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Mean Value Coordinates for Closed Triangular Meshes
Extends mean value coordinates to closed triangular meshes in 3D, enabling smooth and natural cage-based space deformations.
Abstract
This paper generalizes mean value coordinates from closed 2D polygons to closed triangular meshes in 3D, constructing an interpolant that extends function values defined at mesh vertices to the interior. The resulting coordinates are continuous everywhere, smooth on the interior, linear on the triangles, and reproduce linear functions, giving them linear precision. The authors derive a stable closed-form evaluation based on the mean vector of a spherical triangle and handle degenerate and coplanar cases robustly. They demonstrate the coordinates on boundary value interpolation, volumetric texturing, and cage-based surface deformation for character animation, where moving the vertices of an enclosing control mesh induces smooth deformations of the embedded model.
How to read this
- Category
- Method: a cage-based space-deformation coordinate scheme
- Contributions
- Generalizes mean value coordinates from closed 2D polygons to closed triangular meshes in 3D
- Provides an interpolant that is continuous everywhere, smooth on the interior, linear on triangles, and reproduces linear functions (linear precision)
- Derives a stable closed-form evaluation from the mean vector of a spherical triangle with robust handling of degenerate and coplanar cases
- Context
- Extends the 2D mean value coordinates lineage to 3D cages, situating it among barycentric/generalized-coordinate methods for boundary interpolation and cage-based deformation.
- Correctness
- Demonstrated on boundary value interpolation, volumetric texturing, and cage-based character deformation; the coordinates are well-defined for closed triangular cages, but mean value coordinates can be negative, so readers should not assume locality or positivity guarantees.
- Clarity
- Clear and self-contained; a first pass conveys the construction, a second pass clarifies the spherical-triangle derivation and degenerate-case handling.
- How to read it
- First pass for the properties and the deformation use case; a focused second pass on the closed-form evaluation and edge cases is worthwhile if you plan to implement the coordinates.
Builds on
Nothing in the archive, this is a starting point.
Related work
- Green Coordinates for Triquad Cages in 3D 2022 / SIGGRAPH Asia
- Green Coordinates 2008 / SIGGRAPH
- Biharmonic Coordinates 2012 / CGF
- Wires: A Geometric Deformation Technique 1998 / SIGGRAPH
Keywords
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