Clairaut's relation (differential geometry)

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In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that states the product r×cos(θ) is constant in a unit sphere,

where r(t) is the distance from a point on a great circle to the z-axis, and θ(t) is the angle between the tangent vector and the latitudinal circle. The relation remains valid for a geodesic on an arbitrary surface of revolution.

A formal mathematical statement of Clairaut's relation is:[1]

Let γ be a geodesic on a surface of revolution S, let ρ be the distance of a point of S from the axis of rotation, and let ψ be the angle between γ and the meridians of S. Then ρ sin ψ is constant along γ. Conversely, if ρ sin ψ is constant along some curve γ in the surface, and if no part of γ is part of some parallel of S, then γ is a geodesic.

— Andrew Pressley: Elementary Differential Geometry, p. 183

Pressley (p. 185) explains this theorem as an expression of conservation of angular momentum about the axis of revolution when a particle slides along a geodesic under no forces other than those that keep it on the surface.


  • M. do Carmo, Differential Geometry of Curves and Surfaces, page 257.
  1. ^ Andrew Pressley (2001). Elementary Differential Geometry. Springer. p. 183. ISBN 1-85233-152-6.