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In geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the minimum bounding box or smallest bounding box (also known as the minimum enclosing box or smallest enclosing box) for a point set in dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
s is the box
A box (plural: boxes) is a container with rigid sides used for the storage or transportation of its contents. Most boxes have flat, parallel, rectangular sides (typically rectangular prisms). Boxes can be very small (like a matchbox) or v ...
with the smallest measure (area
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
, volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
, or hypervolume in higher dimensions) within which all the points lie. When other kinds of measure are used, the minimum box is usually called accordingly, e.g., "minimum-perimeter bounding box".
The minimum bounding box of a point set is the same as the minimum bounding box of its convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
, a fact which may be used heuristic
A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...
ally to speed up computation.
In the two-dimensional case it is called the ''minimum bounding rectangle
In computational geometry, 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, polygon) or set of objects within its coord ...
''.
Axis-aligned minimum bounding box
The axis-aligned minimum bounding box (or AABB) for a given point set is its minimum bounding box subject to the constraint that the edges of the box are parallel to the (Cartesian) coordinate axes. It is theCartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of ''N'' intervals each of which is defined by the minimal and maximal value of the corresponding coordinate for the points in ''S''.
Axis-aligned minimal bounding boxes are used as an approximate location of an object in question and as a very simple descriptor of its shape. For example, in computational geometry and its applications when it is required to find intersections in the set of objects, the initial check is the intersections between their MBBs. Since it is usually a much less expensive operation than the check of the actual intersection (because it only requires comparisons of coordinates), it allows quickly excluding checks of the pairs that are far apart.
Arbitrarily oriented minimum bounding box
The arbitrarily oriented minimum bounding box is the minimum bounding box, calculated subject to no constraints as to the orientation of the result. Minimum bounding box algorithms based on therotating calipers
In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter (computational geometry), diameter of a set of points.
The method ...
method can be used to find the minimum-area or minimum-perimeter bounding box of a two-dimensional convex polygon in linear time, and of a three-dimensional point set in the time it takes to construct its convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
followed by a linear-time computation. A three-dimensional rotating calipers algorithm can find the minimum-volume arbitrarily-oriented bounding box of a three-dimensional point set in cubic time. Matlab implementations of the latter as well as the optimal compromise between accuracy and CPU time are available..
Object-oriented minimum bounding box
In the case where an object has its ownlocal coordinate system
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies th ...
, it can be useful to store a bounding box relative to these axes, which requires no transformation as the object's own transformation changes.
Digital image processing
Indigital image processing
Digital image processing is the use of a digital computer to process digital images through an algorithm. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing. It allo ...
, the ''bounding box'' is merely the coordinates of the rectangular border that fully encloses a digital image
A digital image is an image composed of picture elements, also known as pixels, each with '' finite'', '' discrete quantities'' of numeric representation for its intensity or gray level that is an output from its two-dimensional functions f ...
when it is placed over a page, a canvas, a screen or other similar bidimensional background.
See also
*Bounding sphere
In mathematics, given a non-empty set of objects of finite extension in d-dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is a d-dimensional solid sphere containing all of these ...
* Bounding volume
In computer graphics and computational geometry, a bounding volume (or bounding region) for a set of objects is a closed region that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency ...
* Minimum bounding rectangle
In computational geometry, 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, polygon) or set of objects within its coord ...
* Darboux integral
In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and onl ...
References
{{reflist Geometry Geometric algorithms