Torsion of a curve

In geometry and kinematics, the torsion of a curve measures how sharply it is twisting out of the osculating plane. 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.

Let r be a space curve parametrized by arc length s and with the unit tangent vector T. If the curvature κ of r at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors

\[\mathbf{N}=\frac{\mathbf{T}'}{\kappa}, \quad \mathbf{B}=\mathbf{T}\times\mathbf{N}\]

respectively, where the prime denotes the derivative of the vector with respect to the parameter s. The torsion τ measures the speed of rotation of the binormal vector at the given point. It is found from the equation

\[\mathbf{B}' = -\tau\mathbf{N}.\]

which means

\[\tau = -\mathbf{N}\cdot\mathbf{B}'.\]

As \(\mathbf{N}\cdot\mathbf{B}= 0\), this is equivalent to \(\tau=\mathbf{N}'\cdot\mathbf{B}\).

Condensed: the full section is in Wikipedia.

• A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature 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 whose curvature and torsion are both constant and non-zero is a helix. The torsion is positive for a right-handed helix and is negative for a left-handed one.

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 R and the vectors

\[\mathbf{r'}(t), \mathbf{r''}(t)\]

are linearly independent.

Then the torsion can be computed from the following formula:

\[\tau = \frac{\det \left( {\mathbf{r}',\mathbf{r}'',\mathbf{r}'''} \right)} {\left\| {\mathbf{r}' \times \mathbf{r}''} \right\|^2} = \frac{\left( {\mathbf{r}' \times \mathbf{r}''} \right)\cdot \mathbf{r}'''} {\left\| {\mathbf{r}' \times \mathbf{r}''} \right\|^2}.\]

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

\[\tau = \frac{x'''\left(y'z''-y''z'\right) + y'''\left(x''z'-x'z''\right) + z'''\left(x'y''-x''y'\right)}{\left(y'z''-y''z'\right)^2 + \left(x''z'-x'z''\right)^2 + \left(x'y''-x''y'\right)^2}.\]