curvature (space curve)

Let $I\subset ℝ$ be an interval, and let $\gamma :I\to {ℝ}^3$ bean arclength parameterization of an oriented space curve, assumed tobe regular, and free of points of inflection. We interpret $\gamma (t)$ asthe trajectory of a particle moving through 3-dimensional space. Let$T(t),N(t),B(t)$ denote the corresponding moving trihedron. Thespeed 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 thecurvature of the space curve. It is invariant with respect toreparameterization, and is therefore a measure of an intrinsic propertyof 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 normalacceleration of a particle to the particle’s speed. This ratiomeasures the degree to which the curve deviates from the straight lineat a particular point. Indeed, one can showthat of all the circles passing through $\gamma (t)$ and lying on theosculating plane, the one of radius $1/\kappa (t)$ serves as the bestapproximation 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 followingdecomposition 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 changeheading the acceleration must be applied along the normal.