Polynomial Cauchy Coordinates for Curved Cages

Polynomial Cauchy Coordinates for Curved Cages

(SIGGRAPH Asia ' Conference Proceedings)
Zhehui Lin
Renjie Chen
Teaser image
Fig. 1. (a) Input shape and its corresponding polygonal cage; (b) Deformation result using Cauchy coordinates [Weber et al. 2009]; (c) Deformation result using our Polynomial Cauchy coordinates; (d) Result of using our coordinates for P2P deformation; (e) Inverse mapping result obtained using our method to recover the original shape from (c). (f1-3) Deformation between curved cages using our method.

Abstract

Barycentric coordinates are widely used in computer graphics, especially in shape deformation. Traditionally, barycentric coordinates are defined for polygonal domains. In this work, we relax this requirement by representing the boundary of the domain using piecewise Bézier curves and extend the complex-valued Cauchy barycentric coordinates [Weber et al. 2009] to the Bézier case. Compared to the existing polynomial 2D Green coordinates [Michel and Thiery 2023], we obtain equivalent results. We further derive a numerical integration formula for the inverse mapping based on Cauchy’s integral formula, enabling deformation between curved cages through an intermediate step. Notably, our approach allows curved cages as input. We also present expressions for the nth-order derivatives of the coordinates, which facilitate constrained deformations based on position constraints. Through extensive experiments, we demonstrate the versatility of our coordinates for interactive deformation.

Citation

@inproceedings{lin2024polynomial, title={Polynomial Cauchy Coordinates for Curved Cages}, author={Lin, Zhehui and Chen, Renjie}, booktitle={SIGGRAPH Asia 2024 Conference Papers}, pages={1--8}, year={2024} }