Check if a parametrisation is suitable

Question

I parametrised the curve C: $y^2=x^3+x^2$ by $r(t) = (x(t), y(t)) = (t^2−1, t^3−t)$

Is there a way to check that $r(t)$ lies on the curve C for all t? How can I be sure that the parametrisation reaches all points of C? Thanks

Answer 1

Yes indeed, your parameterization just works fine! Technically you probably derived the parameterization too and hence there shouldn't be any doubt, but if you were given a curve and a parameterization, then, as @Ispil already pointed out, you can just simply plug in and check whether the equality holds or not, so for $t\in\mathbb R$: $$ y^2=x^3+x^2\iff (t^3-t)^2=(t^2-1)^3+(t^2-1)^2 $$ which holds.

Anyway, the idea behind the parameteriztion of yours is if you start by $$ y^2=x^3+x^2=x^2(x+1) $$ which would yield to a root like $y=x\sqrt{x+1}$ so we want $(x+1)$ actually non-negative - to be well defined - so we take $x=t^2-1$ which gives $y=(t^2-1)\sqrt{(t^2-1)+1}=t^3-t$ and we are done.