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 $y_1(t_1)=y_2(f(t_1))$.

Answer 2

To show that $\{(x,y) \;|\; \exists{t}:x=x_1(t),y=y_1(t)\}$ and $\{(x,y) \;|\; \exists{t}:x=x_2(t),y=y_2(t)\}$ are equal sets (as subsets of $\mathbb{R}^2$), you need a bijection $f$ such that $x_1(t)=x_2(f(t))$ and $y_1(t)=y_2(f(t))$. What one typically means by "traces the same curve" is slightly stronger, in that $f$ should be continuous as well, and perhaps even monotonically increasing, if the sense (e.g., clockwise vs. counterclockwise) of the curve is considered. In your case, the two curves are the same in the stronger sense: the second is simply being traversed twice as fast, and is out of phase with the first at $t=0$.