[Math] How do we prove that two parametric equations are drawing the same thing

Question

For example, if I have
$$\begin {align}
x(t) &= r\sin t\cos t\\
y(t) &= r\sin^2 t\\
\end {align}$$
and
$$\begin {align}
x(t) &= \frac r 2 \cos t\\
y(t) &= \frac r 2 (\sin t + 1)
\end {align}$$
How do we show that the two parametric equations draw the same line?

Answer 1

You should find bijection $t_2=f(t_1)$, so that $x_1(t_1)=x_2(f(t_1))$ and $y_1(t_1)=y_2(f(t_1))$.