Some results concerning the constant astigmatism equation

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.