Torsion of a curve

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. Taken together,the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet-Serret formulas.

Definition

Let "C" be a space curve in a unit-length (or "natural") parametrization and with the unit tangent vector t. If the curvature $kappa$ of "C" at a certain point is not zero then the "principal normal vector" and the "binormal vector" at that point are the unit vectors

: $mathbf\{n\}=frac\{mathbf\{t\}\text{'}\}\{kappa\},\; quad\; mathbf\{b\}=mathbf\{t\}\; imesmathbf\{n\}.$

The torsion $au$ measures the speed of rotation of the binormal vector at the given point. It is found from the equation

: $mathbf\{b\}\text{'}\; =\; -\; aumathbf\{n\}.$

which means

: $au\; =\; -mathbf\{n\}cdotmathbf\{b\}\text{'}.$

"Remark": The derivative of the binormal vector is perpendicular to both the binormal and the tangent, hence it has to be proportional to the principal normal vector. The negative sign is simply a matter of convention: it is a by-product of the historical development of the subject.

The radius of torsion, often denoted by σ, is defined as

: $sigma\; =\; frac\{1\}\; \{\; au\}.$

Properties

* A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve is identically zero then this curve belongs to a fixed plane.

* The curvature and the torsion of a helix are constant. Conversely, any space curve with constant non-zero curvature and constant torsion is a helix. The torsion is positive for a right-handed helix and is negative for a left-handed one.

Alternative description

Let r = r("t") be the parametric equation of a space curve. Assume that this is a regular parametrization and that the curvature of the curve does not vanish. Analytically, r("t") is a three times differentiable function of "t" with values in R³ and the vectors

: $mathbf\{r\text{'}\}(t),\; mathbf\{r"\}(t)$

are linearly independent.

Then the torsion can be computed from the following formula:

:$au\; =$det left( {r',r",r} ight)} over {left| {r' imes r"} ight|^2 = left( {r' imes r"} ight)cdot r} over {left| {r' imes r"} ight|^2.

Here the primes denote the derivatives with respect to "t" and the cross denotes the cross product. For "r" = ("x", "y", "z"), the formula in components is

: $au\; =\; frac\{z$(x'y"-y'x") + z"(xy'-x'y) + z'(x"y-x"'y")}{(x'^2+y'^2+z'^2)(x"^2+y"^2+z"^2)}.

References

Andrew Pressley, "Elementary Differential Geometry", Springer Undergraduate Mathematics Series, Springer-Verlag, 2001 ISBN 1-85233-152-6