclausthal.de/ hormann/papers/barycentric.pdf.Google Scholar
- Khodakovsky, A., Litke, N., and Schroeder, P. 2003. Globally smooth parameterizations with low distortion. ACM Trans. Graph. 22, 3, 350--357. Google ScholarDigital Library
- Kobayashi, K. G., and Ootsubo, K. 2003. t-ffd: free-form deformation by using triangular mesh. In SM '03: Proceedings of the eighth ACM symposium on Solid modeling and application, ACM Press, 226--234. Google ScholarDigital Library
- Loop, C., and Derose, T. 1989. A multisided generalization of Bézier surfaces. ACM Transactions on Graphics 8, 204--234. Google ScholarDigital Library
- MacCracken, R., and Joy, K. I. 1996. Free-form deformations with lattices of arbitrary topology. In SIGGRAPH '96: Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, ACM Press, 181--188. Google ScholarDigital Library
- Malsch, E., and Dasgupta, G. 2003. Algebraic construction of smooth interpolants on polygonal domains. In Proceedings of the 5th International Mathematica Symposium.Google Scholar
- Meyer, M., Lee, H., Barr, A., and Desbrun, M. 2002. Generalized Barycentric Coordinates for Irregular Polygons. Journal of Graphics Tools 7, 1, 13--22. Google ScholarDigital Library
- Schreiner, J., Asirvatham, A., Praun, E., and Hoppe, H. 2004. Inter-surface mapping. ACM Trans. Graph. 23, 3, 870--877. Google ScholarDigital Library
- Sederberg, T. W., and Parry, S. R. 1986. Free-form deformation of solid geometric models. In SIGGRAPH '86: Proceedings of the 13th annual conference on Computer graphics and interactive techniques, ACM Press, 151--160. Google ScholarDigital Library
- Wachspress, E. 1975. A Rational Finite Element Basis. Academic Press, New York.Google Scholar
- Warren, J., Schaefer, S., Hirani, A., and Desbrun, M. 2004. Barycentric coordinates for convex sets. Tech. rep., Rice University.Google Scholar
- Warren, J. 1996. Barycentric Coordinates for Convex Polytopes. Advances in Computational Mathematics 6, 97--108.Google ScholarCross Ref
Index Terms
- Mean value coordinates for closed triangular meshes
Recommendations
Mean Value Coordinates for Closed Triangular Meshes
Seminal Graphics Papers: Pushing the Boundaries, Volume 2Constructing 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 ...
Mean value coordinates for quad cages in 3D
Space coordinates offer an elegant, scalable and versatile framework to propagate (multi-)scalar functions from the boundary vertices of a 3-manifold, often called a cage, within its volume. These generalizations of the barycentric coordinate system ...
Mean value coordinates for closed triangular meshes
SIGGRAPH '05: ACM SIGGRAPH 2005 PapersConstructing 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 ...
Comments