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  2. Tangent vector - Wikipedia

    en.wikipedia.org/wiki/Tangent_vector

    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 ...

  3. Frenet–Serret formulas - Wikipedia

    en.wikipedia.org/wiki/Frenet–Serret_formulas

    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.

  4. Tangential and normal components - Wikipedia

    en.wikipedia.org/wiki/Tangential_and_normal...

    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.

  5. Osculating circle - Wikipedia

    en.wikipedia.org/wiki/Osculating_circle

    Developing the equation for , and grouping the terms in and , we obtain ˙ + ˙ = ¨ + ¨ = ˙ + ˙ Denoting =, the first equation means that is orthogonal to the unit tangent vector at : = The second relation means that = where = = ˙ + ˙ [¨ ¨] is the curvature vector.

  6. Differentiable curve - Wikipedia

    en.wikipedia.org/wiki/Differentiable_curve

    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: = ′ ‖ ′ ‖.

  7. Curvature - Wikipedia

    en.wikipedia.org/wiki/Curvature

    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 ...

  8. Envelope (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Envelope_(mathematics)

    A line is normal to γ at γ(t) if it passes through γ(t) and is perpendicular to the tangent vector to γ at γ(t). Let T denote the unit tangent vector to γ and let N denote the unit normal vector. Using a dot to denote the dot product, the generating family for the one-parameter family of normal lines is given by F : I × R 2 → R where

  9. Differential geometry of surfaces - Wikipedia

    en.wikipedia.org/wiki/Differential_geometry_of...

    Given a tangential vector field X and a tangent vector Y to S at p, the covariant derivative ∇ Y X is a certain tangent vector to S at p. Consequently, if X and Y are both tangential vector fields, then ∇ Y X can also be regarded as a tangential vector field; iteratively, if X , Y , and Z are tangential vector fields, the one may compute ...