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Cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defined as : (A,B;C,D) = \frac where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.) The point is the harmonic conjugate of with respect to and precisely if the cross-ratio of the quadruple is , called the ''harmonic ratio''. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name ''anharmonic ratio''. The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Harmonic Conjugate
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points 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 and . 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: \overline:\overline = \overline:\overline \, . If these segments are now endowed with the ordinary metri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''projective space'') and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "Point at infinity, points at infinity") to Euclidean points, and vice versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translation (geometry), translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in Euclidean geometry, the concept of an angle does not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Of Alexandria
Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known about his life except for what can be found in his own writings, many of which are lost. Pappus apparently lived in Alexandria, where he worked as a Mathematics education, mathematics teacher to higher level students, one of whom was named Hermodorus.Pierre Dedron, J. Itard (1959) ''Mathematics And Mathematicians'', Vol. 1, p. 149 (trans. Judith V. Field) (Transworld Student Library, 1974) The ''Collection'', his best-known work, is a compendium of mathematics in eight volumes, the bulk of which survives. It covers a wide range of topics that were part of the ancient mathematics curriculum, including geometry, astronomy, and mechanics. Pappus was active in a period generally considered one of stagnation in mathematical studies, where, to s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane (mathematics), plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudosphere, pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they local property, locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of event (relativity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Line
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case). There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field ''K'', commonly denoted P1(''K''), as the set of one-dimensional subspaces of a two-dimensional ''K''-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals is a manifold; see '' Real projective line'' for details. Homogeneous coordinates An arbitrary point in the projective line P1(''K'') may be represented by an equivalence class of '' homogeneous coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projectively Extended Real Line
In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standard arithmetic operations extended where possible, and is sometimes denoted by \mathbb^* or \widehat. The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded. The projectively extended real line may be identified with a real projective line in which three points have been assigned the specific values , and . The projectively extended real number line is distinct from the affinely extended real number line, in which and are distinct. Dividing by zero Unlike most mathematical models of numbers, this structure allows division by zero: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an import ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazare Carnot
Lazare Nicolas Marguerite, Comte Carnot (; 13 May 1753 – 2 August 1823) was a French mathematician, physicist, military officer, politician and a leading member of the Committee of Public Safety during the French Revolution. His military reforms, which included the introduction of mass conscription (''levée en masse''), were instrumental in transforming the French Revolutionary Army into an effective fighting force. Carnot was elected to the National Convention in 1792, and a year later he became a member of the Committee of Public Safety, where he directed the French war effort as one of the Ministers of War during the War of the First Coalition. He oversaw the reorganization of the army, imposed discipline, and significantly expanded the French force through the imposition of mass conscription. Credited with France's renewed military success from 1793 to 1794, Carnot came to be known as the "Organizer of Victory". Increasingly disillusioned with the radical politics of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michel Chasles
Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician. Biography He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coalition he was drafted to fight in the defence of Paris in 1814. After the war, he gave up on a career as an engineer or stockbroker in order to pursue his mathematical studies. In 1837 he published the book ''Aperçu historique sur l'origine et le développement des méthodes en géométrie'' ("Historical view of the origin and development of methods in geometry"), a study of the method of reciprocal polars in projective geometry. The work gained him considerable fame and respect and he was appointed Professor at the École Polytechnique in 1841, then he was awarded a chair at the Sorbonne in 1846. A second edition of this book was published in 1875. In 1839, Ludwig Adolph Sohncke (the father of Leonhard Sohncke) translated the original ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Von Staudt
Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic. Life and influence Karl was born in the Free Imperial City of Rothenburg, which is now called Rothenburg ob der Tauber in Germany. From 1814 he studied in Gymnasium in Ausbach. He attended the University of Göttingen from 1818 to 1822 where he studied with Gauss who was director of the observatory. Staudt provided an ephemeris for the orbits of Mars and the asteroid Pallas. When in 1821 Comet Nicollet-Pons was observed, he provided the elements of its orbit. These accomplishments in astronomy earned him his doctorate from University of Erlangen in 1822. Staudt's professional career began as a secondary school instructor in Würzburg until 1827 and then Nuremberg until 1835. He married Jeanette Dreschler in 1832. They had a son Eduard and daughter Mathilda, but Jeanette died in 1848. The book ''Geometrie der Lage'' (184 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Howard Eves
Howard Whitley Eves (10 January 1911, 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics. Eves received his B.S. from the University of Virginia, an M.A. from Harvard University, and a Ph.D. in mathematics from Oregon State University in 1948, the last with a dissertation titled ''A Class of Projective Space Curves'' written under Ingomar Hostetter. He then spent most of his career at the University of Maine, 1954–1976. In later life, he occasionally taught at University of Central Florida. Eves was a strong spokesman for the Mathematical Association of America, which he joined in 1942, and whose Northeast Section he founded. For 25 years he edited the Elementary Problems section of the ''American Mathematical Monthly''. He solved over 300 problems proposed in various mathematical journals. His six volume ''Mathematical Circles'' series, collecting humorous and interesting anecdotes about mathematicians, was recently re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
