Arrangement (space partition)

An arrangement, in the context of geometry, combinatorics, and computer science, refers to the decomposition of a space (typically Euclidean space of some dimension) into cells by a set of intersecting geometric objects. These objects are commonly hyperplanes, lines, curves, surfaces, or higher-dimensional manifolds. The resulting structure defines a partition or subdivision of the space.

More formally, given a finite set of geometric objects (e.g., hyperplanes) H in d-dimensional space, the arrangement A(H) is the cell complex induced by H. Each cell in the arrangement is a maximal connected subset of the space that does not intersect any of the objects in H. Cells can have varying dimensions, ranging from 0-dimensional vertices (intersection points of d objects) to d-dimensional cells (regions bounded by the objects). The arrangement effectively captures all possible spatial relationships determined by the objects in H.

Key properties of an arrangement include the number of cells of each dimension (vertices, edges, faces, etc.), the combinatorial structure of the cell complex (adjacency relationships between cells), and the geometric properties of the individual cells (e.g., volume, shape). The complexity of an arrangement is often measured by the total number of cells it contains.

Arrangements are fundamental structures in various fields. In computational geometry, they are used for solving problems related to range searching, motion planning, and geometric optimization. In combinatorics, arrangements are studied for their combinatorial properties and relationships to other combinatorial objects. They also have applications in fields such as machine learning, robotics, and geographic information systems.

The study of arrangements often involves analyzing the topological and combinatorial properties of the resulting cell complex, developing algorithms for constructing and manipulating arrangements, and applying arrangements to solve specific problems. Variations of arrangements exist, such as pseudoline arrangements (where lines are replaced by curves that behave similarly), and arrangements of other geometric objects beyond hyperplanes.