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T is the unit vector tangent to the curve, pointing in the direction of motion. N is the normal unit vector, the derivative of T with respect to the arclength parameter of the curve, divided by its length. B is the binormal unit vector, the cross product of T and N.
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R n. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of ...
The tangent vector's magnitude ‖ ′ ‖ is the speed at the time t 0. The first Frenet vector e 1 (t) is the unit tangent vector in the same direction, defined at each regular point of γ: = ′ ‖ ′ ‖.
A vector at a particular time t [8] (for instance, a constant acceleration of the particle) is expressed in terms of (,), where and are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. At a slightly later time, the new basis in polar coordinates appears ...
has a length equal to one and is thus a unit tangent vector. If the curve is twice differentiable, that is, if the second derivatives of x and y exist, then the derivative of T(s) exists. This vector is normal to the curve, its length is the curvature κ(s), and it is oriented toward the center of curvature. That is,
It is identical to the frame bundle because, for each unit tangent vector v, there is a unique tangent vector w with det(v, w) = 1. Since the group of rotations in the plane SO(2) acts simply transitively on oriented orthonormal frames in the plane, it follows that it also acts on the frame or circle bundles of M. [7]
are two tangent vectors at point P. Let n(u, v) be the unit normal vector and let (E, F, G) and (L, M, N) be the coefficients of the first and second fundamental forms of this surface, respectively. The Weingarten equation gives the first derivative of the unit normal vector n at point P in terms of the tangent vectors r u and r v:
Remark: The derivative of the binormal vector is perpendicular to both the binormal and the tangent, hence it has to be proportional to the principal normal vector. The negative sign is simply a matter of convention: it is a byproduct of the historical development of the subject.