In classicaldifferential geometry,Clairaut's relation, named afterAlexis Claude de Clairaut, is a formula that characterizes the great circle paths on theunit sphere. The formula states that if${\displaystyle \gamma }$ is a parametrization of a great circle then

${\displaystyle \rho (\gamma (t))\sin \psi (\gamma (t))=\text{constant},}$

where${\displaystyle \rho (P)}$ is the distance from a point${\displaystyle P}$ on thegreat circle to the${\displaystyle z}$-axis, and${\displaystyle \psi (P)}$ is the angle between the great circle and themeridian through the point${\displaystyle P}$.

The relation remains valid for ageodesic on an arbitrarysurface of revolution.

A statement of the general version of Clairaut's relation is:^[1]

Let${\displaystyle \gamma }$ be ageodesic on asurface of revolution${\displaystyle S}$, let${\displaystyle \rho }$ be the distance of a point of${\displaystyle S}$ from theaxis of rotation, and let${\displaystyle \psi }$ be the angle between${\displaystyle \gamma }$ and themeridian of${\displaystyle S}$. Then${\displaystyle \rho \sin \psi }$ is constant along${\displaystyle \gamma }$. Conversely, if${\displaystyle \rho \sin \psi }$ is constant along some curve${\displaystyle \gamma }$ in the surface, and if no part of${\displaystyle \gamma }$ is part of some parallel of${\displaystyle S}$, then${\displaystyle \gamma }$ is a geodesic.

Pressley (p. 185) explains this theorem as an expression ofconservation of angular momentum about theaxis of revolution when a particle moves along a geodesic under no forces other than those that keep it on the surface.

Now imagine a particle constrained to move on a surface of revolution, without external torque around the axis. By conservation of angular momentum:

${\displaystyle r{v}_{\theta }=L,}$

where

• ${\displaystyle r}$ = distance to the axis,
• ${\displaystyle v_{\theta }}$ = component of velocity orthogonal to the meridian,
• ${\displaystyle L}$ = conserved angular momentum around the axis.

But geometrically,

${\displaystyle v_{\theta }=|v|\sin \psi ,}$

If we normalize so the speed${\displaystyle |v|=1}$ (unit speed geodesics), we get:

${\displaystyle r\sin \psi =\frac{L}{|v|}=\text{constant}.}$

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

Thisdifferential geometry-related article is astub. You can help Wikipedia byadding missing information.

• v
• t
• e

Thisgeodesy-related article is astub. You can help Wikipedia byadding missing information.

• v
• t
• e

• Differential geometry
• Differential geometry of surfaces
• Geodesy
• Differential geometry stubs
• Geodesy stubs

• Articles with short description
• Short description matches Wikidata
• All stub articles