Approximation of conformal mappings and novel applications to shape recognition of planar domains

TL;DR

This paper introduces a new method for representing 2D shapes using conformal maps, proving convergence and affirming a conjecture about electrical network approximations, with applications in shape recognition.

Contribution

It presents a convergent approximation scheme for conformal maps of planar domains and confirms Stephenson's conjecture linking Riemann mappings to electrical networks.

Findings

Proved uniform convergence of the approximation scheme.

Confirmed Stephenson's conjecture on electrical network approximation.

Developed a programmable scheme for shape representation.

Abstract

Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this program where we address the case of simply, and doubly-connected planar domains. We prove uniform convergence of our approximation scheme to the appropriate conformal mapping. Along the way, we affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks. In fact, we first treat a more general case. Consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. The resolution of Stephenson's Conjecture then follows by a limiting argument. Our methods involve harmonic mappings and boundary value problems: discrete and analytic in the plane. The scheme we constructed is programmable.