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Comparison Triangle
In metric geometry, comparison triangles are constructions used to define higher bounds on curvature in the framework of locally geodesic metric spaces, thereby playing a similar role to that of higher bounds on sectional curvature in Riemannian geometry. Definitions Comparison triangles Let M_^ = \mathbb^2 be the euclidean plane, M_^ = \mathbb^2 be the unit 2-sphere, and M_^ = \mathbb^2 be the hyperbolic plane. For k > 0, let M_^ and M_^ denote the spaces obtained, respectively, from M_^ and M_^ by multiplying the distance by \frac. For any k\in \R, M_^ is the unique complete, simply-connected, 2-dimensional Riemannian manifold of constant sectional curvature k. Let X be a metric space. Let T be a geodesic triangle in X, i.e. three points p, q and r and three geodesic segments , q/math>, , r/math> and , p/math>. A comparison triangle T* in M_^ for T is a geodesic triangle in M_^ with vertices p', q' and r' such that d(p,q) = d(p',q'), d(p,r) = d(p',r') and d(r,q) = d(r',q') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Geometry
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
A Haefliger
A, or a, is the first letter and the first vowel letter of the Latin alphabet, used in the modern English alphabet, and others worldwide. Its name in English is '' a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version is often written in one of two forms: the double-storey and single-storey . The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English, '' a'' is the indefinite article, with the alternative form ''an''. Name In English, the name of the letter is the ''long A'' sound, pronounced . Its name in most other languages matches the letter's pronunciation in open syllables. History The earliest known ancestor of A is ''aleph''—the first letter of the Phoenician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultralimit
In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces X_n. The concept captures the limiting behavior of finite configurations in the X_n spaces employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence in metric spaces. Ultrafilters An ultrafilter, denoted as ω, on the set of natural numbers is a set of nonempty subsets of (whose inclusion function can be thought of as a measure) which is closed under finite intersection, upwards-closed, and also which, given any subset ''X'' of , contains either ''X'' or . An ultrafilter on is ''non-principal'' if it contains no finite set. Limit of a sequence of points with respect to an ultrafilter In the following, ''ω'' is a non-principal ultrafilter on \mathbb N . If (x_n)_ is a sequence of points in a metric space (''X'',''d'') and ''x''∈ ''X'', then the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Tree
In mathematics, real trees (also called \mathbb R-trees) are a class of metric spaces generalising simplicial Tree (graph theory), trees. They arise naturally in many mathematical contexts, in particular geometric group theory and probability theory. They are also the simplest examples of Gromov hyperbolic spaces. Definition and examples Formal definition A metric space X is a real tree if it is a Geodesic metric space, geodesic space where every triangle is a tripod. That is, for every three points x, y, \rho \in X there exists a point c = x \wedge y such that the geodesic segments [\rho,x], [\rho,y] intersect in the segment [\rho,c] and also c \in [x,y]. This definition is equivalent to X being a "zero-hyperbolic space" in the sense of Gromov (all triangles are "zero-thin"). Real trees can also be characterised by a topology, topological property. A metric space X is a real tree if for any pair of points x, y \in X all topological embeddings \sigma of the segment [0,1] int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a Conceptual metaphor , metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Triangle Inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#Triangle, degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of a triangle then the triangle inequality states that :c \leq a + b , with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (Norm (mathematics), norms): :\, \mathbf u + \mathbf v\, \leq \, \mathbf u\, + \, \mathbf v\, , where the length of the third side has been replaced by the length of the vector sum . When and are real numbers, they can be viewed as vectors in \R^1, and the triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov Product
In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define ''δ''-hyperbolic metric spaces in the sense of Gromov. Definition Let (''X'', ''d'') be a metric space and let ''x'', ''y'', ''z'' ∈ ''X''. Then the Gromov product of ''y'' and ''z'' at ''x'', denoted (''y'', ''z'')''x'', is defined by :(y, z)_ = \frac1 \big( d(x, y) + d(x, z) - d(y, z) \big). Motivation Given three points ''x'', ''y'', ''z'' in the metric space ''X'', by the triangle inequality there exist non-negative numbers ''a'', ''b'', ''c'' such that d(x,y) = a + b, \ d(x,z) = a + c, \ d(y,z) = b + c. Then the Gromov products are (y,z)_x = a, \ (x,z)_y = b, \ (x,y)_z = c. In the case that the points ''x'', ''y'', ''z'' are the outer nodes of a tripod then these Gromov products are the lengths of the edges. In the hyperbolic, spherical or euclidean plane, the Gromov produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Angle
In geometry, an angle of a polygon is formed by two adjacent sides. For a simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than a straight angle ( radians or 180°), then the polygon is called convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.Posamentier, Alfred S., and Lehmann, Ingmar. ''The Secrets of Triangles'', Prometheus Books, 2012. Properties * The sum of the internal angle and the external angle on the same vertex is radians (180°). * The sum of all the internal angles of a simple polygon is radians or degrees, where is the number of sides. The formula can be prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Great Circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non- antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the ''minor arc'', and is the shortest surface-path between them. Its arc length is the great-circle distance between the points (the intrinsic distance on a sphere), and is proportional to the measure of the central angle formed by the two points and the center of the sphere. A great circle is the largest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic Triangle
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. It is a generalization of the notion of a " straight line". The noun ''geodesic'' and the adjective '' 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 geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a 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 geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
