Arrangement (space partition)

In discrete geometry, an arrangement is the decomposition of the d-dimensional linear, affine, or projective space into connected cells of different dimensions, induced by a finite collection of geometric objects, which are usually of dimension one less than the dimension of the space, and often of the same type as each other, such as hyperplanes or spheres.

, the cells in the arrangement are the connected components of sets of the form

the cells are the connected components of the points that belong to every object in

[1] Arrangements in complex vector spaces have also been studied; since complex lines do not partition the complex plane into multiple connected components, the combinatorics of vertices, edges, and cells does not apply to these types of space, but it is still of interest to study their symmetries and topological properties.

Advances in study of more complicated objects, such as algebraic surfaces, contributed to "real-world" applications, such as motion planning and computer vision.