What are equivalent parametric equations?

Question

What are equivalent parametric equations? Is there a fast method to prove that 2 parametric equations are non-equivalent?

Answer 1

Unit Circle: $~(\cos t,\sin t)$, or $(\sin t,\cos t)$, or $\Big(t,\pm\sqrt{1-t^2}\Big)$, or $\bigg(\dfrac{2t}{1+t^2}~,~\dfrac{1-t^2}{1+t^2}\bigg)$, etc.

What do all these various parameterizations of the same geometric shape have in common ?

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad x^2(t)+y^2(t)=1$.