Meusnier's theorem

Meusnier's theorem states that the normal curvature of a surface in a tangent direction equals the curvature of any curve on the surface with that tangent, multiplied by the cosine of the angle between the curve's principal normal and the surface normal.

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Normal curvature: The component of a surface curve’s curvature vector in the direction of the surface normal. It is the surface quantity that Meusnier’s theorem equates to projected curve curvature.
Tangent plane: The plane that best approximates a smooth surface at a point. The theorem fixes a direction within this plane before comparing curvatures.
Jean-Baptiste Meusnier: A French mathematician and engineer whose work included a theorem on curves lying on surfaces. His name is attached to the curvature relation.
Geodesic curvature: The component of a surface curve’s curvature vector tangent to the surface. It contrasts with the normal component isolated by Meusnier’s theorem.
Normal section: A plane curve formed by intersecting a surface with a plane containing the surface normal at a point. Its curvature supplies the normal curvature in the chosen tangent direction.
Surface normal: A vector perpendicular to a surface’s tangent plane at a point. It defines both the normal section and the curvature component used in the theorem.
Differential geometry: The study of geometric structures using calculus and other tools of mathematical analysis. Meusnier’s theorem is an early result in the differential geometry of surfaces.
Geodesic: A curve on a surface whose geodesic curvature vanishes. For a geodesic, the curve’s bending is entirely normal to the surface.
Principal curvature: An extremal value of a surface’s normal curvature at a point, attained in a principal direction. The theorem’s directional normal curvature reaches these extrema in principal directions.
Curvature: A measure of how rapidly a curve or surface changes direction or departs from flatness. Meusnier’s theorem relates two specific meanings of curvature.