# Clairaut's relation

**Clairaut's relation**, named after Alexis Claude de Clairaut, is a formula in classical differential geometry. The formula relates the distance *r*(*t*) from a point on a great circle of the unit sphere to the *z*-axis, and the angle *θ*(*t*) 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 ofSfrom the axis of rotation, and let ψ be the angle between γ and the meridians ofS. 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 ofS, 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.

## References

- M. do Carmo,
*Differential Geometry of Curves and Surfaces*, page 257.

- Andrew Pressley (2001).
*Elementary Differential Geometry*. Springer. p. 183. ISBN 1-85233-152-6.