Statistical Thermodynamics and Differential Geometry of Microstructured Materials

Volume 51 of the series The IMA Volumes in Mathematics and its Applications pp 123-130

A Generalization of a Theorem of Delaunay on Constant Mean Curvature Surfaces

  • Brian SmythAffiliated withDepartment of Mathematics, University of Notre Dame

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In 1841, Delaunay [2] classified all surfaces of revolution of constant mean curvature, with a beautifully simple description in terms of conics. The constancy of the mean curvature is expressed by an ordinary differential equation for the radius of rotation with respect to the meridian length. The resulting equation was, in those days, very familiar as the differential equation governing the roulette of a conic, that is, the locus of the focus of a conic as it rolls in a plane along a line.