Polynomial 2D Green Coordinates for High-order Cages

TL;DR

This paper introduces conformal polynomial coordinates for 2D high-order cages, enabling flexible transformations and deformations of polynomial curves, with applications in shape manipulation.

Contribution

It extends classical Green coordinates to polynomial curves, allowing high-order cage deformations and conformal harmonic transformations in 2D.

Findings

Effective manipulation of Bezier control points for desired deformations

Extension of Green coordinates to polynomial curves of any order

Demonstrated versatility on various 2D deformation tests

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.