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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, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame (TNB frame or TNB basis), together form an orthonormal basis that spans, and are defined as follows: T is the unit vector tangent to the curve, pointing in the direction of motion.
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. ... Tangent vector to a curve/flux line ^ ...
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: = ′ ‖ ′ ‖.
is called the unit tangent vector, so an equivalent definition is that the tangential angle at t is the angle φ such that (cos φ, sin φ) is the unit tangent vector at t. If the curve is parametrized by arc length s , so | x ′( s ), y ′( s ) | = 1 , then the definition simplifies to
Illustration of tangential and normal components of a vector to a surface. In mathematics, given a vector at a point on a curve, that vector can be decomposed uniquely as a sum of two vectors, one tangent to the curve, called the tangential component of the vector, and another one perpendicular to the curve, called the normal component of the vector.
More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point P(s) is a function of the parameter s, which may be thought as the time or as the arc length from a given origin. Let T(s) be a unit tangent vector of the curve at P(s), which is also the derivative of P(s) with ...
Let γ(s) be a regular parametric plane curve, where s is the arc length (the natural parameter).This determines the unit tangent vector T(s), the unit normal vector N(s), the signed curvature k(s) and the radius of curvature R(s) at each point for which s is composed: = ′ (), ′ = (), = | |.