torsion (space curve) Torsion 2002-02-02 01:25:50 2006-03-24 08:20:24 Definition \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} Let $I\subset R$ be an interval and let $\gamma:I\to\reals^3$ be a parameterized space curve, assumed to be \PMlinkname{regular}{SpaceCurve} 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 $\Vert \gamma'(t) \Vert$. In order for a moving particle to escape the osculating plane, it is necessary for the particle to ``roll'' along the axis of its tangent vector, thereby lifting the normal acceleration vector out of the osculating plane. The ``rate of roll'', that is to say the rate at which the osculating plane rotates about the tangent vector, is given by $B(t)\cdot N'(t)$; it is a number that depends on the speed of the particle. The rate of roll relative to the particle's speed is the quantity $$\tau(t) = \frac{B(t)\cdot N'(t)}{\Vert \gamma'(t)\Vert}= \frac{( \gamma'(t)\times \gamma''(t)) \cdot \gamma'''(t)}{\Vert \gamma'(t)\times \gamma''(t)\Vert^2 },$$ called the torsion of the curve, a quantity that is invariant with respect to reparameterization. The torsion $\tau(t)$ is, therefore, a measure of an intrinsic property of the oriented space curve, another real number that can be covariantly assigned to the point $\gamma(t)$.