Mean value coordinates for closed triangular meshes

By Tao Ju, Scott Schaefer, Joe D. Warren. ACM Transactions on Graphics, 2005

713 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Computer Graphics and Computer-Aided Design

View Paper: https://www.semanticscholar.org/paper/9f8f5b70570d97c9334eeed475bcd262d0e8a683

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In Computational Mechanics

In Computer Graphics and Computer-Aided Design

Citations in the corpus

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, background): “On projecting the boundary curve onto the unit circle, an expression for φ(x) that is valid for convex as well as nonconvex domains is obtained in terms of the curve parameter t ∈ [0, 1] [83]:”
Harmonic coordinates for character articulation (2007, neutral): “Following Ju et al. [2005], we require H[ f ](p) to be exact for linear functions.”
Geometric modeling in shape space (2007, background): “2004], coordinates with respect to superposed control structures [Ju et al. 2005], the Poisson equation [Xu et al.”
New perspectives on polygonal and polyhedral finite element methods (2014, methodology): “mean value coordinates to higher-dimensional polytopes have been realized [30, 80, 81], and they have been used in finite element computations [43, 44].”
Accelerated quadratic proxy for geometric optimization (2016, methodology): “, barycentric coordinates, however, any linear or non-linear subspace method can be used instead [Ju et al. 2005; Wang et al. 2015].”
Subspace gradient domain mesh deformation (2006, methodology): “Since [Ju et al. 2005] requires a closed mesh, we close the two open ends of the constrained segment by adding as two virtual vertices (c1 and c2 in Figure 4) the centroids of the boundary curves of the open ends.”
Geometric skinning with approximate dual quaternion blending (2008, methodology): “Another important class of methods is based on generalization of barycentric coordinates [Ju et al. 2005; Joshi et al. 2007].”
Mean value coordinates for arbitrary planar polygons (2006, methodology): “Note that most of these properties (in particular the affine precision and the Lagrange property) can also be derived from the integral construction that was used in [Ju et al. 2005] to derive mean value coordinates in three dimensions.”