curvature (space curve)

Let $I\subset ℝ$ be an interval, and let $\gamma :I\to {ℝ}^3$ be an arclength parameterization of an oriented space curve, assumed to be regular, and free of points of inflection. We interpret $\gamma (t)$ 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

$$v(t)=\parallel {\gamma }^{\prime }(t)\parallel .$$

The quantity

$$\kappa (t)=\frac{\parallel T^{\prime }(t)\parallel }{v(t)}=\frac{\parallel {\gamma }^{\prime }(t)\times {\gamma }^{\prime \prime }(t)\parallel }{{\parallel {\gamma }^{\prime }(t)\parallel }^3}$$

is called the curvature 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 $\gamma (t)$. If one parameterizes the curve with respect to the arclength $s$, one gets the more concise relation that

$$\kappa (s)=\frac{1\cdot \parallel {\gamma }^{\prime \prime }(s)\parallel \cdot \sin \frac{\pi }{2}}{1^3}=\parallel {\gamma }^{\prime \prime }(s)\parallel .$$

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 $\gamma (t)$ and lying on the osculating plane, the one of radius $1/\kappa (t)$ serves as the best approximation to the space curve at the point $\gamma (t)$.

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

$${\gamma }^{\prime }(t)=v(t)T(t).$$

This yields the following decomposition of the acceleration vector:

$${\gamma }^{\prime \prime }(t)=v^{\prime }(t)T(t)+v(t)T^{\prime }(t)=v(t)\left\{{(\log v)}^{\prime }(t)T(t)+\kappa (t)N(t)\right\}.$$

Thus, to change speed, one needs to apply acceleration along the tangent vector; 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