Arrangement (space partition)

For a set ${\displaystyle A}$ of objects in ${\displaystyle \mathbb {R} ^{d}}$, the cells in the arrangement are the connected components of sets of the form ${\displaystyle (\cap X)\setminus \cup (A\setminus X)}$ for subsets ${\displaystyle X}$ of ${\displaystyle A}$. That is, for each ${\displaystyle X}$ the cells are the connected components of the points that belong to every object in ${\displaystyle X}$ and do not belong to any other object. For instance the cells of an arrangement of lines in the Euclidean plane are of three types:

• Isolated points, for which ${\displaystyle X}$ is the subset of all lines that pass through the point.
• Line segments or rays, for which ${\displaystyle X}$ is a singleton set of one line. The segment or ray is a connected component of the points that belong only to that line and not to any other line of ${\displaystyle A}$
• Convex polygons (possibly unbounded), for which ${\displaystyle X}$ is the empty set, and its intersection (the empty intersection) is the whole space. These polygons are the connected components of the subset of the plane formed by removing all the lines in ${\displaystyle A}$.