Find an explicit immersion of $I=(0,1)$ into $S^2$ which is not an embedding.

Question

I am trying to solve this excercise from my instructor's. The fact is: I can provide an example of an immersion (even injective) of an interval in the plane $\mathbb{R}^2$ which is not an embedding. This is the "figure eight map", also known as the Gerono lemniscate $$t\in(-\frac{1}{2}\pi,\frac{3}{2}\pi)\mapsto(\cos{t},\sin{t}\cos{t})\in\mathbb{R}^2$$ Common sense says I can patch this on a sphere, but how to see this explicitely?
Is there a simpler example of such a map?

Answer 1

Take $f : (-\frac{1}{2}\pi,\frac{3}{2}\pi ) \to S^2, f(t) = (\frac{1}{2}\cos t,\frac{1}{2}\cos t \sin t,\sqrt{1- \frac{1}{4}\cos^2 t - \frac{1}{4}\cos^2 t \sin^2 t})$.