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Projective geometry

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Projective geometry is a topic in mathematics. [1]

117 relations: Affine geometry, Affine transformation, Alessandro Padoa, Alfred Clebsch, Alfred North Whitehead, Algebra, Algebraic curve, Algebraic geometry, Algebraic topology, Analytic geometry, Angle, Axiom, Beltrami–Klein model, Bernhard Riemann, Blaise Pascal, Brianchon's theorem, Chern class, Classical group, Compass (drawing tool), Complete quadrangle, Complex number, Complex projective space, Configuration (geometry), Conic section, Corrado Segre, Cross-ratio, David Hilbert, Desargues's theorem, Differential geometry, Dimension, Division ring, Duality (projective geometry), Ellipse, Elliptic geometry, Enumerative geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Erlangen program, Euclidean geometry, Euclidean space, Fano plane, Federigo Enriques, Felix Klein, Filippo Brunelleschi, Finite geometry, Francesco Severi, Gaspard Monge, Geometric transformation, Gino Fano, Girard Desargues, Giuseppe Peano, ..., Grassmann–Cayley algebra, Grassmannian, Group (mathematics), H. F. Baker, Harold Scott MacDonald Coxeter, Homogeneous coordinates, Homography, Howard Eves, Hyperbola, Hyperbolic geometry, Hyperbolic space, Hyperplane, Incidence (geometry), Incidence geometry, Incidence structure, Infinity, Invariant (mathematics), Invariant theory, Italian school of algebraic geometry, Jakob Steiner, Jean-Victor Poncelet, Johannes Kepler, Joseph Diez Gergonne, Joseph Wedderburn, Julius Plücker, Karl Georg Christian von Staudt, Lazare Carnot, Line (geometry), Line coordinates, Linear algebra, Mario Pieri, Mathematics, Max Noether, Möbius transformation, Metric (mathematics), Michel Chasles, Model theory, Non-Desarguesian plane, Non-Euclidean geometry, Ordered geometry, Pappus of Alexandria, Pappus's hexagon theorem, Parabola, Parallel (geometry), Pascal's theorem, Paul Dirac, Perspective (graphical), Point (geometry), Point at infinity, Polyhedron, Projective differential geometry, Projective harmonic conjugate, Projective line, Projective line over a ring, Projective linear group, Projective plane, Projective range, Projective space, Quadric, Quantum mechanics, Real projective plane, Robin Hartshorne, Springer Science+Business Media, Straightedge, Synthetic geometry, Transformation matrix, Translation (geometry). Expand index (67 more) »

Affine geometry

In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting") the metric notions of distance and angle.

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Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

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Alessandro Padoa

Alessandro Padoa (14 October 1868 – 25 November 1937) was an Italian mathematician and logician, a contributor to the school of Giuseppe Peano.

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Alfred Clebsch

Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory.

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Alfred North Whitehead

Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher.

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Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

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Algebraic curve

In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

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Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

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Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

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Analytic geometry

In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.

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In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

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An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

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Beltrami–Klein model

In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or n-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.

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Bernhard Riemann

Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.

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Blaise Pascal

Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, writer and Catholic theologian.

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Brianchon's theorem

In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point.

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Chern class

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles.

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Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals, the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.

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Compass (drawing tool)

A pair of compasses, also known simply as a bow compass, is a technical drawing instrument that can be used for inscribing circles or arcs.

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Complete quadrangle

In mathematics, specifically 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.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers.

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Configuration (geometry)

In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.

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Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane.

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Corrado Segre

Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry.

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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.

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David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.

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Desargues's theorem

In projective geometry, Desargues's theorem, named after Girard Desargues, states: Denote the three vertices of one triangle by and, and those of the other by and.

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Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

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In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

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Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

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Duality (projective geometry)

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept.

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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.

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Elliptic geometry

Elliptic geometry is a geometry in which Euclid's parallel postulate does not hold.

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Enumerative geometry

In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.

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Ergebnisse der Mathematik und ihrer Grenzgebiete

Ergebnisse der Mathematik und ihrer Grenzgebiete/A Series of Modern Surveys in Mathematics is a series of scholarly monographs published by Springer Science+Business Media.

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Erlangen program

The Erlangen program is a method of characterizing geometries based on group theory and projective geometry.

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Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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Fano plane

In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane of order 2.

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Federigo Enriques

Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.

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Felix Klein

Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.

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Filippo Brunelleschi

Filippo Brunelleschi (1377 – April 15, 1446) was an Italian designer and a key figure in architecture, recognised to be the first modern engineer, planner and sole construction supervisor.

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Finite geometry

A finite geometry is any geometric system that has only a finite number of points.

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Francesco Severi

Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician.

