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.