In classical

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the

unit sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...

. The formula states that if γ is a parametrization of a great circle then :

\rho(\gamma(t)) \sin \psi(\gamma(t)) = \text,\,

where ''ρ''(''P'') is the distance from a point ''P'' on the

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...

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 In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...

on an arbitrary

surface of revolution A surface of revolution is a Surface (mathematics), surface in Euclidean space created by rotating a curve (the ''generatrix'') one full revolution (unit), revolution around an ''axis of rotation'' (normally not Intersection (geometry), intersec ...

. A statement of the general version of Clairaut's relation is: Pressley (p. 185) explains this theorem as an expression of

conservation of angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

about the axis of revolution when a particle moves 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. Differential geometry Differential geometry of surfaces Geodesy {{geodesy-stub