Parametric cubic curves

A parametric cubic curve is a curve whose x, y, z coordinates are each a cubic polynomial in a parameter u (0 ≤ u ≤ 1):

x(u) = a_x u³ + b_x u² + c_x u + d_x y(u) = a_y u³ + b_y u² + c_y u + d_y z(u) = a_z u³ + b_z u² + c_z u + d_z

In matrix form: P(u) = [u³ u² u 1] · C, where C is the 4×3 matrix of coefficients, found from end points and tangents (Hermite) or control points (Bezier).

Why parametric equations are needed

1. A curve may be multivalued (a circle has two y values per x), but each coordinate is a single-valued function of u.
2. A vertical tangent has infinite slope in y = f(x) form, but is handled easily as dx/du = 0.
3. They are independent of the coordinate system; they work equally for 2D and 3D curves.
4. Tangent vectors are simply P'(u) = [3u² 2u 1 0]·C.
5. Curves can be joined smoothly by matching end points and derivatives.

Why cubic? Lower degrees (linear, quadratic) cannot control both end points and end tangents, and cannot have an inflection. Higher degrees need more computation and cause unwanted wiggles. Cubic has four coefficients per coordinate, matching two end points plus two end tangents, and gives C² smooth joins.