curvature: Quick Facts

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The curvature at each point of a line is defined to be 1/r, where r is the radius of the osculating, or “kissing,” circle that best approximates the line at the given point.

The normal, or perpendicular, at each point of a surface defines the corresponding tangent plane, and vice versa.

A grid representing spacetime is distorted by three spheres of different sizes, illustrating how massive objects curve spacetime according to general relativity.

Two diagrams illustrate intrinsic curvature: on the left, a dome-like bowl shape representing positive curvature; on the right, a saddle shape representing negative curvature.

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Topics

curvature tensor Gaussian curvature osculating circle surface curve tangent slope length of a curve conic section catenary spiral orthogonal trajectory algebraic surface Lissajous figure rhumb line

Key People

Gaspard Monge, detail of an oil painting by Jean Naigeon, 1811; in the Museum of Fine Arts, Beaune, France.

Gaspard Monge, count de Péluse

French mathematician and public official