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Gaspard Monge

Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818) was a French mathematician, the inventor of descriptive geometry (the mathematical basis of technical drawing), and the father of differential geometry.

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Geometric transformation

A geometric transformation is any bijection of a set having some geometric structure to itself or another such set.

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Gino Fano

Gino Fano (5 January 18718 November 1952) was an Italian mathematician, best known as the founder of finite geometry.

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Girard Desargues

Girard Desargues (21 February 1591 – September 1661) was a French mathematician and engineer, who is considered one of the founders of projective geometry.

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Giuseppe Peano

Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist.

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Grassmann–Cayley algebra

In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product.

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In mathematics, the Grassmannian is a space which parametrizes all -dimensional linear subspaces of the n-dimensional vector space.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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H. F. Baker

Henry Frederick Baker FRS FRSE (3 July 1866 – 17 March 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as solitons), and Lie groups.

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Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 – March 31, 2003) was a British-born Canadian geometer.

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Homogeneous coordinates

In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.

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In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive.

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Howard Eves

Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics.

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In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set.

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Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

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Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

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In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

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Incidence (geometry)

In geometry, an incidence relation is a binary relation between different types of objects that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used.

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Incidence geometry

In mathematics, incidence geometry is the study of incidence structures.

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Incidence structure

In mathematics, an abstract system consisting of two types of objects and a single relationship between these types of objects is called an incidence structure.

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Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

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Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

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Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions.

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Italian school of algebraic geometry

In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885–1935) done internationally in birational geometry, particularly on algebraic surfaces.

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Jakob Steiner

Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry.

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Jean-Victor Poncelet

Jean-Victor Poncelet (1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique.

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Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer, and astrologer.

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Joseph Diez Gergonne

Joseph Diez Gergonne (19 June 1771 at Nancy, France – 4 May 1859 at Montpellier, France) was a French mathematician and logician.

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Joseph Wedderburn

Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882, Forfar, Angus, Scotland – 9 October 1948, Princeton, New Jersey) was a Scottish mathematician, who taught at Princeton University for most of his career.

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Julius Plücker

Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist.

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Karl Georg Christian 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.

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Lazare Carnot

Lazare Nicolas Marguerite, Count Carnot (13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician.

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Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

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Line coordinates

In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point.

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Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

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Mario Pieri

Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry.

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Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Max Noether

Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions.

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Möbius transformation

In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0.

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Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

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Michel Chasles

Michel Floréal Chasles (15 November 1793 – 18 December 1880) was a French mathematician.

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Model theory

In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

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Non-Desarguesian plane

In mathematics, a non-Desarguesian plane, named after Girard Desargues, is a projective plane that does not satisfy Desargues' theorem, or in other words a plane that is not a Desarguesian plane.

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Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

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Ordered geometry

Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement.

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Pappus of Alexandria

Pappus of Alexandria (Πάππος ὁ Ἀλεξανδρεύς; c. 290 – c. 350 AD) was one of the last great Greek mathematicians of Antiquity, known for his Synagoge (Συναγωγή) or Collection (c. 340), and for Pappus's hexagon theorem in projective geometry.

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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.

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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped.

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Parallel (geometry)

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.

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Pascal's theorem

In projective geometry, Pascal's theorem (also known as the hexagrammum mysticum theorem) states that if six arbitrary points are chosen on a conic (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in three points which lie on a straight line, called the Pascal line of the hexagon.

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Paul Dirac

Paul Adrien Maurice Dirac (8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century.

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Perspective (graphical)

Perspective (from perspicere "to see through") in the graphic arts is an approximate representation, generally on a flat surface (such as paper), of an image as it is seen by the eye.

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Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

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Point at infinity

In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.

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In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

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Projective differential geometry

In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group.

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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: The point D does not depend on what point L is taken initially, nor upon what line through C is used to find M and N. This fact follows from Desargues theorem.

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Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.

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Projective line over a ring

In mathematics, the projective line over a ring is an extension of the concept of projective line over a field.

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Projective linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V).

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Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

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Projective range

In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion.

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Projective space

In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.

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In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

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Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.

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Robin Hartshorne

Robin Cope Hartshorne (born March 15, 1938) is an American mathematician.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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A straightedge or straight edge is a tool with a straight edge, used for drawing straight lines, or checking their straightness.

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Synthetic geometry

Synthetic geometry (sometimes referred to as axiomatic or even pure geometry) is the study of geometry without the use of coordinates or formulas.

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Transformation matrix

In linear algebra, linear transformations can be represented by matrices.

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Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

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Algebraic projective geometry, Axioms of projective geometry, Projective Geometry.


[1] https://en.wikipedia.org/wiki/Projective_geometry

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