# expressions for curvature and torsion

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

 $\displaystyle\kappa(t)$ $\displaystyle=\frac{\|\alpha^{\prime}(t)\times\alpha^{\prime\prime}(t)\|}{\|% \alpha^{\prime}(t)\|^{3}};$ $\displaystyle\tau(t)$ $\displaystyle=\frac{(\alpha^{\prime}(t)\times\alpha^{\prime\prime}(t))\cdot% \alpha^{\prime\prime\prime}(t)}{\|\alpha^{\prime}(t)\times\alpha^{\prime\prime% }(t)\|^{2}}.$

## References

John McCleary, Geometry from a Differentiable Viewpoint, Cambridge University Press, 1994.

Title expressions for curvature and torsion ExpressionsForCurvatureAndTorsion 2013-03-22 15:38:14 2013-03-22 15:38:14 Simone (5904) Simone (5904) 7 Simone (5904) Theorem msc 53A04 Torsion CurvatureOfACurve