Path connected subset $\mathbb{R}^2$

Question

We all know that $\mathbb{R}^2$ is path connected. Consider the subset $S=\{(x,y):\mid \mid (x,y)\mid \mid \geq c, \text{for fixed $c$ in $\mathbb{R}$}\}.$ It can be geometrically visualize that $S$ is path connected.

But I need a precise path between any two points of $S$. I think the path should be circular, but I do not know that how to proceed.

Answer 1

Suppose that the points are given by $x=r_1 e^{i\theta_1},y=r_2 e^{i\theta_2}$. Then a path can be given by $$\gamma(t)=((1-t)r_1+tr_2)e^{i((1-t)\theta_1+t\theta_2)}$$ for $t\in[0,1]$.