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# 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.

## Mathematics Subject Classification

53A04*no label found*

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## Attached Articles

## Corrections

amalmagate by rmilson ✓

arclength parameterization by rspuzio ✓

emphasis on defined terms by Mathprof ✓