Two-dimensional shapes and lemniscates

TL;DR

This paper establishes a one-to-one correspondence between polynomial lemniscates of degree n and nth roots of Blaschke products, linking geometric shapes with complex function theory to advance shape analysis.

Contribution

It introduces a novel correspondence between shapes defined by polynomial lemniscates and roots of Blaschke products, enriching the mathematical framework for shape fingerprinting.

Findings

Lemniscates approximate all Jordan curves in Hausdorff metric.

Roots of Blaschke products approximate all orientation-preserving circle diffeomorphisms.

The correspondence provides a natural link between shape geometry and complex analysis.

Abstract

A shape in the plane is an equivalence class of sufficiently smooth Jordan curves, where two curves are equivalent if one can be obtained from the other by a translation and a scaling. The fingerprint of a shape is an equivalence of orientation preserving diffeomorphisms of the unit circle, where two diffeomorphisms are equivalent if they differ by right composition with an automorphism of the unit disk. The fingerprint is obtained by composing Riemann maps onto the interior and exterior of a representative of a shape in a suitable way. In this paper, we show that there is a one-to-one correspondence between shapes defined by polynomial lemniscates of degree n and nth roots of Blaschke products of degree n. The facts that lemniscates approximate all Jordan curves in the Hausdorff metric and roots of Blaschke products approximate all orientation preserving diffeomorphisms of the circle in the C^1-norm suggest that lemniscates and roots of Blaschke products are natural objects to study in the theory of shapes and their fingerprints.