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Green Coordinates for Triquad Cages in 3D
Extends Green coordinates to quadrilateral cage faces in 3D, providing quasi-conformal deformations with closed-form expressions for production cages.
Abstract
We introduce Green coordinates for triquad cages in 3D. Based on Green’s third identity, Green coordinates allow defining the harmonic deformation of a 3D point inside a cage as a linear combination of its vertices and face normals. Using appropriate Neumann boundary conditions, the resulting deformations are quasi-conformal in 3D, and thus best-preserve the local deformed geometry, in that volumetric conformal 3D deformations do not exist unless rigid. Most coordinate systems use cages made of triangles, yet quads are in general favored by artists as those align naturally onto important geometric features of the 3D shapes, such as the limbs of a character, without introducing arbitrary asymmetric deformations and representation. While triangle cages admit per-face constant normals and result in a single Green normal-coordinate per triangle, the case of quad cages is at the same time more involved (as the normal varies along non-planar quads) and more flexible (as many different mathematical models allow defining the smooth geometry of a quad interpolating its four edges). We consider bilinear quads, and we introduce a new Neumann boundary condition resulting in a simple set of four additional normal-coordinates per quad. Our coordinates remain quasi-conformal in 3D, and we demonstrate their superior behavior under non-trivial deformations of realistic triquad cages.
How to read this
- Category
- Method: a cage-based deformation coordinate system
- Contributions
- Green coordinates extended to triquad (bilinear quad) cages in 3D
- A new Neumann boundary condition giving a simple set of closed-form coordinates per quad
- Quasi-conformal volumetric deformations that best-preserve local deformed geometry
- Context
- Directly extends Lipman et al.'s Green Coordinates (2008) from triangle cages to quad cages, grounded in Green's third identity and harmonic deformation theory.Builds on: Green Coordinates
- Correctness
- Builds on the established result that volumetric conformal 3D deformations do not exist unless rigid, so it targets quasi-conformality; the method assumes bilinear (non-planar) quads and the varying normal makes the formulation more involved, a limitation to expect when reading the derivation.
- Clarity
- Mathematically dense; a first pass conveys why quads are favored and what quasi-conformal buys you, but the boundary-condition derivation needs a careful second or third pass.
- How to read it
- First pass for the motivation (quads align to character features) and the quasi-conformal claim; reserve a slow second pass for the Neumann condition and closed-form expressions if you intend to implement.
Builds on
- Green Coordinates 2008
Built upon by
Nothing yet.
Related work
- Green Coordinates 2008 / SIGGRAPH
- Mean Value Coordinates for Closed Triangular Meshes 2005 / SIGGRAPH
- Biharmonic Coordinates 2012 / CGF
- Somigliana Coordinates: an Elasticity-Derived Approach for Cage Deformation 2023 / SIGGRAPH
Keywords
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