How to parameterise the curve $ x^2 = 4y, 3x^3 = 8z$?

Question

As per title, I'm unsure how to parameterise the given curve? Are there different methods? I'm unsure about parameterisation in general, I just tend to remember specific formulas.

Answer 1

How about: $$ r(t) = \left[ \begin{matrix} x \\ y \\ z \end{matrix} \right] = \left[ \begin{matrix} t \\ \frac{t^2}{4} \\ \frac{3t^3}{8} \\ \end{matrix} \right] $$

for $t \in \mathbb{R}$.

More complicated it gets if you want to parameterise by the arc length $s$, with $ds^2=dx^2 + dy^2 + dz^2$.