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Projective Harmonic Conjugate
In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line through meet at respectively. If and meet at , and meets at , then is called the harmonic conjugate of with respect to . The point does not depend on what point is taken initially, nor upon what line through is used to find and . This fact follows from Desargues theorem. In real projective geometry, harmonic conjugacy can also be defined in terms of the cross-ratio as . Cross-ratio criterion The four points are sometimes called a harmonic range (on the real projective line) as it is found that always divides the segment ''internally'' in the same proportion as divides ''externally''. That is: :, AC, :, BC, = , AD, :, DB, \, . If these segments are now endowed with the ordinary metric interpretation of real num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Quadrangle
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four Point (geometry), points in a Plane (geometry), plane, no three of which are Collinearity, on a common line, and of the six Line (geometry), lines connecting the six pairs of points. Duality (projective geometry), Dually, a ''complete quadrilateral'' is a system of four lines, no three of which pass through the same point, and the six points of Line–line intersection, intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used. Diagonals The six lines of a complete quadrangle meet in pairs to form three additional points called the ''diagonal points'' of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connected by line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversive Geometry
Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotions from any exterior source. An inversive heat source would be a heat source where all the heat remains within the object and is not subject to any format of transference Transference (german: Übertragung) is a phenomenon within psychotherapy in which the "feelings, attitudes, or desires" a person had about one thing are subconsciously projected onto the here-and-now Other. It usually concerns feelings from a ... or externalisation. Is the opposite of Transversive activities and objects which suggest by their very nature that the outcome is transferred to the secondary source. Psychoanalytic terminology Emotion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pole And Polar
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into its polar line and each line in the plane into its pole. Properties Pole and polar have several useful properties: * If a point P lies on the line ''l'', then the pole L of the line ''l'' lies on the polar ''p'' of point P. * If a point P moves along a line ''l'', its polar ''p'' rotates about the pole L of the line ''l''. * If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points. * If a point lies on the conic section, its polar is the tangent through this point to the conic section. * If a point P lies on its own polar line, then P is on the conic section. * Each line has, with respect to a non-degenerated conic section, exactly one pole. Special case of circles The pole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luis Santaló
Luís Antoni Santaló Sors (October 9, 1911 – November 22, 2001) was a Spanish mathematician. He graduated from the University of Madrid and he studied at the University of Hamburg, where he received his Ph.D. in 1936. His advisor was Wilhelm Blaschke. Because of the Spanish Civil War, he moved to Argentina as a professor in the National University of the Littoral, National University of La Plata and University of Buenos Aires. His work with Blaschke on convex sets is now cited in its connection with Mahler volume. Blaschke and Santaló also collaborated on integral geometry. Santaló wrote textbooks in Spanish on non-Euclidean geometry, projective geometry, and tensors. Works Luis Santaló published in both English and Spanish: ''Introduction to Integral Geometry'' (1953) Chapter I. Metric integral geometry of the plane including densities and the isoperimetric inequality. Ch. II. Integral geometry on surfaces including Blaschke's formula and the isoperimetric inequalit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skew Lines
In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar. General position If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines. After the first three points have been chosen, the fourth point will define a non-skew line if, and only if, it is coplanar with the first three points. However, the plane through the first three points forms a subset of measure zero of the cube, and the probability that the fourth point lies on this plane is zero. If it does not, the lines defined by the points will be skew. Similarly, in three-dimensional space a very small perturbati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pencil (mathematics)
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane. Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle. Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two dimensional nature of such a pencil, it is sometimes referred to as a ''flat pencil''. Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line. A more common term for this set is a ''range'' of points. Penci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrastigm
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a ''complete quadrilateral'' is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used. Diagonals The six lines of a complete quadrangle meet in pairs to form three additional points called the ''diagonal points'' of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connected by lines; the line segments connecting these pairs are called ''diagonals''. For points and lines in the Euclidean plane, the diag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Court Publishing
The Open Court Publishing Company is a publisher with offices in Chicago and LaSalle, Illinois. It is part of the Carus Publishing Company of Peru, Illinois. History Open Court was founded in 1887 by Edward C. Hegeler of the Matthiessen-Hegeler Zinc Company, at one time the largest producer of zinc in the United States. Hegeler intended for the firm to serve the purpose of discussing religious and psychological problems on the principle that the scientific world-conception should be applied to religion. Its first managing editor was Paul Carus, Hegeler's son-in-law through his marriage to engineer Mary Hegeler Carus.Fields 1992, pg. 138 For the first 80 years of its existence, the company had its offices in the Hegeler Carus Mansion. Open Court specializes in philosophy, science, and religion. It was one of the first academic presses in the country, as well as one of the first publishers of inexpensive editions of the classics. It also published the journals ''Open Court'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the '' American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to records on JSTOR. Mission and Vision The mission of the MAA is to advance the understanding of mathematics and its impact on our world. We envision a society that values the power and beauty of mathematics and fully realizes its potential to promote human flourishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wesley Young
John Wesley Young (17 November 1879, Columbus, Ohio – 17 February 1932, Hanover, New Hampshire) was an American mathematician who, with Oswald Veblen, introduced the axioms of projective geometry, coauthored a 2-volume work on them, and proved the Veblen–Young theorem. He was a proponent of Euclidean geometry and held it to be substantially "more convenient to employ" than non-Euclidean geometry. His lectures on algebra and geometry were compiled in 1911 and released as ''Lectures on Fundamental Concepts of Algebra and Geometry.'' John Wesley Young was born in 1879 to William Henry Young and Marie Louise Widenhorn Young. William Henry Young was from West Virginia and was of Native American parentage. After serving in the Civil War he was appointed to the consulate in Germany. He taught Mathematics at Ohio University. Marie Louise Widehorn Young was born in Paris, France and spoke French and German fluently. John Wesley grew up in both Europe and America due to his father's prof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Parallelogramm
Fano is a town and ''comune'' of the province of Pesaro and Urbino in the Marche region of Italy. It is a beach resort southeast of Pesaro, located where the ''Via Flaminia'' reaches the Adriatic Sea. It is the third city in the region by population after Ancona and Pesaro. History An ancient town of Marche, it was known as Fanum Fortunae after a temple of Fortuna located there. Its first mention in history dates from 49 BC, when Julius Caesar held it, along with Pisaurum and Ancona. Caesar Augustus established a '' colonia'', and built a wall, some parts of which remain. In 2 AD Augustus also built an arch (which is still standing) at the entrance to the town. In January 271, the Roman Army defeated the Alamanni in the Battle of Fano that took place on the banks of the Metauro river just inland of Fano. Fano was destroyed by Vitiges' Ostrogoths in AD 538. It was rebuilt by the Byzantines, becoming the capital of the maritime Pentapolis ("Five Cities") that included also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
