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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] |
Projective Configuration
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. Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book ''Geometrie der Lage'', in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book ''Anschauliche Geometrie'', reprinted in English as . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be ''realizable'' in that geometry), or as a type of abstract incidence geometry. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pencil Of Circles
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] |
Circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an obtuse triangle, the minimum bounding circle has the longest sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthocenter
In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the ''extended base'' of the altitude. The intersection of the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of orthogonal projection. Altitudes can be used in the computation of the area of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the trigonometri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point. Similarly, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conic
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a ''focus'', and some particular line, called a ''directrix'', are in a fixed ratio, called the ''eccentricity''. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of deg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the greatest mathematicians and physicists and among the most influential scientists of all time. He was a key figure in the philosophical revolution known as the Enlightenment. His book (''Mathematical Principles of Natural Philosophy''), first published in 1687, established classical mechanics. Newton also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for developing infinitesimal calculus. In the , Newton formulated the laws of motion and universal gravitation that formed the dominant scientific viewpoint for centuries until it was superseded by the theory of relativity. Newton used his mathematical description of gravity to derive Kepler's laws of planetary motion, account for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called '' Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plane by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mario Pieri
Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry. Biography Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore Pincherle. Obtaining a scholarship, Pieri transferred to ''Scuola Normale Superiore'' in Pisa. There he took his degree in 1884 and worked first at a technical secondary school in Pisa. When an opportunity arose at the military academy in Turin to teach projective geometry, Pieri moved there and, by 1888, he was also an assistant instructor in the same subject at the University of Turin. By 1891, he had become ''libero docente'' at the university, teaching elective courses. Pieri continued to teach in Turin until 1900 when, through competition, he was awarded the position of ''extraordinary professor'' at University of Catania on the island of S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Range
In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the lines of a pencil. A projectivity is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of projective harmonic conjugates. Indeed, three points on a projective line determine a fourth by this relation. Application of a projectivity to this quadruple results in four points likewise in the harmonic relation. Such a quadruple of points is termed a harmonic range. In 1940 Julian Coolidge described this structure and identified its originator: :Two fundamental one-dimensional forms such as point ranges, pencils of lines, or of planes are defined as projective, when their membe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
