Tangent Vector and Tangent Line

Consider a fixed point X and a moving point P on a curve. As point P moves toward X, the vector from X to P approaches the tangent vector at X. The line that contains the tangent vector is the tangent line.

Computing the tangent vector at a point is very simple. Recall from your calculus knowledge that the derivative of parametric curve f(u) is the following:

f'(u) = ( f'(u), g'(u), h'(u) )
where f'(u) = df/du, g'(u) = dg/du and h'(u) = dh/du.

In general, the length of the tangent vector f'(u) is not one, and normalization is required. That is, the unit-length tangent vector at parameter u, or at point f(u), is

f'(u) / | f'(u) |
where | x | is the length of vector x. The tangent line at f(u) is either
f(u) + tf'(u)
or if unit-length direction vector is preferred
f(u) + t(f'(u)/|f'(u)|)
where t is a parameter.