PlanetMath: curvature (space curve)

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curvature (space curve)

(Definition)

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 denote the corresponding moving trihedron. The speed of this particle is given by

$\displaystyle v(t) = \Vert \gamma'(t) \Vert.$

The quantity

$\displaystyle \kappa(t) = \frac{\Vert T'(t)\Vert}{v(t)} = \frac{\Vert \gamma'(t)\times \gamma''(t)\Vert} {\Vert \gamma'(t)\Vert^3}$

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

, one gets the more concise relation that

$\displaystyle \kappa(s) = \frac{1\cdot\Vert\gamma''(s)\Vert\cdot\sin\frac{\pi}{2}}{1^3} = \Vert\gamma''(s)\Vert.$

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

$\displaystyle \gamma'(t) = v(t) T(t).$

This yields the following decomposition of the acceleration vector:

$\displaystyle \gamma''(t) = v'(t) T(t) + v(t) T'(t) = v(t) \left\{ (\log v)'(t)\, T(t) + \kappa(t)\, N(t)\right\}.$

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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"curvature (space curve)" is owned by slider142. [ full author list (3) | owner history (2) ]

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Other names:

curvature

Attachments:: curvature (plane curve) (Topic) by rspuzio
example of curvature (space curve) (Example) by bloftin

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Cross-references: tangent vector, vector, decomposition, derivative, best approximation, radius, osculating plane, passing through, circles, line, straight, degree, normal, ratio, relation, point, real number, property, measure, reparameterization, invariant, moving trihedron, trajectory, points of inflection, regular, oriented space curve, arclength parameterization, interval
There are 28 references to this entry.

This is version 9 of curvature (space curve), born on 2002-02-02, modified 2006-12-01.
Object id is 1632, canonical name is CurvatureOfACurve.
Accessed 9953 times total.

Classification:

AMS MSC:

53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

Pending Errata and Addenda

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