curvature (space curve)

Let I be an intervalMathworldPlanetmathPlanetmath, and let γ:I3 be an arclength parameterization of an oriented space curve, assumed to be regularPlanetmathPlanetmathPlanetmath, and free of points of inflection. We interpret γ(t) as the trajectory of a particle moving through 3-dimensional space. Let T(t),N(t),B(t) denote the corresponding moving trihedron. The speed of this particle is given by


The quantity


is called the curvatureMathworldPlanetmathPlanetmath of the space curve. It is invariant with respect to reparameterization, and is therefore a measure of an intrinsic property of the curve, a real number geometrically assigned to the point γ(t). If one parameterizes the curve with respect to the arclength s, one gets the more concise relationMathworldPlanetmath that


Physically, curvature may be conceived as the ratio of the normal acceleration of a particle to the particle’s speed. This ratio measures the degree to which the curve deviates from the straight line at a particular point. Indeed, one can show that of all the circles passing through γ(t) and lying on the osculating plane, the one of radius 1/κ(t) serves as the best approximation to the space curve at the point γ(t).

To treat curvature analytically, we take the derivativePlanetmathPlanetmath of the relation


This yields the following decomposition of the acceleration vector:


Thus, to change speed, one needs to apply acceleration along the tangent vectorMathworldPlanetmath; to change heading the acceleration must be applied along the normal.

Title curvature (space curve)
Canonical name CurvaturespaceCurve
Date of creation 2013-03-22 12:14:58
Last modified on 2013-03-22 12:14:58
Owner slider142 (78)
Last modified by slider142 (78)
Numerical id 13
Author slider142 (78)
Entry type Definition
Classification msc 53A04
Synonym curvature
Related topic SpaceCurve
Related topic Torsion
Related topic ExpressionsForCurvatureAndTorsion
Related topic SerretFrenetFormulas