Curvature is a measure of the amount by which a curve, a surface, or any other manifold deviates from a straight line, a plane, or a hyperplane (the multidimensional equivalent of a plane). For a plane curve, the curvature at a given point has a magnitude equal to one over the radius of an osculating circle (a circle that "kisses," or just touches, the curve at the given point) and is a vector pointing in the direction of that circle's center. The smaller the radius r of the osculating circle, the greater the magnitude of the curvature (1/r) will be. A straight line has zero curvature everywhere; a circle of radius r has a curvature of magnitude 1/r everywhere.
Another way to define the curvature at any point P on a plane curve is the difference between the angle θ made by the tangent at that point with a fixed straight line, and the angle θ + δθ made by the tangent at the adjacent point P' with the same straight line. The curvature at P may therefore be defined as the rate of change, k, or θ at that point. In terms of differential calculus, for the curve y = f(x)
Types of curvature
A minimal surface, like that of a soap film, has a mean curvature of zero. In the case of higher-dimensional manifolds, curvature is defined in terms of a curvature tensor, which describes what happens to a vector that is transported around a small loop of the manifold.
Related categories GEOMETRY
SPACE AND TIME
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