Minimum bounding rectangle - AbsoluteAstronomy.com

The minimum bounding rectangle (MBR), also known as bounding box or envelope, is an expression of the maximum extents of a 2-dimensional object (e.g. point, line, polygon) within its 2-D (x, y) coordinate system

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

, in other words min(x), max(x), min(y), max(y). The MBR is a 2-dimensional case of the minimum bounding box

Minimum bounding box

The minimum or smallest bounding or enclosing box for a point set in N dimensions is the box with the smallest measure within which all the points lie...

.

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

Geography

Geography is the science that studies the lands, features, inhabitants, and phenomena of Earth. A literal translation would be "to describe or write about the Earth". The first person to use the word "geography" was Eratosthenes...

feature or dataset, for either display, first-approximation spatial query, or spatial indexing purposes.

The degree to which an "overlapping rectangle

Rectangle

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

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 and longitude 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

C-squares is a system of geocodes that provides a basis for simple spatial indexing of geographic features or data. It was devised by Tony Rees of CSIRO Marine and Atmospheric Research in 2001-2, and described in the literature in 2003...

.

MBRs are also an essential prerequisite for the R-tree

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 significant use in both research and real-world applications...

method of spatial indexing.

MBRs 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

Geospatial metadata is a type of metadata that is applicable to objects that have an explicit or implicit geographic extent, in other words, are associated with some position on the surface of the Globe...

, i.e. metadata

Metadata

The term metadata is an ambiguous term which is used for two fundamentally different concepts . Although the expression "data about data" is often used, it does not apply to both in the same way. Structural metadata, the design and specification of data structures, cannot be about data, because at...

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

Dublin Core

The Dublin Core metadata terms are a set of vocabulary terms which can be used to describe resources for the purposes of discovery. The terms can be used to describe a full range of web resources: video, images, web pages etc and physical resources such as books and objects like artworks...

metadata scheme, "Bounding Coordinates" in the (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

ISO/TC 211 is a standard technical committee formed within ISO, tasked with covering the areas of digital geographic information and geomatics. It is responsible for preparation of a series of International Standards and Technical Specifications numbered in the range starting at 19101...

). It is also (as "boundingBox") an element in Geography Markup Language

Geography Markup Language

The Geography Markup Language is the XML grammar defined by the Open Geospatial Consortium to express geographical features. GML serves as a modeling language for geographic systems as well as an open interchange format for geographic transactions on the Internet...

(GML), that is utilised by a range of Web Service specifications from the Open Geospatial Consortium

Open Geospatial Consortium

The Open Geospatial Consortium , an international voluntary consensus standards organization, originated in 1994. In the OGC, more than 400 commercial, governmental, nonprofit and research organizations worldwide collaborate in a consensus process encouraging development and implementation of open...

(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 the Geospatial Methods site is also well worth investigating. See also documentation for specific spatially-enabled databases, e.g.,.