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.
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