Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Conformal Surface Splines

Yousuf Soliman, Ulrich Pinkall, Peter Schröder

TL;DR

A family of boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems and demonstrate the applicability of this framework to geometric modeling, mathematical visualization, and form finding.

Abstract

We introduce a family of boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems. Our free boundary conditions fix the metric on the boundary, up to a global scale, and admit a discretization compatible with discrete conformal equivalence. We also introduce constraints on the conformal scale factor, enforcing rigidity of the geometry in regions of interest, and describe how in the presence of point constraints the conformal class encodes knot points of the spline that can be directly manipulated. To control the tangent planes, we introduce flux constraints balancing the internal material stresses. The collection of these point constraints provide intuitive controls for exploring a subspace of conformal immersions interpolating a fixed set of points in space. We demonstrate the applicability of our framework to geometric modeling, mathematical visualization, and form finding.

Conformal Surface Splines

TL;DR

A family of boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems and demonstrate the applicability of this framework to geometric modeling, mathematical visualization, and form finding.

Abstract

We introduce a family of boundary conditions and point constraints for conformal immersions that increase the controllability of surfaces defined as minimizers of conformal variational problems. Our free boundary conditions fix the metric on the boundary, up to a global scale, and admit a discretization compatible with discrete conformal equivalence. We also introduce constraints on the conformal scale factor, enforcing rigidity of the geometry in regions of interest, and describe how in the presence of point constraints the conformal class encodes knot points of the spline that can be directly manipulated. To control the tangent planes, we introduce flux constraints balancing the internal material stresses. The collection of these point constraints provide intuitive controls for exploring a subspace of conformal immersions interpolating a fixed set of points in space. We demonstrate the applicability of our framework to geometric modeling, mathematical visualization, and form finding.

Paper Structure

This paper contains 21 sections, 8 theorems, 73 equations, 23 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Discrete Conformal Equivalence
  5. Discrete Willmore Energy
  6. Discrete Conformal Variational Problems
  7. Conformal Constraints
  8. Conformal Knot Points
  9. Total Torsion
  10. Flux Constraints
  11. Scale Constraints
  12. Conservation Laws
  13. Quaternions, Circles, and Spheres
  14. Free Boundary Conditions
  15. Numerics and Evaluation
  16. ...and 6 more sections

Key Result

Theorem 1

A smooth conformal immersion $f:\Sigma\to\mathbb{R}^3$ of a compact Riemann surface $\Sigma$ is a critical point of $\mathcal{F}$ under all infinitesimal conformal variations if and only if there exists a holomorphic quadratic differential $q\in H^0(K^2)$ so that Here, $\mathop{\mathrm{grad}}\limits\mathcal{F}(f)\in\Omega^2(\Sigma)$ is the gradient 2-form of $\mathcal{F}$ defined so that for all

Figures (23)

  • Figure 1: Manipulating the scale factor at the vertices of an icosahedron inscribed in a sphere simultaneously can be used to construct conformal Christmas ornaments.
  • Figure 2: Designing ribbons with conformal surface splines results in a large solution space of interpolating surfaces parameterized by the conformal structure of the immersion punctured at the user-specified point constraints. Fixing the metric on the boundary, up to a scale, provides effective boundary control without explicit specification of the extrinsic geometry. Top row: stretching the conformal type of a rectangular strip results in surfaces that resemble extrusions of planar elastic curves with length constraints. Bottom row: shearing the conformal type produces more complicated deformations.
  • Figure 3: The additional conformal parameters that arise when specifying a surface with point constraints correspond to their positions in the parameter domain. Pulling these conformal knots together in $M$ results in a pulling more material between the points extrinsically.
  • Figure 4: From top to bottom. Row 1: unconstrained Willmore minimization. Row 2: Willmore minimization under a soft conformal constraint. Bubbling occurs even though the surface remains approximately conformal throughout the flow. Row 3: Willmore minimization under area, volume, and conformality constraints. All three of these constraints ensure both the stability of the thin tube around a centerline and the total torsion of the curve. Row 4: Willmore minimization under area and volume constraints. The stability of the thin tube persists, but the total torsion of the curve is not conserved.
  • Figure 5: Prescribing the normal vector at an isolated point in Willmore minimization is not well-posed, illustrated by the sequence of Willmore surfaces converging to a flat disk with a shrinking boundary component encoding the desired normal direction.
  • ...and 18 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1: Bohle:2008:CWS
  • Theorem 2: Soliman:2021:CWS
  • Definition 4
  • Lemma 1: Muller:2021:DCT Lemma 9, 10
  • Proposition 1
  • proof
  • Remark 1
  • ...and 8 more