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

  • Brian Smyth
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 51)


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.


Curvature Surface Quadratic Differential Isometric Immersion Vertical Translation Holomorphic Quadratic Differential 
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Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Brian Smyth
    • 1
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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