Twist (geometry)

In differential geometry, winding or torsion is a measure of the deviation of a curve from a flat course. The turn describes the local behavior of the curve together with the curvature and, like the curvature, occurs as a coefficient in the Frenetian formulas .

definition

Torsion and corresponding rotation of the binormal vector.

The curve under consideration is parameterized by the arc length s:

${\displaystyle {\vec {r}}={\vec {r}}(s)\,.}$

For a curve point , the tangent unit vector ('direction of the curve') is obtained by deriving it according to s ${\displaystyle {\vec {r}}(s)}$

${\displaystyle {\vec {t}}(s)={\vec {r}}\,'(s)\,.}$

The direction of curvature of the curve is obtained by deriving and normalizing again as the main normal unit vector

${\displaystyle {\vec {n}}(s)={\frac {{\vec {t}}\,'(s)}{|{\vec {t}}\,'(s)|}}={\frac {{\vec {r}}\,''(s)}{|{\vec {r}}\,''(s)|}}\,.}$

In order to obtain a measure for the 'speed of rotation' of um , the binormal unit vector is calculated using the vector product ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {t}}}$

${\displaystyle {\vec {b}}(s)={\vec {t}}(s)\times {\vec {n}}(s)}$

set. The turn (torsion) of the curve at the point s now results as its change in direction, projected onto , i.e. by the scalar product ${\displaystyle \tau (s)\,}$ ${\displaystyle {\vec {n}}}$

${\displaystyle \tau (s)=-{\vec {b}}\,'(s)\cdot {\vec {n}}(s)\,.}$

Geometric meaning: The torsion is a measure of the change in direction of the binormal unit vector. The greater the torsion, the faster the binormal unit vector rotates around the axis given by the tangential vector as a function of . There are some (partly animated) graphic illustrations for this . ${\displaystyle \displaystyle \tau (s)}$ ${\displaystyle {\vec {b}}(s)}$ ${\displaystyle \displaystyle s}$

calculation

The definition of the turn given above is not particularly suitable for the practical calculation, since a parameterization through the arc length is assumed. The following formula refers to a curve in three-dimensional space ( ), which as a function r of any parameter t (in practice usually time) in the form ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle {\vec {r}}={\vec {r}}(t)}$

given is:

${\displaystyle \tau (t)={\frac {\left({\vec {r}}\,'(t)\times {\vec {r}}\,''(t)\right)\cdot {\vec {r}}\,'''(t)}{\left|{\vec {r}}\,'(t)\times {\vec {r}}\,''(t)\right|^{2}}}}$

In the case of a flat curve, there is nothing to calculate since the winding has the value 0. Note that the sign for practical calculations of the torsion is purely a matter of convention. For example, do Carmo gives the torsion with a negative sign.

Designations

With the sign convention defined above is called a curve with sinistral or leftward , is so one speaks dextral or right manoeuvrable curves. In the older literature you left agile curves also called hop agile , very agile and wine nimble , because the tendrils of vine plants and hop along such curves grow. ${\displaystyle \tau <0}$ ${\displaystyle \tau >0}$

literature

1. ↑ Manfredo P. do Carmo: Differential geometry of curves and surfaces (= Vieweg study. Advanced course in mathematics 55). Vieweg & Sohn, Braunschweig et al. 1983, ISBN 3-528-07255-5 .
2. ↑ Wolfgang Kühnel : Differential Geometry. Curves - surfaces - manifolds. 4th revised edition. Friedr. Vieweg & Sohn Verlag, Wiesbaden 2008, ISBN 978-3-8348-0411-2 , Section 2.8: Space curves

Web links

Commons : Torsion graphic illustrations  - collection of images, videos and audio files

• Create animated illustrations of the torsion yourself ( Maple worksheet)