curvature (space curve)
Let be an interval, and let be an arclength parameterization of an oriented space curve, assumed to be regular, and free of points of inflection. We interpret 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
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 . If one parameterizes the curve with respect to the arclength , one gets the more concise relation 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 and lying on the osculating plane, the one of radius serves as the best approximation to the space curve at the point .
To treat curvature analytically, we take the derivative of the relation
This yields the following decomposition of the acceleration vector:
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)|
|Date of creation||2013-03-22 12:14:58|
|Last modified on||2013-03-22 12:14:58|
|Last modified by||slider142 (78)|