Determine if 3 parametric curves have same plot

Question

I have to determine if those three parametric curves have the same plot:

$\gamma_1(t) = (\cos(t), \sin (t))$ for $t \in \mathbb{R}$

$\gamma_2(t) = (\cos(t), \sin (t))$ for $t \in [0,2\pi]$

$\gamma_3(t) = (\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2})$ for $t \in \mathbb{R}$

I don't have any idea on how to resolve this, checked several times on my lessons but I don't get how to calculate parametric plot. Any help (not answer of course, I want to understand this question :)) please?

Answer 1

It is clear that both $\gamma_1$ and $\gamma_2$ are unit circles centered in the origin. The difference is that $\gamma_1$ just goes around once, while $\gamma_3$ go on and on. And you can check that the trace of $\gamma_3$ is contained in the unit circle by checking that the functions $x(t) = (1-t^2)/(1+t^2)$ and $y(t) = 2t/(1+t^2)$ satisfy the relation $$x(t)^2+y(t)^2 = 1.$$ (familiar?)

Answer 2

Hints:

1. What is different about $\gamma_1$ and $\gamma_2$? Does this difference change how the plot looks? Why or why not? (Bonus hint: Don't let your head spin around in circles.)
2. The curve $\gamma_3$ is more different, but the main question is this: Do the reals map to the same locus of points as the other two functions?

Answer 3

All three plots are the same. The first two are simply the parametrization of the unit circle by trigonometric functions with period $2\pi$. The last one is a rational parametrization of the unit circle.