# Minimum bounding box > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Minimum_bounding_box > Markdown URL: https://mediated.wiki/source/Minimum_bounding_box.md > Source: https://en.wikipedia.org/wiki/Minimum_bounding_box > Source revision: 1249919681 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{short description|Smallest box which encloses some set of points}} thumb|A sphere enclosed by its axis-aligned minimum bounding box (in 3 dimensions) In [geometry](/source/geometry), the '''minimum bounding box''' or '''smallest bounding box''' (also known as the '''minimum enclosing box''' or '''smallest enclosing box''') for a point set {{mvar|S}} in {{mvar|N}} [dimension](/source/dimension)s is the [box](/source/hyperrectangle) with the smallest [measure](/source/Measure_(mathematics)) ([area](/source/area), [volume](/source/volume), or [hypervolume](/source/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](/source/convex_hull), a fact which may be used [heuristic](/source/heuristic)ally to speed up computation.{{cite web | author = Toussaint, G. T. | title = Solving geometric problems with the rotating calipers | publisher=Proc. MELECON '83, Athens |url=http://web.cs.swarthmore.edu/~adanner/cs97/s08/pdf/calipers.pdf |year = 1983 }} In the two-dimensional case it is called the ''[minimum bounding rectangle](/source/minimum_bounding_rectangle)''. ==Axis-aligned minimum bounding box== The [axis-aligned](/source/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](/source/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 as an approximate location of an object in question and as a very simple descriptor of its shape. For example, in [computational geometry](/source/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](/source/Minimum_bounding_box_algorithms) based on the [rotating calipers](/source/rotating_calipers) 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](/source/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.{{citation |author=Joseph O'Rourke |title=Finding minimal enclosing boxes |publisher=[Springer Netherlands](/source/Springer_Netherlands) |journal=Parallel Programming |year=1985}} Matlab implementations of the latter as well as the optimal compromise between accuracy and CPU time are available.{{cite web |url=https://github.com/chadogome/OptimalOBB |title=Matlab implementation of several minimum-volume bounding box algorithms |last1=Chang |first1=Chia-Tche |last2=Gorissen |first2=Bastien |last3=Melchior |first3=Samuel |website=[GitHub](/source/GitHub) |date=2018}}. ==Object-oriented minimum bounding box== In the case where an object has its own [local coordinate system](/source/local_coordinate_system), 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](/source/digital_image_processing), the ''bounding box'' is merely the coordinates of the rectangular border that fully encloses a [digital image](/source/digital_image) when it is placed over a page, a canvas, a screen or other similar bidimensional background. ==See also== * [Bounding sphere](/source/Bounding_sphere) * [Bounding volume](/source/Bounding_volume) * [Minimum bounding rectangle](/source/Minimum_bounding_rectangle) * [Darboux integral](/source/Darboux_integral) ==References== {{reflist}} Category:Geometry Category:Geometric algorithms --- Adapted from the Wikipedia article [Minimum bounding box](https://en.wikipedia.org/wiki/Minimum_bounding_box) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Minimum_bounding_box?action=history)). 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