Clairaut's relation

Clairaut's relation states that along a geodesic on a surface of revolution, the distance from the axis multiplied by the sine of the angle with a meridian is constant.

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Geodesic: A curve that locally minimizes distance and has zero intrinsic acceleration. Clairaut's relation describes how a geodesic travels across a surface of revolution.
Riemannian metric: A smoothly varying inner product on tangent spaces that defines lengths and angles. The metric defines the geodesics and angle measured in Clairaut's relation.
Geodesics on a sphere: The shortest paths on a sphere, which lie along great circles. On a sphere, Clairaut's relation identifies the great-circle paths and their latitude behavior.
Alexis Claude Clairaut: A French mathematician whose work included the theory of curves and the figure of the Earth. The relation bears his name and arose from his work on curves on surfaces of revolution.
Surface of revolution: A surface generated by rotating a plane curve around an axis in its plane. Rotational symmetry supplies the axis and meridians used in the relation.
Meridian (geography): A curve on a surface of revolution formed by intersecting it with a plane containing the axis. The angle in the relation is measured between the geodesic and a meridian.
Geodesics on a cylinder: Locally shortest curves on a cylinder, which unwrap into straight lines in a plane. The cylinder provides a simple example where the conserved product constrains a geodesic's pitch.
Geodesy: The science of measuring Earth’s shape, orientation, and gravitational field. The geometry of rotationally symmetric Earth models provides a practical setting for the relation.
Killing vector field: A vector field whose flow preserves a Riemannian metric. The rotational symmetry generates a conserved quantity along each geodesic.
Parallel (geometry): A circular cross-section of a surface of revolution perpendicular to its axis. The distance to the axis is constant along each parallel.