Torsion of curves

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[edit] Differential geometry of curves

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. It is analogous to curvature in two dimensions. Given a function r(t) with values in R3, the torsion at a given value of t is

\tau = {{\det \left( {r',r'',r'''} \right)} \over {\left\| {r' \times r''} \right\|^2}} = {{\left( {r' \times r''} \right)\cdot r'''} \over {\left\| {r' \times r''} \right\|^2}}.

Here the primes denote the derivatives of r with respect to t; if the cross product in the denominator is zero, the torsion τ is defined to be zero as well.

If function is defined in parametric form, then its torsion is:

F[x,y,z]=\frac{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}{(x'^2+y'^2+z'^2)(x''^2+y''^2+z''^2)}

The torsion of a curve will be zero if and only if the curve sits inside a fixed plane. It is positive for right-handed helix and negative for left-handed ones.

[edit] See also

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