# Clairaut's relation (differential geometry)

In classical differential geometry, **Clairaut's relation**, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ is a parametrization of a great circle then

where *ρ*(*P*) is the distance from a point *P* on the great circle to the *z*-axis, and *ψ*(*P*) is the angle between the great circle and the meridian through the point *P*.

The relation remains valid for a geodesic on an arbitrary surface of revolution.

A statement of the general version 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 meridian 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 moves along a geodesic under no forces other than those that keep it on the surface.