Tangent Vector

For a curve with Position Vector , the unit tangent vector is defined by

 (1) (2) (3)

where is a parameterization variable and is the Arc Length. For a function given parametrically by , the tangent vector relative to the point is therefore given by
 (4) (5)

To actually place the vector tangent to the curve, it must be displaced by . It is also true that
 (6) (7) (8)

where is the Normal Vector, is the Curvature, and is the Torsion.

See also Curvature, Normal Vector, Tangent, Tangent Bundle, Tangent Plane, Tangent Space, Torsion (Differential Geometry)

References

Gray, A. Tangent and Normal Lines to Plane Curves.'' §5.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 85-90, 1993.