constructing an explicit homotopy

Question

I can see that the paths $(\cos(\pi s), \sin(\pi s))$ and $(\cos(\pi s), -\sin(\pi s))$ in $\mathbb{R}^2 \setminus \{0\}$ are 'homotopic' But can't construct an explicit homotopy between them.

Could anyone suggest me an explicit homotoy function?

Here I mean just homotopy, not path-homotopy

Answer 1

You can explicitly construct a "retraction" of the first path to the constant path based at $(1,0)$ ($s=0$), by putting $\gamma_t(s) = (\cos(\pi s t), \sin(\pi s t))$. You can do the same for the second path. Now do the first "retraction", then the second one in reverse direction.