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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.
Here the vectors N, B and the torsion are not well defined. If the torsion is always zero then the curve will lie in a plane. A curve may have nonzero curvature and zero torsion. For example, the circle of radius R given by r(t) = (R cos t, R sin t, 0) in the z = 0 plane has zero torsion and curvature equal to 1/R. The converse, however, is false.
Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system. The intrinsic quantities used most often are arc length s {\displaystyle s} , tangential angle θ {\displaystyle \theta } , curvature κ {\displaystyle \kappa } or radius of curvature , and, for 3 ...
The torsion tensor thus is related to, although distinct from, the torsion of a curve, as it appears in the Frenet–Serret formulas: the torsion of a connection measures a dislocation of a developed curve out of its plane, while the torsion of a curve is also a dislocation out of its osculating plane.
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].
where c ∈ ℝ n is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝ n are perpendicular vectors of length ρ (that is, a · a = b · b = ρ 2 and a · b = 0), and h : ℝ → ℝ is an arbitrary function which is twice differentiable at t.
It expresses the condition that the torsion of ∇ is zero, and as such is also called torsion-freeness. [7] There are alternative characterizations. [8] An extension of the fundamental theorem states that given a pseudo-Riemannian manifold there is a unique connection preserving the metric tensor, with
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]
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