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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] |
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] |
Concentric
In geometry, two or more objects are said to be ''concentric'' when they share the same center. Any pair of (possibly unalike) objects with well-defined centers can be concentric, including circles, spheres, regular polygons, regular polyhedra, parallelograms, cones, conic sections, and quadrics. Geometric objects are '' coaxial'' if they share the same axis (line of symmetry). Geometric objects with a well-defined axis include circles (any line through the center), spheres, cylinders, conic sections, and surfaces of revolution. Concentric objects are often part of the broad category of '' whorled patterns'', which also includes '' spirals'' (a curve which emanates from a point, moving farther away as it revolves around the point). Geometric properties In the Euclidean plane, two circles that are concentric necessarily have different radii from each other.. However, circles in three-dimensional space may be concentric, and have the same radius as each other, but nevert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Points At Infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring. In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point). In the ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Polyhedron
In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space. The Platonic solids and Archimedean solids have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra can be represented as ideal polyhedra – a polyhedron can be ideal only when it can be represented in Euclidean geometry with all its vertices on a circumscribed sphere. Using linear programming, it is possible to test whether a polyhedron has an ideal version, in polynomial time. Every tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometimes called ideal vertices. All ideal triangles are congruent. Properties Ideal triangles have the following properties: * All ideal triangles are congruent to each other. * The interior angles of an ideal triangle are all zero. * An ideal triangle has infinite perimeter. * An ideal triangle is the largest possible triangle in hyperbolic geometry. In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties: * Any ideal triangle has area π. Distances in an ideal triangle * The inscribed circle to an ideal triangle has radius r=\ln\sqrt = \frac \ln 3 = \operatorname\frac = 2 \operatorname(2- \sqrt) = = \operatorname\frac\sqrt = \operatorname\frac\sqrt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperboloid Model
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperboloid in (''n''+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and ''m''-planes are represented by the intersections of (''m''+1)-planes passing through the origin in Minkowski space with ''S''+ or by wedge products of ''m'' vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the ''n''-sphere is embedded in (''n''+1)-dimensional Euclidean space. Other models of hyperbolic space can be thought of as map projections of ''S''+: the Beltrami–Klein model is the projection of ''S''+ through the origin onto a plane perpe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc (geometry)
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geometry), point. This is the definition that appeared more than 2000 years ago in Euclid's Elements, Euclid's ''Elements'': "The [curved] line is […] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which […] will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image (mathematics), image of an interval (mathematics), interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this artic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Sphere
In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -sphere of unit radius in -dimensional Euclidean space; the unit circle is a special case, the unit -sphere in the Euclidean plane, plane. An (Open set, open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center. A sphere or ball with unit radius and center at the origin (mathematics), origin of the space is called ''the'' unit sphere or ''the'' unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation (geometry), translation and scaling (geometry), scaling, so the study of spheres in general can often be reduced to the study of the unit sphere. The unit sphere is often used as a model for spherical geometry because it has constant sectional cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferdinand Karl Schweikart
Ferdinand Karl Schweikart (1780–1857) was a German jurist and amateur mathematician who developed an ''astral geometry'' before the discovery of non-Euclidean geometry. Life and work Schweikart, son of an attorney in Hesse, was educated in the school of his town. He went to the high school in Hanau and Waldeck before entering in 1796 to study law in the university of Marburg, where he attended lectures of the mathematics professor J.K.F. Hauff. He was awarded a doctorate in law at the university of Jena in 1798. After practicing as a lawyer for a few years in Erbach, he was, from 1803 to 1807, instructor of the youngest prince of Hohenlohe-Ingelfingen. From 1809, he was university professor of jurisprudence successively at the universities of Giessen (1809-1812), Kharkiv (1812-1816), Marburg (1816-1821) and Königsberg (1821 afterwards)., 1909. Meyers Großes Konversations-Lexikon. But Schweikart is best remembered for his works on mathematics: in 1807 he published Die Theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. ''Perpendicular'' is also used as a noun: a perpendicular is a line which is perpendicular to a given line or plane. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either simple polygon, simple (not self-intersecting), or complex polygon, complex (self-intersecting, or crossed). Simple quadrilaterals are either convex polygon, convex or concave polygon, concave. The Internal and external angle, interior angles of a simple (and Plane (geometry), planar) quadrilateral ''ABCD'' add up to 360 Degree (angle), degrees, that is :\angle A+\angle B+\angle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal Triangle
In hyperbolic geometry an ideal triangle is a hyperbolic triangle whose three vertices all are ideal points. Ideal triangles are also sometimes called ''triply asymptotic triangles'' or ''trebly asymptotic triangles''. The vertices are sometimes called ideal vertices. All ideal triangles are congruent. Properties Ideal triangles have the following properties: * All ideal triangles are congruent to each other. * The interior angles of an ideal triangle are all zero. * An ideal triangle has infinite perimeter. * An ideal triangle is the largest possible triangle in hyperbolic geometry. In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties: * Any ideal triangle has area π. Distances in an ideal triangle * The inscribed circle to an ideal triangle has radius r=\ln\sqrt = \frac \ln 3 = \operatorname\frac = 2 \operatorname(2- \sqrt) = = \operatorname\frac\sqrt = \operatorname\frac\sqrt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
