How to find the points in which a given curve intersects itself?

Question

Apologies in advance for my lack of knowledge with *tex.

Hi everyone and thanks for any sort of help!

I am given the following parametric curve:

$(t^2\cos t, t^2\sin t,t^2), \text{where} -2\pi \le t \le 2\pi $

There's one known intersection of the curve with itself at $(4\pi^2,0,4\pi^2)$ and I know that there is at least one other intersection within the given domain, but I have no idea how to find it.

I've gone ahead and graphed the curve on desmos: Graph

I searched the internet for a bit and found other examples, but they involve simple linear equations.

Answer 1

Hint: If $t$ and $s$ are two times when the curve is at the same point, then $t^2 = s^2$. So $t = \pm s$. The first case is silly. In the second case $\sin t = \sin s = \sin(-t) $, which implies...

Answer 2

The Hint of AreaMan is perfect for the solution (+1). I suggest a visual interpretation that can help.

Your graph is the projection of the curve in the $(x,y)$ plane, so self intersection of the curve are also self intersections in this projection. The projection of the point $(4\pi^2,0,4\pi^2)$ is the point near $x=40$ in the graph, that is reached when $t=2\pi$ and when $t=-2\pi$ ( note that the coordinate $z$ is the same for opposite values of $t$).

The other point where the curve self intersect is projected at the point near $x=-10$, that is reached whan $t=\pm \pi$ and has coordinates $(-\pi^2,0,\pi^2)$.