# How to mathematically color the regions bounded by a parametric curve?

Usually, if an implicit equation $F(x, y) = 0$ defines a curve (or curves) on the x-y plane, then we can use the inequalities $F(x, y) < 0$ or $F(x, y) > 0$ to color the regions bounded by the curve (or curves). In this way, we can make interesting pictures.

Suppose $(x, y) = (f(t), g(t))$ defines a parametric curve (an example picture) on the plane. How to color the regions (an example picture) bounded by the curve without converting the parametric equation to an implicit equation?

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Given a closed curve $\gamma: t\mapsto{\bf z}(t)$ $\ (0\leq t\leq T)$ in the plane under mild technical conditions for each point $c=(a,b)\notin\{{\bf z}(t)\ |\ t\in [0,T]\}$ the winding number $n_\gamma(c)$ is defined by $$n_\gamma(c):={1\over 2\pi}\int_0^T {(y(t)-b)\dot x(t)-(x(t)-a)\dot y(t)\over (x(t)-a)^2+(y(t)-b)^2}\ dt ={1\over 2\pi i}\int_\gamma{dz\over z-c}\ .$$ This number is always an integer (this is a miracle). Color the point $c$ green if $n_\gamma(c)$ is odd and white otherwise.
You can use the winding number to discriminate among the regions. If we use complex numbers to represent the plane, the winding number of the closed parametric curve $z = \gamma(t)$, $0 \le t \le 1$, around a point $a$ not on the curve is $$n(\gamma;a) = \frac{1}{2\pi i} \oint_\gamma \frac{dz}{z-a} = \frac{1}{2\pi i} \int_0^1 \frac{\gamma'(t)\ dt}{\gamma(t)-a}$$ which should take integer values (so you can use numerical methods to calculate this approximately and round to the nearest integer).