Parameterized function crosses own path

[mariush]

Hi!

Given a function $$r:\mathbb{R} \rightarrow \mathbb{R}^2, r(t) = (f_1(t), f_2(t))$$, is there a way to analytically determine if there are points (x1, x2) where r(t) = (x1, x2) for multiple t-values?

Lets say i was to find such points for the function $$r(t) = (t^3-t, 3t^2 + 1)$$

How should i go about finding the points without having to plot the graf?

Thanks!

 

[CompuChip]

In theory, you could assume that given some time t, the graph crosses the same point r(t) after time a again, leading to the equation
r(t) = r(t - a).

This leads to two equations in t and a.
The question would then be if there is a non-trivial solution (i.e. $a \neq 0$).

 

[mariush]

Exactly!

Thanks a lot :)