Let's say you have a simple parametric equation where $x= \sin(3t)$ and $y=\cos(7t).$ This is a pretty simple parametric equation that generates a relatively complicated Lissajous figure.
You can check it out here pretty easily: www.wolframalpha.com/input/?i=parametric+plot
(If I presented the entire web address for the equations, then it would be kind of a mess in this posting)
But, what I'm trying to do is find the intersection points. I count $32$ of them. But what if I want to do it without having to tediously count each one visually off the graph$?$ How would I do that$?$ What's the general approach$?$ The reason I'm asking is because I want to make some figures out of wood on a CNC machine, and I need to know the locations of the intersections so I can correctly raise or lower the cutting tool.
What's the general approach$?$ I just used this Lissajous figure as an example, but what would I do if I had an parametric equation like $$x = \sin(3t)-\cos(3t^2)$$ $$y = \tan(t^4) + \sin(6t)$$ and I have no idea how the curve looks$?$
TYIA