I have a set of inequalities that represent a closed region in Cartesian space (the problem is in $\mathbb{R}^2$ but can be extended to $\mathbb{R}^n$ with hyperplanes instead of lines ). It is possible (but not necessary) that the region represented by the inequalities is actually a union of two or more disjoint regions. Is there some method to partition the set into smaller sets with each output set uniquely describing a connected, closed region?
For example, in the image, the region bounded by the lines is $R_1 \cup R_2$ , defined by the the set of inequalities
$\left(\begin{array}{c} a_1x+b_1y+c_1 > 0\\ -a_2x-b_2y-c_2 > 0\\ a_3x+b_3y+c_3 > 0\\ a_4x+b_4y+c_4 > 0\\ -a_5x-b_5y-c_5 > 0\end{array}\right)$
What I would like is a method to split this set into two sets each defining $R_1$ and $R_2$ respectively as
$R_1 = \left(\begin{array}{c} a_1x+b_1y+c_1 > 0\\ -a_2x-b_2y-c_2 > 0\\ a_3x+b_3y+c_3 > 0\end{array}\right)$ and $ R_2 = \left(\begin{array}{c} a_1x+b_1y+c_1 > 0\\ a_4x+b_4y+c_4 > 0\\ -a_5x-b_5y-c_5 > 0\end{array}\right)$