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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.
Thus the torsion tensor is a tensor: a (bilinear) function of two input vectors v and w that produces an output vector (,). It is skew symmetric in the arguments v and w , a reflection of the fact that traversing the circuit in the opposite sense undoes the original displacement, in much the same way that twisting a screw in opposite directions ...
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
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]
denotes the cross product, which produces a vector that is perpendicular both to r and to F following the right-hand rule, is the angle between the force vector and the lever arm vector. The SI unit for torque is the newton-metre (N⋅m). For more on the units of torque, see § Units.
Then it follows immediately from the definitions that M is torsion-free (if the ring R is not a domain then torsion is considered with respect to the set S of non-zero-divisors of R). In particular, any free abelian group is torsion-free and any vector space over a field K is torsion-free when viewed as a module over K.
where κ n−1 is last Frenet curvature (the torsion of the curve) and sgn is the signum function. The minimum total absolute curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2 π for the unknot, but by the Fáry–Milnor theorem it is at least 4 π for any other knot. [2]