|
Minimum Bounding Box
In geometry, the minimum or smallest bounding or enclosing box for a point set in dimensions is the box with the smallest measure (area, volume, 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, a fact which may be used heuristically 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 (three-dimensional case), etc. In the two-dimensional case it is called the minimum bounding rectangle. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 coordinate system; in other words , , , . The MBR is a 2-dimensional case of the minimum bounding box. MBRs are frequently used as an indication of the general position of a geographic feature or dataset, for either display, first-approximation spatial query, or spatial indexing purposes. The degree to which an "overlapping rectangles" 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" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with a history stretching back to antiquity. Analysis of algorithms, Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points. For such sets, the difference between O(''n''2) and O(''n'' log ''n'') may be the difference between days and seconds of computation. The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (Computer-aided design, CAD/Compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. The definition of the Darboux integral has the advantage of being easier to apply in computations or proofs than that of the Riemann integral. Consequently, introductory textbooks on calculus and real analysis often develop Riemann integration using the Darboux integral, rather than the true Riemann integral. Moreover, the definition is readily extended to defining Riemann–Stieltjes integration. Darboux integrals are named after their inventor, Gaston Darboux (1842–1917). Definition The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real-valued ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
