[Paper Review] Approximation of Polyhedral Surface Uniformization

Yaron Lipman|arXiv (Cornell University)|Jan 27, 2013

Advanced Numerical Analysis Techniques10 references3 citations

TL;DR

This paper presents a provably convergent algorithm for approximating the conformal uniformization of disk-type polyhedral surfaces to an equilateral triangle domain using simplicial maps on progressively refined triangulations. By constructing convex spaces of quasiconformal maps with bounded dilatation, the method ensures local uniform convergence to the true uniformization map, providing the first such convergence guarantee for general polyhedral surfaces with arbitrary polygonal faces.

ABSTRACT

We present a constructive approach for approximating the conformal map (uniformization) of a polyhedral surface to a canonical domain in the plane. The main tool is a characterization of convex spaces of quasiconformal simplicial maps and their approximation properties. As far as we are aware, this is the first algorithm proved to approximate the uniformization of general polyhedral surfaces.

Motivation & Objective

• To develop a constructive algorithm that approximates the conformal uniformization map of a general polyhedral surface to a canonical domain in the plane.
• To establish convergence guarantees for the approximation of the uniformization map under refinement of the surface mesh.
• To characterize a convex space of quasiconformal simplicial maps that ensures stability and convergence to the true conformal structure.
• To provide a framework that works for polyhedral surfaces with arbitrary-shaped polygonal faces, not restricted to specific geometric forms.
• To fix the uniformization map uniquely via boundary vertex constraints mapping to the corners of an equilateral triangle domain.

Proposed method

• The method uses regular subdivision of the polyhedral surface by connecting mid-edges of each triangle, creating a sequence of increasingly refined triangulations $\mathcal{S}^0 \prec \mathcal{S}^1 \prec \cdots \prec \mathcal{S}^q$.
• Simplicial maps $\Phi^q$ are constructed as piecewise-affine, continuous mappings from $\mathcal{S}^q$ to the complex plane $\mathbb{C}$, forming the space $\mathcal{F}^{\mathcal{S}^q}$.
• A convex space $U^q$ of $K$-quasiconformal homeomorphisms is defined such that each $\Phi^q \in U^q$ maps $\mathcal{S}$ onto the equilateral triangle domain $\mathcal{T}$ with uniform dilatation bound $K$ independent of refinement level $q$.
• The construction relies on knowing the argument of the derivative $\Psi'$ of the true uniformization map $\Psi$ up to an error of $\pm(\frac{\pi}{2} - \varepsilon)$, enabling stable map construction.
• The method ensures that $\Phi^q \circ \Psi^{-1}$ converges uniformly to the identity on compact subsets of the interior of $\mathcal{T}$, proving local uniform convergence.
• Geometric analysis using complex power maps $z \mapsto z^\gamma$ and binomial expansions is used to prove that no triangle flips under the map, ensuring orientation preservation.

Experimental results

Research questions

• RQ1 Can a constructive algorithm be designed to approximate the conformal uniformization of a general polyhedral surface with arbitrary polygonal faces?
• RQ2 Is it possible to prove convergence of such an approximation to the true uniformization map under mesh refinement?
• RQ3 What conditions on the derivative of the uniformization map ensure the stability and non-flipping of simplicial approximations?
• RQ4 How can the uniformization map be uniquely fixed among all conformal equivalences using only boundary vertex constraints?
• RQ5 What role does quasiconformal theory play in constructing a convex space of approximating maps with guaranteed convergence?

Key findings

• The algorithm constructs a sequence of simplicial maps $\Phi^q$ on increasingly refined triangulations $\mathcal{S}^q$ that converge locally uniformly to the true uniformization map $\Psi$.
• Each map $\Phi^q$ is a $K$-quasiconformal homeomorphism onto the equilateral triangle domain $\mathcal{T}$, with $K$ independent of the refinement level $q$.
• The convergence is proven via the uniform convergence of $\Phi^q \circ \Psi^{-1}$ to the identity map on compact subsets of $\mathcal{T}$'s interior.
• The method guarantees no triangle flipping under the map, as shown by bounding the real part of complex power maps using binomial expansions and angle constraints.
• The condition $\theta \leq 60.4^\circ$ ensures that $\mathrm{Re}(1+\xi)^\gamma - 1 > 0$, which prevents orientation reversal in critical triangle types.
• The proof relies on bounding coefficients in the binomial expansion of $z^\gamma$ and showing that $\Upsilon(\theta) > 0$ for $\theta \leq 60.4^\circ$, ensuring non-flipping behavior.