Parametric and non-parametric curve representation

Non-parametric representation gives a direct relation between coordinates: explicit $y=f(x)$ or implicit $f(x,y)=0$. Parametric representation expresses each coordinate as a function of an independent parameter $u$: $x=x(u)$, $y=y(u)$, $z=z(u)$, with $u$ in a range, usually $0\le u\le1$.

Why parametric is preferred

1. Curves can be closed, multi-valued or have vertical tangents.
2. Easy to generate points at equal steps of $u$ for display and tool paths.
3. Tangent vector is simply $d\mathbf P/du$.
4. Segments are bounded and can be joined with continuity conditions.
5. Same equation works in 2D and 3D, and with any axes.

Circle of radius 5

Non-parametric (implicit): $x^2+y^2=25$; explicit form needs two branches $y=\pm\sqrt{25-x^2}$.

Parametric:

$$x = 5\cos\theta,\qquad y = 5\sin\theta,\qquad 0\le\theta\le 2\pi$$

For example, $\theta=0$ gives $(5,0)$ directly, where the explicit form has an infinite slope ($dy/dx\to\infty$), while the parametric tangent $(dx/d\theta,\ dy/d\theta)=(0,\ 5)$ stays finite.