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Let γ(s) be a regular parametric plane curve, where s is the arc length (the natural parameter).This determines the unit tangent vector T(s), the unit normal vector N(s), the signed curvature k(s) and the radius of curvature R(s) at each point for which s is composed: = ′ (), ′ = (), = | |.
In 1659 van Heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a curve (i.e., an integral). As an example of his method, he determined the arc length of a semicubical parabola, which required finding the area under a parabola. [9]
Any regular curve may be parametrized by the arc length (the natural parametrization). From the point of view of a theoretical point particle on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist.
The parametrization is regular for the given values of the parameters if the vectors , are linearly independent. The tangent plane at a regular point is the affine plane in R 3 spanned by these vectors and passing through the point r ( u , v ) on the surface determined by the parameters.
The arc length of one branch between x = x 1 and x = x 2 is a ln y 1 / y 2 . The area between the tractrix and its asymptote is π a 2 / 2 , which can be found using integration or Mamikon's theorem .
A space curve; the vectors T, N, B; and the osculating plane spanned by T and N. In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space, or the geometric properties of the curve itself irrespective of any motion.
A curve can be described, and thereby defined, by a pair of scalar fields: curvature and torsion , both of which depend on some parameter which parametrizes the curve but which can ideally be the arc length of the curve.
Differentiable curve#Length and natural parametrization To a section : This is a redirect from a topic that does not have its own page to a section of a page on the subject. For redirects to embedded anchors on a page, use {{ R to anchor }} instead .