Tangent Vector

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

			(1)
			(2)
			(3)

where  is a parameterization variable,  is the arc length, and an overdot denotes a derivative with respect to , . 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,  is the torsion, and  is the scalar triple product.

REFERENCES:

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.