In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in aRiemannian manifold. This article deals primarily with the first concept.
The canonical example of extrinsic curvature is that of a circle, which everywhere has curvature equal to the reciprocal of its radius.
For a plane curve C, the mathematical definition of curvature uses a parametric representation of C with respect to the arc length parametrization. It can be computed given any regular parametrization by a more complicated formula given below.
Let γ(s) be a regular parametric curve, where s is the arc length, or natural parameter. This determines the unit tangent vector T(s), the unit normal vector N(s), the curvature κ(s), the oriented or signed curvature k(s), and the radius of curvature R(s) at each point:
The curvature of a straight line is identically zero. The curvature of a circle of radius R is constant, i.e. it does not depend on the point and is equal to the reciprocal of the radius: Thus for a circle, the radius of curvature is simply its radius. Straight lines and circles are the only plane curves whose curvature is constant. Given any curve C and a point P on it where the curvature is non-zero, there is a unique circle which most closely approximates the curve near P, the osculating circle at P. The radius of the osculating circle is the radius of curvature of C at this point.
