computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

, the minimum bounding rectangle (MBR), also known as bounding box (BBOX) or envelope, is an expression of the maximum extents of a two-dimensional object (e.g. point,

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

) or set of objects within its

coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...

; in other words , , , . The MBR is a 2-dimensional case of the

minimum bounding box In geometry, the minimum or smallest bounding or enclosing box for a point set in dimensions is the box with the smallest measure (area, volume, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure ...

. MBRs are frequently used as an indication of the general position of a

geographic Geography (from Greek: , ''geographia''. Combination of Greek words ‘Geo’ (The Earth) and ‘Graphien’ (to describe), literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, a ...

feature or dataset, for either display, first-approximation spatial query, or spatial indexing purposes. The degree to which an "overlapping

rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...

s" query based on MBRs will be satisfactory (in other words, produce a low number of " false positive" hits) will depend on the extent to which individual spatial objects occupy (fill) their associated MBR. If the MBR is full or nearly so (for example, a mapsheet aligned with axes of

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...

and

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

will normally entirely fill its associated MBR in the same coordinate space), then the "overlapping rectangles" test will be entirely reliable for that and similar spatial objects. On the other hand, if the MBR describes a dataset consisting of a diagonal line, or a small number of disjunct points (patchy data), then most of the MBR will be empty and an "overlapping rectangles" test will produce a high number of false positives. One system that attempts to deal with this problem, particularly for patchy data, is

c-squares C-squares (acronym for the ''Concise Spatial QUery And REpresentation System'') is a system of spatially unique, location-based identifiers (geocodes) for areas on the surface of the earth, represented as cells from a latitude- and longitude-base ...

. MBRs are also an essential prerequisite for the

R-tree R-trees are tree data structures used for spatial access methods, i.e., for indexing multi-dimensional information such as geographical coordinates, rectangles or polygons. The R-tree was proposed by Antonin Guttman in 1984 and has found sig ...

method of

spatial index A spatial database is a general-purpose database (usually a relational database) that has been enhanced to include spatial data that represents objects defined in a geometric space, along with tools for querying and analyzing such data. Most s ...

ing.

As spatial metadata

Owing to their simplicity of expression and ease of use for searching, MBRs (frequently as "bounding box" or "bounding coordinates") are also commonly included in relevant standards for

geospatial metadata Geospatial metadata (also geographic metadata) is a type of metadata applicable to geographic data and information. Such objects may be stored in a geographic information system (GIS) or may simply be documents, data-sets, images or other object ...

, i.e.

metadata Metadata is "data that provides information about other data", but not the content of the data, such as the text of a message or the image itself. There are many distinct types of metadata, including: * Descriptive metadata – the descriptive ...

that describes spatial (geographic) objects; examples includ
DCMI Box
as an extension to the

Dublin Core 220px, Logo image of DCMI, which formulates Dublin Core The Dublin Core, also known as the Dublin Core Metadata Element Set (DCMES), is a set of fifteen "core" elements (properties) for describing resources. This fifteen-element Dublin Core has ...

metadata scheme, "Bounding Coordinates" in th
(U.S.) FGDC metadata standard
and "Geographic Bounding Box" in the (2003–current) ISO 19115 Metadata Standard for geographic information (

ISO/TC 211 ISO/TC 211 is a standard technical committee formed within ISO, tasked with covering the areas of digital geographic information (such as used by geographic information systems) and geomatics. It is responsible for preparation of a series of Inte ...

). It is also (as "boundingBox") an element in Geography Markup Language (GML), that is utilised by a range of Web Service specifications from the Open Geospatial Consortium (OGC). In the ISO 19107 Spatial Schema (ISO/TC 211), MBR appears as the datatype GM_Envelope that is returned by the envelope() operation on the root class GM_Object. Web-accessible articles that deal further with the concept of the MBR include "Unlocking the Mysteries of the Bounding Box" by Douglas R. Caldwell, and "Geographic Database Search Interfaces and the Equatorial Cylindrical Equidistant Projection" by Ross S. Swick and Kenneth W. Knowles. The section on "searching" on th
Geospatial Methods
site is also well worth investigating. See also documentation for specific spatially enabled databases, e.g.ESRI, 1993. Understanding GIS: The Arc/Info method. John Wiley and Sons

References

External links

Geospatial Methods website
Cartography Geometric algorithms