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Polygon Triangulation
In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is . Triangulations may be viewed as special cases of planar straight-line graphs. When there are no holes or added points, triangulations form maximal outerplanar graphs. Polygon triangulation without extra vertices Over time, a number of algorithms have been proposed to triangulate a polygon. Special cases It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex ''n''-gon by non-intersecting diagonals is the (''n''−2)nd Catalan number, which equals :\frac, a formula found by Leonhard Euler. A monotone polygon can be triangulated in linear time with either the algorithm of A. Fournier and D.Y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fan Triangulation
In computational geometry, a fan triangulation is a simple way to triangulate a polygon by choosing a vertex and drawing edges to all of the other vertices of the polygon. Not every polygon can be triangulated this way, so this method is usually only used for convex polygons. Properties Aside from the properties of all triangulations, fan triangulations have the following properties: * All convex polygons, but not all polygons, can be fan triangulated. * Polygons with only one concave vertex can always be fan triangulated, as long as the diagonals are drawn from the concave vertex. * It can be known if a polygon can be fan triangulated by solving the Art gallery problem, in order to determine whether there is at least one vertex that is visible from every point in the polygon. * The triangulation of a polygon with n vertices uses n - 3 diagonals, and generates n - 2 triangles. * Generating the list of triangles is trivial if an ordered list of vertices is available, and can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algorith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ear (mathematics)
In geometry, a vertex (in plural form: vertices or vertexes) is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called "convex" if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angles); otherwise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two Ears Theorem
In geometry, the two ears theorem states that every simple polygon with more than three vertices has at least two ears, vertices that can be removed from the polygon without introducing any crossings. The two ears theorem is equivalent to the existence of polygon triangulations. It is frequently attributed to Gary H. Meisters, but was proved earlier by Max Dehn. Statement of the theorem An ear of a polygon is defined as a vertex such that the line segment between the two neighbors of lies entirely in the interior of the polygon. The two ears theorem states that every simple polygon has at least two ears. Ears from triangulations An ear and its two neighbors form a triangle within the polygon that is not crossed by any other part of the polygon. Removing a triangle of this type produces a polygon with fewer sides, and repeatedly removing ears allows any simple polygon to be triangulated. Conversely, if a polygon is triangulated, the weak dual of the triangulation (a graph wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Godfried Toussaint
Godfried Theodore Patrick Toussaint (1944 – July 2019) was a Canadian computer scientist, a professor of computer science, and the head of the Computer Science Program at New York University Abu Dhabi (NYUAD) in Abu Dhabi, United Arab Emirates. He is considered to be the father of computational geometry in Canada. He did research on various aspects of computational geometry, discrete geometry, and their applications: pattern recognition (k-nearest neighbor algorithm, cluster analysis), motion planning, visualization (computer graphics), knot theory (stuck unknot problem), linkage (mechanical) reconfiguration, the art gallery problem, polygon triangulation, the largest empty circle problem, unimodality (unimodal function), and others. Other interests included meander (art), compass and straightedge constructions, instance-based learning, music information retrieval, and computational music theory. He was a co-founder of the Annual ACM Symposium on Computational Geometry, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ACM Transactions On Graphics
''ACM Transactions on Graphics'' (TOG) is a bimonthly peer-reviewed scientific journal that covers the field of computer graphics. It was established in 1982 and is published by the Association for Computing Machinery. TOG publishes two special issues for ACM SIGGRAPH's conference proceedings. Starting in 2003, all papers accepted for presentation at the annual SIGGRAPH conference are printed in a special summer issue of the journal. Beginning in 2008, papers presented at SIGGRAPH Asia are printed in a special November/December issue. The editor-in-chief is Carol O'Sullivan (Trinity College Dublin). According to the ''Journal Citation Reports'', the journal had a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 5.414. The journal ranks 1st in computer gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alain Fournier
Alain Fournier (1943–2000) was a computer graphics researcher. Biography Alain Fournier was born on November 5, 1943, in Lyon, France. He was married twice, first to Beverly Bickle (married 1968, divorced 1984) and later to Adrienne Drobnies, with whom he had one daughter, Ariel. Fournier's early training was in chemistry, culminating in a B.Sc. from INSA, France, in 1965. After emigrating from France to Montreal, Quebec, Canada in the 1970s, he co-wrote a textbook on chemistry, and taught the subject in Quebec. His career in computer graphics spanned only about 20 years. In 1980 he received a Ph.D. in computer science from the University of Texas at Dallas under the supervision of Zvi Meir Kedem, and with Donald Fussell and Loren Carpenter reported the results of his Ph.D. work on stochastic modelling in a seminal paper in 1980. He then went on to an outstanding academic career, first at the University of Toronto as part of the Dynamic Graphics Project and subsequently at th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Polygon
In geometry, a polygon ''P'' in the plane is called monotone with respect to a straight line ''L'', if every line orthogonal to ''L'' intersects the boundary of ''P'' at most twice. Similarly, a polygonal chain ''C'' is called monotone with respect to a straight line ''L'', if every line orthogonal to ''L'' intersects ''C'' at most once. For many practical purposes this definition may be extended to allow cases when some edges of ''P'' are orthogonal to ''L'', and a simple polygon may be called monotone if a line segment that connects two points in ''P'' and is orthogonal to ''L'' lies completely in ''P''. Following the terminology for monotone functions, the former definition describes polygons strictly monotone with respect to ''L''. Properties Assume that ''L'' coincides with the ''x''-axis. Then the leftmost and rightmost vertices of a monotone polygon decompose its boundary into two monotone polygonal chains such that when the vertices of any chain are being traversed in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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