torsion (space curve)

Let $I\subset R$ be an interval and let $\gamma :I\to {ℝ}^3$ be aparameterized space curve, assumed to beregular (http://planetmath.org/SpaceCurve) and free of points of inflection. Weinterpret $\gamma (t)$ as the trajectory of a particle moving through3-dimensional space. Let $T(t),N(t),B(t)$ denote the correspondingmoving trihedron. The speed of this particle is given by $\parallel {\gamma }^{\prime }(t)\parallel$.

In order for a moving particle to escape the osculating plane, it isnecessary for the particle to “roll” along the axis of its tangentvector, thereby lifting the normal acceleration vector out of theosculating plane. The “rate of roll”, that is to say the rate atwhich the osculating plane rotates about the tangent vector, is givenby $B(t)\cdot N^{\prime }(t)$; it is a number that depends on thespeed of the particle. The rate of roll relative to the particle’sspeed is the quantity

$$\tau (t)=\frac{B(t)\cdot N^{\prime }(t)}{\parallel {\gamma }^{\prime }(t)\parallel }=\frac{({\gamma }^{\prime }(t)\times {\gamma }^{\prime \prime }(t))\cdot {\gamma }^{\prime \prime \prime }(t)}{{\parallel {\gamma }^{\prime }(t)\times {\gamma }^{\prime \prime }(t)\parallel }^2},$$

called the torsion of the curve, a quantitythat is invariant with respect to reparameterization. The torsion$\tau (t)$ is, therefore, a measure of an intrinsic property of theoriented space curve, another real number that can be covariantlyassigned to the point $\gamma (t)$.