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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 ...
From just the curvature and torsion, the vector fields for the tangent, normal, and binormal vectors can be derived using the Frenet–Serret formulas. Then, integration of the tangent field (done numerically, if not analytically) yields the curve.
the tangent vector at point p of the i th coordinate curve. The ∂ i are a natural basis for the tangent space at point p, and the X i the corresponding coordinates for the vector field X = X i ∂ i. When both curvature and torsion vanish, the connection defines a pre-Lie algebra structure on the space of global sections of the tangent bundle.
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]
The rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group A is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted T(A). A group is called ...