Lower degree polynomials provide little flexibility in controlling the shape of the curve. Higher degree polynomials provide flexibility but introduces wiggles. They also require too much computation. Hence cubic polynomials are perfect.
Equations for Parametric cubic polynomials:
x(u) = axu³ + bxu² + cxu + dx
y(u) = ayu³ + byu² + cyu + dy
z(u) = azu³ + bzu² + czu + dz