specify that any polygon must be at least n nanometers from any other polygon. This is converted into a geometry query by enlarging a polygon by n/2 at all sides and query to find all intersecting polygons. In probability and statistical learning theory The number of components in a space partition plays a central role in some results in probability theory. See Growth function for more details. In Geography and GIS There are many studies and applications where Geographical Spatial Reality is partitioned by hydrological criteria, administrative criteria, mathematical criteria or many others. In the context of Cartography and GIS - Geographic Information System, is common to identify cells of the partition by standard codes. For example the for HUC code identifying hydrographical basins and sub-basins, ISO 3166-2 codes identifying countries and its subdivisions, or arbitrary DGGs - discrete global grids identifying quadrants or locations. Data structures Common space-partitioning systems include: BSP trees Quadtrees Octrees k-d trees Bins R-trees Number of components Suppose the n-dimensional Euclidean space is partitioned by hyperplanes that are -dimensional. What is the number of components in the partition? The largest number of components is attained when the hyperplanes are in general position, i.e, no two are parallel and no three have the same intersection. Denote this maximum number of components by . Then, the following recurrence relation holds: - when there are no dimensions, there is a single point. - when there are no hyperplanes, all the space is a single component. And its solution is: if if (consider e.g. perpendicular hyperplanes; each additional hyperplane divides each existing component to 2). which is upper-bounded as: See also Binary space partitioning Discrete global grid Polygon partition Tessellation References Computer graphics Geometric algorithms