page_title: Curvature
section_title: Frenet–Serret formulas for plane curves
hierarchical_section_title: Curvature / Plane curves / Frenet–Serret formulas for plane curves
context_page_description: In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
  For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. 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. The curvature of a straight line is zero. The curvature of a curve at a point is normally a scalar quantity, that is, it is expressed by a single real number.
  For surfaces, that are embedded in a Euclidean space, the concept of curvature is more complex, as depending on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
  For Riemannian manifolds that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space.
context_section_description: The expression of the curvature In terms of arc-length parametrization is essentially the first Frenet–Serret formula
  where the primes refer to the derivatives with respect to the arc length s, and N(s) is the normal unit vector in the direction of T′(s).
  As planar curves have zero torsion, the second Frenet–Serret formula provides the relation
  For a general parametrization by a parameter t, one needs expressions involving derivatives with respect to t. As these are obtained by multiplying by  the derivatives with respect to s, one has, for any proper parametrization
caption: The vectors T and N at two points on a plane curve, a translated version of the second frame (dotted), and the change in T: δT. δs is the distance between the points. In the limit dT/ds will be in the direction N and the curvature describes the speed of rotation of the frame.