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Minimum bounding rectangle

Overview

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 mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex...

, 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 which all the points lie within...

.

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

Geography

Geography is the study of the Earth and its lands, features, inhabitants, and phenomena. 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 geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle. A simple rectangle with vertices ABCD would be denoted as ....

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.

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Encyclopedia

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 mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex...

, 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 which all the points lie within...

.

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

Geography

Geography is the study of the Earth and its lands, features, inhabitants, and phenomena. 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 geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle. A simple rectangle with vertices ABCD would be denoted as ....

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 that are similar to B-trees, but are used for spatial access methods i.e., for indexing multi-dimensional information; for example, the coordinates of geographical data...

method of spatial index

Spatial index

Spatial indexes are used by spatial databases to optimize spatial queries. Indexes used by non-spatial databases cannot effectively handle features such as how far two points differ and whether points fall within a spatial area of interest...

ing.

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

Metadata is "data about data", of any sort in any media. Metadata is text, voice, or image that describes what the audience wants or needs to see or experience. The audience could be a person, group, or software program...

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

Dublin Core

The Dublin Core metadata element set is a standard for cross-domain information resource description. It defines conventions for describing things online in ways that make them easy to find. Dublin Core is widely used to describe digital materials such as video, sound, image, text, and composite...

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 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

Founded in 1994, the Open Geospatial Consortium is an international voluntary consensus standards organization. In the OGC, more than 370+ commercial, governmental, nonprofit and research organizations worldwide collaborate in an open consensus process encouraging development and implementation of...

(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. , .