Intuitively, the curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve. The curvature of a differentiable curve was originally defined through osculating circles.
In Tractatus de configurationibus qualitatum et motuum, the 14th-century philosopher and mathematician Nicole Oresme introduces the concept of curvature as a measure of departure from straightness for circles he has the curvature as being inversely proportional to the radius and he attempts to extend this idea to other curves as a continuously varying magnitude. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.įor Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.įor surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. The curvature of a straight line is zero. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. Smaller circles bend more sharply, and hence have higher curvature. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.įor curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. In mathematics, curvature is any of several strongly related concepts in geometry. A migrating wild-type Dictyostelium discoideum cell whose boundary is colored by curvature.
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