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Perspective (geometry)
Two figures in a plane are perspective from a point ''O'', called the center of perspectivity, if the lines joining corresponding points of the figures all meet at ''O''. Dually, the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in projective geometry where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions. Terminology The line which goes through the points where the figure's corresponding sides intersect is known as the axis of perspectivity, perspective axis, homology axis, or archaically, perspectrix. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the center of perspectivity, perspective center, homology center, pole, or arch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues Theorem Alt
Girard Desargues (; 21 February 1591September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry. Desargues' theorem, the Desargues graph, and the crater Desargues (crater), Desargues on the Moon are named in his honour. Biography Born in Lyon, Desargues came from a family devoted to service to the French crown. His father was a royal civil law notary, notary, an investigating commissioner of the Seneschal, Seneschal's court in Lyon (1574), the collector of the tithes on ecclesiastical revenues for the city of Lyon (1583) and for the diocese of Lyon. Girard Desargues worked as an architecture, architect from 1645. Prior to that, he had worked as a tutor and may have served as an engineer and technical consultant in the entourage of Cardinal Richelieu, Richelieu. Yet his involvement in the Siege of La Rochelle, though alleged by Ch. Weiss in ''Biographie Universelle'' (1842), has never been testified. As an architect, Des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Duality
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to ''space duality'' and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that determines which points lie on which lines. These set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, Academic journal, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, Technology, technical, and engineering subject areas, abandoning its literary interests. Wiley's son Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perspective (graphical)
Linear or point-projection perspective () is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, generally on a flat surface, of an image as it is seen by the eye. Perspective drawing is useful for representing a three-dimensional scene in a two-dimensional medium, like paper. It is based on the optical fact that for a person an object looks N times (linearly) smaller if it has been moved N times further from the eye than the original distance was. The most characteristic features of linear perspective are that objects appear smaller as their distance from the observer increases, and that they are subject to , meaning that an object's dimensions parallel to the line of sight appear shorter than its dimensions perpendicular to the line of sight. All objects will recede to points in the distance, usually along the horizon line, but also above and below the horiz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvilinear Perspective
Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces, for which (straight) lines on the 3D object are projected to curves on the 2D surface that are typically not straight (hence the qualifier "curvilinear"). It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book ''La Perspective curviligne'',Albert Flocon and André Barre, ''La Perspective curviligne'', Flammarion, Éditeur, Paris, 1968 which was translated into English in 1987 as ''Curvilinear Perspective: From Visual Space to the Constructed Image'' and published by the University of California Press.Albert Flocon and André Barre, ''Curvilinear Perspective: From Visual Space to the Constructed Image'', (Robert Hansen, translator), University of California Press, Berkeley and Los Angeles, California, 1987 Curvilinear perspective is sometimes colloquially called fisheye perspective, by analogy to a fisheye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reye Configuration
In geometry, the Reye configuration, introduced by , is a Configuration (geometry), configuration of 12 Point (geometry), points and 16 Line (geometry), lines. Each point of the configuration belongs to four lines, and each line contains three points. Therefore, in the notation of configurations, the Reye configuration is written as . It is symmetric (both point and line transitive) and has 576 automorphisms. Realization The Reye configuration can be realized in three-dimensional projective space by taking the lines to be the 12 edges and four long diagonals of a cube, and the points as the eight vertices of the cube, its center, and the three points where groups of four parallel cube edges meet the plane at infinity. Two regular tetrahedron, tetrahedra may be inscribed within a cube, forming a stella octangula; these two tetrahedra are perspective figures to each other in four different ways, and the other four points of the configuration are their centers of perspectivity. Thes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Configuration
In geometry, the Pappus configuration is a configuration of nine points and nine lines in the Euclidean plane, with three points per line and three lines through each point. History and construction This configuration is named after Pappus of Alexandria. Pappus's hexagon theorem states that every two triples of collinear points and (none of which lie on the intersection of the two lines) can be completed to form a Pappus configuration, by adding the six lines , , , , , and , and their three intersection points , , and . These three points are the intersection points of the "opposite" sides of the hexagon . According to Pappus' theorem, the resulting system of nine points and eight lines always has a ninth line containing the three intersection points , , and , called the ''Pappus line''. The Pappus configuration can also be derived from two triangles and that are in perspective with each other (the three lines through corresponding pairs of points meet at a single crossin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus's Hexagon Theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that *given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear, lying on the ''Pappus line''. These three points are the points of intersection of the "opposite" sides of the hexagon AbCaBc. It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes. If one considers a pappian plane containing a hexagon as just described but with sides Ab and aB parallel and also sides Bc and bC parallel (so that the Pappus line u is the line at infinity), one gets the ''affine version'' of Pappus's theorem shown in the second diagram. If the Pappus line u and the lines g,h have a point in common, one gets the so-called little version of Pappus's theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Hexagon
Pappus may refer to: * Pappus (botany), a structure within certain flowers * ''Pappus'' (bug), a genus of insects in the tribe Mirini * Pappus of Alexandria, Greek mathematician ** Pappus's hexagon theorem, often just called 'Pappus's theorem', a theorem named for Pappus of Alexandria ** Pappus's centroid theorem, another theorem named for Pappus of Alexandria ** Pappus configuration, a geometric configuration related to 'Pappus's theorem' ** Pappus graph, a graph related to the pappus configuration See also * Papus (other) * Pappu, an Indian male given name ** Pappu (cinematographer) Sudheesh Pappu (1977/1978 – 14 November 2022), professionally credited mononymously as Pappu, was an Indian cinematographer who worked in Malayalam cinema. Early life Pappu was born in Thrippunithura, Ernakulam, Kerala. He was schooled in Ke ... (1977–2022), Indian cinematographer * ''Pappu'' (film), 1980 Indian film {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues Configuration
In geometry, the Desargues configuration is a Configuration (geometry), configuration of ten points and ten lines, with three points per line and three lines per point. It is named after Girard Desargues. The Desargues configuration can be constructed in two dimensions from the points and lines occurring in Desargues's theorem, in three dimensions from five planes in general position, or in four dimensions from the 5-cell, the four-dimensional regular simplex. It has a large group of symmetries, taking any point to any other point and any line to any other line. It is also self-dual, meaning that if the points are replaced by lines and vice versa using projective duality, the same configuration results. Graph (discrete mathematics), Graphs associated with the Desargues configuration include the Desargues graph (its graph of point-line incidences) and the Petersen graph (its graph of non-incident lines). The Desargues configuration is one of ten different configurations with ten p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
