Rolling construction for anisotropic Delaunay surfaces

Miyuki Koiso and Bennett Palmer

Vol. 234 (2008), No. 2, 345–378

DOI: 10.2140/pjm.2008.234.345

Abstract

Anisotropic Delaunay surfaces are surfaces of revolution that have 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 that is rolled without slipping along a line. This generalizes the Delaunay’s classical construction for surfaces of revolution with constant mean curvature. Our result is given as a corollary of a new geometric description of the rolling curve of a general plane curve. Also, we characterize anisotropic Delaunay curves by using their isothermic self-duality.

Keywords

roulette, rolling curve, anisotropic, Delaunay surface

Mathematical Subject Classification 2000

Primary: 58E12

Secondary: 49Q10

Milestones

Received: 12 February 2007

Revised: 25 September 2007

Published: 1 February 2008