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.

**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.

© 1996-9

1999-05-26