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Plane Sweep
In computational geometry, a sweep line algorithm or plane sweep algorithm is an algorithmic paradigm that uses a conceptual ''sweep line'' or ''sweep surface'' to solve various problems in Euclidean space. It is one of the key techniques in computational geometry. The idea behind algorithms of this type is to imagine that a line (often a vertical line) is swept or moved across the plane, stopping at some points. Geometric operations are restricted to geometric objects that either intersect or are in the immediate vicinity of the sweep line whenever it stops, and the complete solution is available once the line has passed over all objects. History This approach may be traced to scanline algorithms of rendering in computer graphics, followed by exploiting this approach in early algorithms of integrated circuit layout design, in which a geometric description of an IC was processed in parallel strips, because the entire description could not fit into memory. Applications Applicati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-balancing Binary Search Tree
In computer science, a self-balancing binary search tree (BST) is any node-based binary search tree that automatically keeps its height (maximal number of levels below the root) small in the face of arbitrary item insertions and deletions.Donald Knuth. ''The Art of Computer Programming'', Volume 3: ''Sorting and Searching'', Second Edition. Addison-Wesley, 1998. . Section 6.2.3: Balanced Trees, pp.458–481. These operations when designed for a self-balancing binary search tree, contain precautionary measures against boundlessly increasing tree height, so that these abstract data structures receive the attribute "self-balancing". For height-balanced binary trees, the height is defined to be logarithmic \mathcal O(\log n) in the number n of items. This is the case for many binary search trees, such as AVL trees and red–black trees. Splay trees and treaps are self-balancing but not height-balanced, as their height is not guaranteed to be logarithmic in the number of items. S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 analogous to rotating a spring-loaded vernier caliper around the outside of a convex polygon. Every time one blade of the caliper lies flat against an edge of the polygon, it forms an antipodal pair with the point or edge touching the opposite blade. The complete "rotation" of the caliper around the polygon detects all antipodal pairs; the set of all pairs, viewed as a graph, forms a thrackle. The method of rotating calipers can be interpreted as the projective dual of a sweep line algorithm in which the sweep is across slopes of lines rather than across - or -coordinates of points. History The rotating calipers method was first used in the dissertation of Michael Shamos in 1978. Shamos uses this method to generate all antipodal pai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Operations On Polygons
Boolean operations on polygons are a set of Boolean operations (AND, OR, NOT, XOR, ...) operating on one or more sets of polygons in computer graphics. These sets of operations are widely used in computer graphics, CAD, and in EDA (in integrated circuit physical design and verification software). Algorithms * Greiner–Hormann clipping algorithm * Vatti clipping algorithm * Sutherland–Hodgman algorithm (special case algorithm) * Weiler–Atherton clipping algorithm (special case algorithm) Uses in software Early algorithms for Boolean operations on polygons were based on the use of bitmaps. Using bitmaps in modeling polygon shapes has many drawbacks. One of the drawbacks is that the memory usage can be very large, since the resolution of polygons is proportional to the number of bits used to represent polygons. The higher the resolution is desired, the more the number of bits is required. Modern implementations for Boolean operations on polygons tend to use plane swee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delaunay Triangulation
In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle of any triangle in DT(P). Delaunay triangulations maximize the minimum of all the angles of the triangles in the triangulation; they tend to avoid sliver triangles. The triangulation is named after Boris Delaunay for his work on this topic from 1934. For a set of points on the same line there is no Delaunay triangulation (the notion of triangulation is degenerate for this case). For four or more points on the same circle (e.g., the vertices of a rectangle) the Delaunay triangulation is not unique: each of the two possible triangulations that split the quadrangle into two triangles satisfies the "Delaunay condition", i.e., the requirement that the circumcircles of all triangles have empty interiors. By considering circumscribed spheres, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fortune's Algorithm
Fortune's algorithm is a sweep line algorithm for generating a Voronoi diagram from a set of points in a plane using O(''n'' log ''n'') time and O(''n'') space. Section 7.2: Computing the Voronoi Diagram: pp.151–160. It was originally published by Steven Fortune in 1986 in his paper "A sweepline algorithm for Voronoi diagrams." Algorithm description The algorithm maintains both a sweep line and a ''beach line'', which both move through the plane as the algorithm progresses. The sweep line is a straight line, which we may by convention assume to be vertical and moving left to right across the plane. At any time during the algorithm, the input points left of the sweep line will have been incorporated into the Voronoi diagram, while the points right of the sweep line will not have been considered yet. The beach line is not a straight line, but a complicated, piecewise curve to the left of the sweep line, composed of pieces of parabolas; it divides the portion of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Voronoi Diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed there is a corresponding region, called a Voronoi cell, consisting of all points of the plane closer to that seed than to any other. The Voronoi diagram of a set of points is dual to that set's Delaunay triangulation. The Voronoi diagram is named after mathematician Georgy Voronoy, and is also called a Voronoi tessellation, a Voronoi decomposition, a Voronoi partition, or a Dirichlet tessellation (after Peter Gustav Lejeune Dirichlet). Voronoi cells are also known as Thiessen polygons. Voronoi diagrams have practical and theoretical applications in many fields, mainly in science and technology, but also in visual art. The simplest case In the simplest case, shown in the first picture, we are given a finite set of points in the Euclid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bentley–Ottmann Algorithm
In computational geometry, the Bentley–Ottmann algorithm is a sweep line algorithm for listing all ''crossings'' in a set of line segments, i.e. it finds the ''intersection points'' (or, simply, ''intersections'') of line segments. It extends the Shamos–Hoey algorithm, a similar previous algorithm for testing whether or not a set of line segments has any crossings. For an input consisting of n line segments with k crossings (or intersections), the Bentley–Ottmann algorithm takes time \mathcal((n + k) \log n). In cases where k = \mathcal\left(\frac \right), this is an improvement on a naïve algorithm that tests every pair of segments, which takes \Theta(n^2). The algorithm was initially developed by ; it is described in more detail in the textbooks , , and . Although asymptotically faster algorithms are now known by and , the Bentley–Ottmann algorithm remains a practical choice due to its simplicity and low memory requirements. Overall strategy The main idea of the Bentle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '' Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment Intersection
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point or does not exist (if the lines are parallel). Other types of geometric intersection include: * Line–plane intersection * Line–sphere intersection * Intersection of a polyhedron with a line * Line segment intersection * Intersection curve Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloi ... [...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. 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 (CAD/ CAM), but many problems in computational geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Ian Shamos
Michael Ian Shamos (born April 21, 1947) is an American mathematician, attorney, book author, journal editor, consultant and company director. He is (with Franco P. Preparata) the author of ''Computational Geometry'' (Springer-Verlag, 1985), which was for many years the standard textbook in computational geometry, and is known for the Shamos–Hoey sweep line algorithm for line segment intersection detection and for the rotating calipers technique for finding the width and diameter of a geometric figure. His publications also include works on electronic voting, the game of billiards, and intellectual property law in the digital age. He was a fellow of Sigma Xi (1974–83), had an IBM Fellowship at Yale University (1974–75), was SIAM National Lecturer (1977–78), distinguished lecturer in computer science at the University of Rochester (1978), visited McGill University (1979), and belonged to the Duquesne University Law Review (1980–81). He won the first annual Black & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
