What about the following parametric equation? If we are willing to use complex numbers to represent points in the plane, we could use $$\prod_{k=0}^{n-1} \left[t\exp(2\pi ik/n) +(1-t)\exp(2\pi i(k+1)/n)\right]=0,\qquad 0 \le t<1$$

The following is probably not in the spirit of the game, but what about a parametric equation? If we are willing to use complex numbers to represent points in the plane, we could use $$z=t\exp(2\pi ik/n) +(1-t)\exp(2\pi i(k+1)/n),\qquad 0 \le t<1,\quad k=0, 1, \dots, n-1$$