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Poincaré Disk Model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group , the quotient of the special unitary group SU(1,1) by its center . Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. The Poincaré ball model is the similar model for ''3'' or ''n''-dimensional hyperbolic geometry in which the points of the geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion. Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ideal Point
In hyperbolic geometry, an ideal point, omega point or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line ''l'' and a point ''P'' not on ''l'', right- and left-limiting parallels to ''l'' through ''P'' converge to ''l'' at ''ideal points''. Unlike the projective case, ideal points form a boundary, not a submanifold. So, these lines do not intersect at an ideal point and such points, although well-defined, do not belong to the hyperbolic space itself. The ideal points together form the Cayley absolute or boundary of a hyperbolic geometry. For instance, the unit circle forms the Cayley absolute of the Poincaré disk model and the Klein disk model. The real line forms the Cayley absolute of the Poincaré half-plane model. Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point. Properties * The hyperbolic distance between an ideal point and any other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cayley–Klein Metric
In mathematics, a Cayley–Klein metric is a metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of distance" where he calls the quadric the absolute. The construction was developed in further detail by Felix Klein in papers in 1871 and 1873, and subsequent books and papers. The Cayley–Klein metrics are a unifying idea in geometry since the method is used to provide metrics in hyperbolic geometry, elliptic geometry, and Euclidean geometry. The field of non-Euclidean geometry rests largely on the footing provided by Cayley–Klein metrics. Foundations The algebra of throws by Karl von Staudt (1847) is an approach to geometry that is independent of metric. The idea was to use the relation of projective harmonic conjugates and cross-ratios as fundamental to the measure on a line. Another important insight was the Laguerre formula by Edmond Laguerre (1853), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Orthogonal Circles
In geometry, two circles are said to be orthogonal if their respective tangent lines at the points of intersection are perpendicular (meet at a right angle). A straight line through a circle's center is orthogonal to it, and if straight lines are also considered as a kind of generalized circles, for instance in inversive geometry, then an orthogonal pair of lines or line and circle are orthogonal generalized circles. In the conformal disk model of the hyperbolic plane, every geodesic is an arc of a generalized circle orthogonal to the circle of ideal points bounding the disk. See also * Orthogonality * Radical axis * Power center (geometry) * Apollonian circles * Bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal coordinates, orthogonal coordinate system based on the Apollonian circles.Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 19 ... References * * * * Circles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geodesics
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection (mathematics), connection. It is a generalization of the notion of a "Line (geometry), straight line". The noun ''wikt:geodesic, geodesic'' and the adjective ''wikt:geodetic, geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any Ellipsoidal geodesic, ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's Planetary surface, surface. For a spherical Earth, it is a line segment, segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a Distance (graph theory), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Droites DisquePoincare
Les Droites () is a mountain in the Mont Blanc massif in the French Alps and is the lowest of the List of Alpine Four-thousanders, 4000-metre peaks in the Alps. The mountain has two summits: * West summit (3,984 m), first ascent by W. A. B. Coolidge, Christian Almer and Ulrich Almer on 16 July 1876 * East summit (4,000 m), first ascent by Thomas Middlemore and John Oakley Maund with Henri Cordier (mountaineer), Henri Cordier, Johann Jaun and Andreas Maurer on 7 August 1876 The north face of the mountain rises some 1,600 m from the Argentière basin at an average angle of 60°, and is the steepest face on the 10-km-long ridge that stretches from the Aiguille Verte to Mont Dolent. The first route to be made on it was via the central couloir on the north-east flank by Bobi Arsandaux and Jacques Lagarde on 31 July 1930. The north spur was first climbed in 1972 by France, French alpinist Nicolas Jaeger. The dangers of climbing this face were highlighted on an episode of the Discovery C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hans Reichenbach
Hans Reichenbach (; ; September 26, 1891 – April 9, 1953) was a leading philosopher of science, educator, and proponent of logical empiricism. He was influential in the areas of science, education, and of logical empiricism. He founded the ''Gesellschaft für empirische Philosophie'' (Society for Empirical Philosophy) in Berlin in 1928, also known as the " Berlin Circle". Carl Gustav Hempel, Richard von Mises, David Hilbert and Kurt Grelling all became members of the Berlin Circle. In 1930, Reichenbach and Rudolf Carnap became editors of the journal '' Erkenntnis''. He also made lasting contributions to the study of empiricism based on a theory of probability; the logic and the philosophy of mathematics; space, time, and relativity theory; analysis of probabilistic reasoning; and quantum mechanics. In 1951, he authored ''The Rise of Scientific Philosophy'', his most popular book. Early life Hans was the second son of a Jewish merchant, Bruno Reichenbach, who had conve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rudolf Carnap
Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. Biography Carnap's father rose from being a poor ribbon-weaver to be the owner of a ribbon-making factory. His mother came from an academic family; her father was an educational reformer and her oldest brother was the archaeologist Wilhelm Dörpfeld. As a ten-year-old, Carnap accompanied Wilhelm Dörpfeld on an expedition to Greece. Carnap was raised in a profoundly religious Protestant family, but later became an atheist. He began his formal education at the Barmen Gymnasium (school), Gymnasium and the Gymnasium in Jena. From 1910 to 1914, he attended the University of Jena, intending to write a thesis in physics. He also intently studied Immanuel Kant's ''Critique of Pure Reason'' during a course taught by Bruno Bauch, and was one of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Coefficient Of Dilatation
Thermal expansion is the tendency of matter to increase in length, area, or volume, changing its size and density, in response to an increase in temperature (usually excluding phase transitions). Substances usually contract with decreasing temperature (thermal contraction), with rare exceptions within limited temperature ranges (''negative thermal expansion''). Temperature is a monotonic function of the average molecular kinetic energy of a substance. As energy in particles increases, they start moving faster and faster, weakening the intermolecular forces between them and therefore expanding the substance. When a substance is heated, molecules begin to vibrate and move more, usually creating more distance between themselves. The relative expansion (also called strain) divided by the change in temperature is called the material's coefficient of linear thermal expansion and generally varies with temperature. Prediction If an equation of state is available, it can be used to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
