![Minimum bounding rectangle](https://upload.wikimedia.org/wikipedia/commons/5/5b/Minimum_bounding_rectangle.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the minimum or smallest bounding or 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 coor ...
s is the box with the smallest
measure (
area
Area is the quantity that expresses the extent of a region on the plane or on a curved 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 ope ...
,
volume
Volume is a measure of occupied 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). Th ...
, or
hypervolume
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
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, a fact which may be used
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
ally to speed up computation.
The terms "box" and "hyperrectangle" come from their usage in the
Cartesian coordinate system, where they are indeed visualized as a rectangle (two-dimensional case),
rectangular parallelepiped
In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...
(three-dimensional case), etc.
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 coordin ...
.
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 the
Cartesian product 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 to 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 In computational geometry, the smallest enclosing box problem is that of finding the oriented minimum bounding box enclosing a set of points. It is a type of bounding volume. "Smallest" may refer to volume, area, perimeter, ''etc.'' of the box.
I ...
based on the
rotating 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 of a set of points.
The method is so named because the idea is an ...
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 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 own
local coordinate system
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...
, 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
In
digital 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 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 an d-dimensional solid sphere containing all of the ...
*
Bounding volume
*
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 coordin ...
*
Darboux integral
In the branch of mathematics known as 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 functi ...
References
{{reflist
Geometry
Geometric algorithms