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:

$r(t) \cos \theta(t) = \text{constant}.\,$

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

References

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

See also

• Clairaut's theorem