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Green Coordinates
Derives shape-preserving cage coordinates from Green's integral identity, inducing quasi-conformal mappings for realistic mesh deformation.
Abstract
We introduce Green Coordinates for closed polyhedral cages. The coordinates are motivated by Green's third integral identity and respect both the vertices position and faces orientation of the cage. We show that Green Coordinates lead to space deformations with a shape-preserving property. In particular, in 2D they induce conformal mappings, and extend naturally to quasi-conformal mappings in 3D. In both cases we derive closed-form expressions for the coordinates, yielding a simple and fast algorithm for cage-based space deformation. We compare the performance of Green Coordinates with those of Mean Value Coordinates and Harmonic Coordinates and show that the advantage of the shape-preserving property is not achieved at the expense of speed or simplicity. We also show that the new coordinates extend the mapping in a natural analytic manner to the exterior of the cage, allowing the employment of partial cages.
How to read this
- Category
- Method: shape-preserving cage-based deformation coordinates
- Contributions
- Green Coordinates for closed polyhedral cages, derived from Green's third integral identity, respecting both cage vertex positions and face orientations
- A shape-preserving property yielding conformal mappings in 2D and quasi-conformal mappings in 3D, with closed-form coordinate expressions
- Natural analytic extension of the deformation to the cage exterior, enabling partial cages
- Context
- Advances cage-based space deformation beyond Mean Value Coordinates (Ju et al.) and Harmonic Coordinates by adding a shape-preserving (quasi-conformal) guarantee.Builds on: Mean Value Coordinates for Closed Triangular Meshes
- Correctness
- Compared against Mean Value and Harmonic Coordinates and shown to gain shape preservation without sacrificing speed or simplicity; readers should note the coordinates depend on face orientation and that quasi-conformality is a property of the mapping, not a guarantee against all cage-design artifacts.
- Clarity
- Mathematically grounded but well-motivated; a first pass conveys the shape-preserving advantage, a second pass clarifies the derivation from Green's identity.
- How to read it
- Read the comparison figures against MVC and Harmonic Coordinates first to grasp the benefit, then a second pass on the derivation and closed-form expressions if implementing cage deformation.
Builds on
Related work
- Green Coordinates for Triquad Cages in 3D 2022 / SIGGRAPH Asia
- Harmonic Coordinates for Character Articulation 2007 / SIGGRAPH
- Mean Value Coordinates for Closed Triangular Meshes 2005 / SIGGRAPH
- Biharmonic Coordinates 2012 / CGF
Keywords
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