In classical differential geometry, Clairaut's relation, named after

Alexis Claude de Clairaut Alexis Claude Clairaut (; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had out ...

, is a formula that characterizes the great circle paths on the

unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...

. 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 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 surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...

. 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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...

about the

axis of revolution In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the '' axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is th ...

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