A Reciprocal Transformation for the Constant Astigmatism Equation

[摘要] We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation $z_{yy} + (1/z)_{xx} + 2 = 0$. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter $\lambda = \pm1$. It is also a nonlocal symmetry.

[关键词] constant astigmatism equation;exact solution;constant astigmatism surface;orthogonal equiareal pattern;reciprocal transformation;sine-Gordon equation