Variational Green and Biharmonic Coordinates for 2D Polynomial Cages

ACM Transaction on Graphics (SIGGRAPH ' Proceedings)
Élie Michel
Alec Jacobson
Siddhartha Chaudhuri
Jean-Marc Thiery
Teaser image
We enable the use of rest cages made of polynomial curves for conformal cage-based deformation (a). Previous methods were limited to straight rest cages, making it cumbersome to fit some rest shapes like the bird’s beak (b). Furthermore, we provide the derivatives of our coordinates so they can be used in variational solvers. Note that Green coordinate foster conformal regularity over strict interpolation of the cage transform..

Abstract

We present closed-form expressions for Green and biharmonic coordinates with respect to polynomial curved 2D cages, enabling reliable cage-based image deformation both to and from a curved cage. We further provide closed-form expressions for first- and second-order derivatives of these coordinates with respect to the encoded position. This enables the use of variational solvers for interacting with the 2D shape at arbitrary points while keeping the fast decoding strength of cage-based deformation, which we illustrate for a variety of elastic deformation energies.

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Citation

@article{michel25variational, title = {Variational Green and Biharmonic Coordinates for 2D Polynomial Cages}, author = {Michel, Élie and Jacobson, Alec and Chaudhuri, Siddhartha and Thiery, Jean-Marc}, journal = {ACM Transaction on Graphics (SIGGRAPH '25 Proceedings)}, year = {2025}, publisher = {ACM}, doi = {10.1145/3731421}, }