Orthogonal ring patterns in the plane

TL;DR

This paper introduces orthogonal ring patterns, a generalization of circle patterns involving pairs of concentric circles, and explores their properties, equations, and applications including analogues of classical functions and boundary condition-based constructions.

Contribution

It extends circle pattern theory to orthogonal ring patterns, establishes their governing equations, and develops a variational principle for their construction.

Findings

Orthogonal ring patterns are governed by the same equations as circle patterns.

Existence of a one-parameter family interpolating between circle patterns and their duals.

Constructed ring pattern analogues of classical functions like Doyle spiral, Erf, and $z^{eta}$.

Abstract

We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that orthogonal ring patterns are governed by the same equation as circle patterns. For every ring pattern there exists a one parameter family of patterns that interpolates between a circle pattern and its dual. We construct ring pattern analogues of the Doyle spiral, Erf and $z^{\alpha}$ functions. We also derive a variational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions.