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Polynomial 2D Green Coordinates for High-order Cages

Shibo Liu, Ligang Liu, Xiao-Ming Fu

TL;DR

This work addresses 2D shape deformation using high-order cages whose edges are polynomial Bézier curves. It extends classical Green and Cauchy coordinates to provide closed-form, conformal mappings for high-order cages by leveraging a residue-theorem-based integration, enabling both cage-based and point-to-point deformations with interactive editing. Key contributions include closed-form expressions for Cauchy coordinates and their derivatives on arbitrary-degree cage edges, extensions to Polynomial Green coordinates and rational polynomial cages, and an efficient C++ implementation with real-time performance and robust comparisons to prior methods. The results demonstrate smoother, more conformal deformations than prior straight-edge or non-closed-form approaches, and the framework enables direct manipulation of Bézier control points to achieve desired deformations, with clear pathways toward 3D extensions in future work.

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.

Polynomial 2D Green Coordinates for High-order Cages

TL;DR

This work addresses 2D shape deformation using high-order cages whose edges are polynomial Bézier curves. It extends classical Green and Cauchy coordinates to provide closed-form, conformal mappings for high-order cages by leveraging a residue-theorem-based integration, enabling both cage-based and point-to-point deformations with interactive editing. Key contributions include closed-form expressions for Cauchy coordinates and their derivatives on arbitrary-degree cage edges, extensions to Polynomial Green coordinates and rational polynomial cages, and an efficient C++ implementation with real-time performance and robust comparisons to prior methods. The results demonstrate smoother, more conformal deformations than prior straight-edge or non-closed-form approaches, and the framework enables direct manipulation of Bézier control points to achieve desired deformations, with clear pathways toward 3D extensions in future work.

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.
Paper Structure (37 sections, 3 theorems, 20 equations, 14 figures, 1 algorithm)

This paper contains 37 sections, 3 theorems, 20 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Cage coordinates
  4. Coordinates for high-order cages
  5. High-order meshes and cages
  6. Method
  7. Revisiting Cauchy coordinates
  8. Cauchy transform
  9. Cages
  10. Deriving Cauchy coordinates
  11. 2D Cauchy coordinates and their derivatives for high-order cages
  12. Enhanced deformation interactivity
  13. Holomorphicity on the boundary
  14. Derivation
  15. Closed-form solutions
  16. ...and 22 more sections

Key Result

Theorem 1

Suppose that $\partial \Omega$ is smooth and $u$ is a holomorphic function on $\Omega$ that extends to be a continuous function on $\Omega\cup \partial \Omega$. If the boundary values of $u$ are smooth on an open arc $\Gamma$ in $\partial\Omega$, then all the derivatives of $u$ extend continuously f

Figures (14)

  • Figure 1: Deformation via our coordinates and their derivatives. Upper: input images and cubic cages. Bottom: cage-based deformation from cubic to 7th-order cages (left) and point-to-point deformation (right).
  • Figure 2: Deformations using linear and high-order input cages. Upper: a cage with 72 linear segments exhibits self-intersections, preventing deformation. Middle: deformation via a 120-segment polygonal cage under Cauchy coordinates. Bottom: deformation with a 24-segment cubic cage using our coordinates. High-order input cages lead to more intuitive deformations with many fewer parameters.
  • Figure 3: Linear cage (a1) vs. quadratic rational cage (b1) with the same number of control points. Deformation using our coordinates (b2) is smoother than the linear cage-based deformation via Cauchy coordinates (a2) (see the difference in the red boxes).
  • Figure 4: The deformation of a high-order cage. An original curve $\mathbf{e}_j$ is deformed into $f^{\text{new}(\mathbf{e}_j)}$ and any point $z$ inside the cage is deformed to $u(z)$.
  • Figure 5: Polynomial Cauchy coordinates mapping a linear cage (a1, with non-endpoint control points on edges shown as solid dots) to a cubic cage lead to non-smoothness near the boundaries (a2). Our cubic input cage tightly encloses the shape (b1), generating smooth deformation (b2).
  • ...and 9 more figures

Theorems & Definitions (4)

  • Theorem 1: Theorem 3.3 in Bell2015cauchytransform
  • Lemma 1
  • Theorem 2
  • proof