Green Coordinates (patent pending) |
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Abstract: |
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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. |
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Technical Report: |
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