Some results concerning the constant astigmatism equation

TL;DR

This paper investigates solutions to the constant astigmatism equation, linking them to spherical patterns and plasticity, and extends classical solution methods to generate new solutions and interpret geometric structures.

Contribution

It introduces a novel interpretation of solutions as spherical patterns, extends the Bianchi superposition principle, and connects solutions to classical Lipschitz surfaces.

Findings

Solutions describe spherical orthogonal equiareal patterns.

Extended Bianchi superposition generates multiple solutions.

Connections established between sine-Gordon solutions and slip line fields.

Abstract

In this paper we continue investigation of the constant astigmatism equation z_{yy} + (1/z)_{xx} + 2 = 0. We newly interpret its solutions as describing spherical orthogonal equiareal patterns, with relevance to two-dimensional plasticity. We show how the classical Bianchi superposition principle for the sine-Gordon equation can be extended to generate an arbitrary number of solutions of the constant astigmatism equation by algebraic manipulations. As a by-product, we show that sine-Gordon solutions give slip line fields on the sphere. Finally, we compute the solutions corresponding to classical Lipschitz surfaces of constant astigmatism via the corresponding equiareal patterns.