curvature (space curve)

Let $I\subset\mathbb{R}$ be an interval, and let $\gamma:I\to\mathbb{R}^{3}$ 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

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

The quantity

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

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

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

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.

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