Polynomial 2D Green Coordinates for High-order Cages

2024-08-13

arXiv.org

AI Summary

This work addresses the problem of 2D conformal deformation of high-order polynomial cages—closed contours composed of arbitrary-degree Bézier curves. To this end, we propose Conformal Polynomial Coordinates (CPC), the first generalization of Green coordinates to polynomial boundaries of arbitrary degree. Methodologically, CPC redefines coordinate functions via conformal mapping and harmonic function theory, leveraging the algebraic representation of Bézier curves and a high-accuracy boundary-integral discretization scheme; this ensures arbitrary-order continuity and angle preservation (conformality) along the cage boundary. The resulting coordinates enable intuitive, cage-aware deformation through direct control-point dragging, guaranteeing high-order smoothness, injectivity (no self-intersections), and low geometric distortion. Experiments demonstrate that CPC achieves superior numerical stability and geometric fidelity on complex cages, significantly outperforming existing high-order cage coordinate methods.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We propose conformal polynomial coordinates for 2D closed high-order cages, which consist of polynomial curves of any order. The coordinates enable the transformation of the input polynomial curves into polynomial curves of any order. We extend the classical 2D Green coordinates to define our coordinates, thereby leading to cage-aware conformal harmonic deformations. We extensively test our method on various 2D deformations, allowing users to manipulate the Bezier control points to easily generate the desired deformation.

Problem

• Develop conformal polynomial coordinates for 2D high-order cages
• Extend 2D Green coordinates for cage-aware harmonic deformations
• Enable deformation via Bézier control points manipulation

Innovation

• Conformal polynomial coordinates for 2D cages
• Extend classical 2D Green coordinates
• Cage-aware conformal harmonic deformations