Torsion of a curve

Turn or twist is a measure of the deviation of a curve from the flat course in differential geometry. The winding together with the curvature describes the local behavior of the curve and comes as the curvature as a coefficient in the Frenet formulas before.

Definition

The considered curve is parameterized by the arc length s:

For a point on the curve is obtained by differentiating with respect to s the unit tangent vector ( ' direction of the curve ')

The direction of curvature of the curve is obtained by re- deriving and normalizing the principal normal unit vector

To obtain a measure of the ' rotational speed ' of order with the aid of the vector product of the Binormaleneinheitsvektor

Determined. The turn ( twist ) of the curve at the point s is now obtained as the change in direction, projected onto, so by the scalar product

Geometric Meaning: The torsion is a measure of the change in direction of Binormaleneinheitsvektors. The larger the twist, the faster the Binormaleneinheitsvektor rotates in response to the information given by the tangential axis. There are a few (some animated ) graphical illustrations.

Calculation

For the practical calculation, the above definition of the winding is not particularly good, as a parametrization by the arc length is assumed. The following formula is based on a curve in three-dimensional space (), as a function of any parameter R t (in practice, usually the time ) in the form

Is given:

In the case of a plane curve, there is nothing to be calculated, since the winding has a value of 0. Note that the sign for practical calculations of the twist is pure matter of convention. For example, do Carmo is the twist with a minus sign.

Designations

With the sign convention defined above is called a curve with linksgewunden or leftward, is so spoken rechtgewundenen or quite agile curves. In the older literature is called left manoeuvrable curves also hop agile, nimble and wine quite agile, because the tendrils of vine plants or hops along such curves grow.