Rolling construction for anisotropic Delaunay surfaces

Abstract

Anisotropic Delaunay surfaces are surfaces of revolution with constant anisotropic mean curvature. We show how the generating curves of such surfaces can be obtained as the trace of a point held in a fixed position relative to a curve which is rolled without slipping along a line. This generalizes the classical construction for surfaces of revolution with constant mean curvature due to Delaunay [1]. Our result is given as a corollary of a new geometric description of the rolling curve of a general plane curve. Moreover, we characterize anisotropic Delaunay curves by using their isothermic self-duality.