Tangent Vector

For a curve with radius vector r(t), the unit tangent vector T^^(t) is defined by


where t is a parameterization variable, s is the arc length, and an overdot denotes a derivative with respect to t, x^.=dx/dt. For a function given parametrically by (f(t),g(t)), the tangent vector relative to the point (f(t),g(t)) is therefore given by


To actually place the vector tangent to the curve, it must be displaced by (f(t),g(t)). It is also true that


where N is the normal vector, kappa is the curvature, tau is the torsion, and [A,B,C] is the scalar triple product.

See also

Binormal Vector, Curvature, Manifold Tangent Vector, Normal Vector, Tangent, Tangent Bundle, Tangent Plane, Tangent Space, Torsion Explore this topic in the MathWorld classroom

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Gray, A. "Tangent and Normal Lines to Plane Curves." §5.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 108-111, 1997.

Referenced on Wolfram|Alpha

Tangent Vector

Cite this as:

Weisstein, Eric W. "Tangent Vector." From MathWorld--A Wolfram Web Resource.

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