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Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
On the example of a torus knot, the tangent vector T, the normal vector N, and the binormal vector B, along with the curvature κ(s), and the torsion τ(s) are displayed. At the peaks of the torsion function the rotation of the Frenet–Serret frame ( T , N , B ) around the tangent vector is clearly visible.
The torsion tensor is a bilinear map of two input vectors ,, that produces an output vector (,) representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are ,.
In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the displacement vector and the force vector. The direction of the torque can be determined by using the right hand grip rule : if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force ...
The Darboux vector provides a concise way of interpreting curvature κ and torsion τ geometrically: curvature is the measure of the rotation of the Frenet frame about the binormal unit vector, whereas torsion is the measure of the rotation of the Frenet frame about the tangent unit vector. [2]
The vector ˙ is the angular acceleration. Again, note that all quantities are defined in the rotating reference frame. Again, note that all quantities are defined in the rotating reference frame. In orthogonal principal axes of inertia coordinates the equations become
Indeed, if is a vector of unit length on a Riemannian -manifold, then (,) is precisely () times the average value of the sectional curvature, taken over all the 2-planes containing . There is an ( n − 2 ) {\displaystyle (n-2)} -dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full ...
Torsion of a square section bar Example of torsion mechanics. In the field of solid mechanics, torsion is the twisting of an object due to an applied torque. [1] [2] Torsion could be defined as strain [3] [4] or angular deformation, [5] and is measured by the angle a chosen section is rotated from its equilibrium position. [6]