I am looking at the following question
Consider $f(x,y,z) = (\sqrt{x^2 + y^2}-r)^2 + z^2 - a^2$ the zero set of this gives a torus. The torus has an interior hole of radius $1$ and rotating circle of radius $1, (a=1, r=2)$. Then consider a curve $z(t) = (t, m\cdot t)$ so that $x(z(t))$ is a curve on the torus,
-Prove that this curve on the torus has a self-intersection if and only if the number m is a rational number.
What do I need to show to demonstrate the curve has a self intersection? To me it seems that the curve would have the same value at two different places. So, there is some rational number m that when plugged into $x(z(t))$ will have the same value at different times?
I have found $x(u,v)$ to be $((r+acosv)cosu, (r+acosv)sinu, asinv)$
Any advice on how to show this?