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Polynomial 2D Biharmonic Coordinates for High-order Cages

Shibo Liu, Ligang Liu, Xiao-Ming Fu

TL;DR

This work addresses 2D shape deformation using closed-form biharmonic coordinates for high-order cages and derives analytic expressions by applying a high-order boundary element method. By accommodating polynomial cage edges of arbitrary order, the method extends classical biharmonic coordinates and provides a flexible deformation space that preserves boundary alignment while offering boundary-value and derivative control, via analytically computed boundary integrals involving $G_1$, $G_2$, and $\Delta f$. The approach enables deformations from curved cages to curved curves and supports a tunable balance between conformality and interpolation through energies and weights, demonstrated on diverse 2D deformations. While effective in 2D, the framework invites extension to 3D and incorporation of variational or artist-friendly controls for broader applications.

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.

Polynomial 2D Biharmonic Coordinates for High-order Cages

TL;DR

This work addresses 2D shape deformation using closed-form biharmonic coordinates for high-order cages and derives analytic expressions by applying a high-order boundary element method. By accommodating polynomial cage edges of arbitrary order, the method extends classical biharmonic coordinates and provides a flexible deformation space that preserves boundary alignment while offering boundary-value and derivative control, via analytically computed boundary integrals involving , , and . The approach enables deformations from curved cages to curved curves and supports a tunable balance between conformality and interpolation through energies and weights, demonstrated on diverse 2D deformations. While effective in 2D, the framework invites extension to 3D and incorporation of variational or artist-friendly controls for broader applications.

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.
Paper Structure (32 sections, 1 theorem, 16 equations, 12 figures)

This paper contains 32 sections, 1 theorem, 16 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Interpolating coordinates
  4. Non-interpolating coordinates
  5. Extension to high-order cages
  6. Boundary element method in graphics
  7. method
  8. Boundary element method
  9. BEM overview
  10. Biharmonic Dirichlet problem
  11. Boundary integral equations
  12. Biharmonic Coordinates for linear cages
  13. Polynomial Biharmonic Coordinates
  14. Reformulating boundary integral equations
  15. High-order elements
  16. ...and 17 more sections

Key Result

Lemma 1

For $p(t) = \prod_{i=1}^{n} (t - \omega_i)^{n_i}$, the following holds:

Figures (12)

  • Figure 1: Deformation using our coordinates. Given an input image and a polygonal cage (a1), our coordinates enable deformations to be high-order cages with polynomial curves of different orders (3 in (a2) and 4 in (a3)). Moreover, we can transform a cubic input high-order cage (b1) into cubic (b2).
  • Figure 2: Comparison with Cubic MVC and PolyGC on the Pants shape with a polygonal cage (a). (b) Cubic MVC Li2013 interpolates the boundary but causes severe visual artifacts. (c) PolyGC MichelThiery2023 produces conformal deformation but misses the boundary alignment. (d) Our coordinates provide a favorable trade-off between boundary alignment and deformation distortion control.
  • Figure 3: Given an input shape enclosed by a cubic high-order cage (a), our coordinates (c) lead to better alignment between the cage and the shape boundary than liu2024polygc (b).
  • Figure 4: Using the same number of control points, deforming the curved cage with 10 cubic curves (c) achieves smoother and more intuitive editing than deforming the polygonal cage having 40 straight segments (b).
  • Figure 5: High-order boundary element method. The curved boundary is composed of a series of cubic elements $\mathbf{c}_i$, on which $\mathbf{g}_1$ is a cubic polynomial function.
  • ...and 7 more figures

Theorems & Definitions (1)

  • Lemma 1