Polynomial 2D Biharmonic Coordinates for High-order Cages

TL;DR

This paper introduces a method to compute biharmonic coordinates for 2D high-order cages, allowing flexible polynomial deformations of shapes, which enhances intuitive control and geometric alignment.

Contribution

It derives closed-form biharmonic coordinate expressions for high-order 2D cages using the boundary element method, enabling advanced shape manipulation.

Findings

Effective deformation of 2D shapes demonstrated

Enhanced control over boundary and interior geometry

Practical application on various deformation scenarios

Abstract

We derive closed-form expressions of biharmonic coordinates for 2D high-order cages, enabling the transformation of the input polynomial curves into polynomial curves of any order. Central to our derivation is the use of the high-order boundary element method. We demonstrate the practicality and effectiveness of our method on various 2D deformations. In practice, users can easily manipulate the Bezier control points to perform the desired intuitive deformation, as the biharmonic coordinates provide an enriched deformation space and encourage the alignment between the boundary cage and its interior geometry.