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 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] Template:Quotation 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.
References
- M. do Carmo, Differential Geometry of Curves and Surfaces, page 257.