expressions for curvature and torsion

For a regular (http://planetmath.org/Curve), parameterized curve $\alpha :(a,b)\to {ℝ}^3$, not necessarily unit speed, the curvature $\kappa (t)$ and torsion $\tau (t)$ are given, respectively, by

$\kappa (t)$ $={\displaystyle \frac{\parallel {\alpha }^{\prime }(t)\times {\alpha }^{\prime \prime }(t)\parallel }{{\parallel {\alpha }^{\prime }(t)\parallel }^3}};$

$\tau (t)$ $={\displaystyle \frac{({\alpha }^{\prime }(t)\times {\alpha }^{\prime \prime }(t))\cdot {\alpha }^{\prime \prime \prime }(t)}{{\parallel {\alpha }^{\prime }(t)\times {\alpha }^{\prime \prime }(t)\parallel }^2}}.$

Title	expressions for curvature and torsion
Canonical name	ExpressionsForCurvatureAndTorsion
Date of creation	2013-03-22 15:38:14
Last modified on	2013-03-22 15:38:14
Owner	Simone (5904)
Last modified by	Simone (5904)
Numerical id	7
Author	Simone (5904)
Entry type	Theorem
Classification	msc 53A04
Related topic	Torsion
Related topic	CurvatureOfACurve