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surfaces | A Wisdom Archive on surfaces | | surfaces A selection of articles related to surfaces | |
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| ARTICLES RELATED TO surfaces | | | |  surfaces: Encyclopedia II - Carl Friedrich Gauss - Biography
Carl Friedrich Gauss - Early years.
Gauss was born in Brunswick, in the Duchy of Brunswick-Lüneburg (now part of Lower Saxony, Germany), as the only son of uneducated lower-class parents. According to legend, his gifts became apparent at the age of three when he corrected, in his head, an error his father had made on paper while calculating finances. Another story has it that in elementary school his teacher tried to occupy pupils by making them add up the integers from 1 to 100. The young Gauss produced the cor ...
See also:Carl Friedrich Gauss, Carl Friedrich Gauss - Biography, Carl Friedrich Gauss - Early years, Carl Friedrich Gauss - Middle years, Carl Friedrich Gauss - Later years death and afterwards, Carl Friedrich Gauss - Family, Carl Friedrich Gauss - Personality, Carl Friedrich Gauss - Commemorations Read more here: » Carl Friedrich Gauss: Encyclopedia II - Carl Friedrich Gauss - Biography |
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| | | | | surfaces: Encyclopedia II - Michael Polanyi - Early life Michael was born into a Jewish family in Budapest. His older brother Karl become a famous economist. Their father was an engineer and entrepreneur whose volatile fortunes in railway speculation motivated Polanyi to seek financial stability through a career in medicine. He graduated in 1913, and shortly afterwards served as a physician in the Austro-Hungarian army during World War I, but was hospitalised, and during his convalescence wrote what became a doctorate in physical chem ...
See also:Michael Polanyi, Michael Polanyi - Early life, Michael Polanyi - Physical chemistry, Michael Polanyi - Philosophy of science, Michael Polanyi - Economics, Michael Polanyi - Honours, Michael Polanyi - Knowledge, Michael Polanyi - Bibliography Read more here: » Michael Polanyi: Encyclopedia II - Michael Polanyi - Early life |
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| | | | | | surfaces: Encyclopedia II - Laplace operator - Laplace-Beltrami operator The Laplacian can be extended to functions defined on surfaces, or more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace-Beltrami operator. One defines it, just as the Laplacian, as the divergence of the gradient. To be able to find a formula for this operator, one will need to first write the divergence and the gradient on a manifold.
If g denotes the (pseudo)-metric tensor on the manifold, one finds that the vol ...
See also:Laplace operator, Laplace operator - Definition, Laplace operator - Coordinate expressions, Laplace operator - Identities, Laplace operator - Laplace-Beltrami operator, Laplace operator - Laplace-de Rham operator, Laplace operator - Properties Read more here: » Laplace operator: Encyclopedia II - Laplace operator - Laplace-Beltrami operator |
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| | | | | | surfaces: Encyclopedia II - Unifying theories in mathematics - Mathematical theories The term theory is used informally within mathematics to mean a self-consistent body of definitions, axioms, theorems, examples, and so on. (Examples include group theory, Galois theory, control theory, and K-theory.) In particular there is no connotation of hypothetical. Thus the term unifying theory is more like sociological term used to study the actions of mathematicians. It may assume nothing conjectural, that would be analogous to an undiscovered scientific link. There is really no cognate within mathematics to ...
See also:Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory Read more here: » Unifying theories in mathematics: Encyclopedia II - Unifying theories in mathematics - Mathematical theories |
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| | | | | surfaces: Encyclopedia II - Computer vision - State of the art The field of computer vision can be characterized as immature and diverse. Even though earlier work exists, it was not until the late 1970's that a more focused study of the field started when computers could manage the processing of large data sets such as images. However, these studies usually originated from various other fields, and consequently there is no standard formulation of the "computer vision problem". Also, and to an even larger extent, there is no standard formulation of how computer vision problems should be solved. Instead, ...
See also:Computer vision, Computer vision - State of the art, Computer vision - Examples of applications for computer vision, Computer vision - Typical tasks of computer vision, Computer vision - Object Recognition, Computer vision - Optical Character Recognition, Computer vision - Tracking, Computer vision - Scene interpretation, Computer vision - Egomotion, Computer vision - Computer Vision Systems, Computer vision - Image acquisition, Computer vision - Preprocessing, Computer vision - Feature extraction, Computer vision - Registration, Computer vision - Related Fields, Computer vision - A University Video Communication on Model-Based Computer Vision, Computer vision - Applications Read more here: » Computer vision: Encyclopedia II - Computer vision - State of the art |
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| | | | | | surfaces: Encyclopedia II - Curvature - Curvature of plane curves For a plane curve C, the curvature at a given point P has a magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve at the given point), and is a vector pointing in the direction of that circle's center. The smaller the radius r of the osculating circle, the larger the magnitude of the curvature (1/r) will be; so that where a curve is "nearly straight", the curvature will be close to zero, and where the curve undergoes ...
See also:Curvature, Curvature - Curvature of plane curves, Curvature - Local expressions, Curvature - Example, Curvature - Curvature of space curves, Curvature - Curvature of surfaces in 3-space, Curvature - Curvature of space Read more here: » Curvature: Encyclopedia II - Curvature - Curvature of plane curves |
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| | | | | surfaces: Encyclopedia II - Attractor - Types of attractors Attractors are parts of the phase space of the dynamical system. Until the 1960s, as evidenced by textbooks of that era, attractors were thought of as being geometrical subsets of the phase space: points, lines, surfaces, volumes. The (topologically) wild sets that had been observed were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.
Two simple attractors are the fixed point and the limit cycle. There can be many other geometri ...
See also:Attractor, Attractor - Motivation and definition, Attractor - Mathematical definition, Attractor - Types of attractors, Attractor - Fixed point, Attractor - Limit cycle, Attractor - Limit tori, Attractor - Strange attractor, Attractor - Partial differential equations Read more here: » Attractor: Encyclopedia II - Attractor - Types of attractors |
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| | | | | surfaces: Encyclopedia II - Curvature - Curvature of surfaces in 3-space For two-dimensional surfaces embedded in R3, there are two kinds of curvature: Gaussian curvature and mean curvature. To compute these at a given point of the surface, consider the intersection of the surface with a plane containing a fixed normal vector at the point. This intersection is a plane curve and has a curvature; if we vary the plane, this curvature will change, and there are two extremal values - the maximal and the minimal curvature, called the principal curvatures, k1 and ...
See also:Curvature, Curvature - Curvature of plane curves, Curvature - Local expressions, Curvature - Example, Curvature - Curvature of surfaces in 3-space, Curvature - Curvature of space Read more here: » Curvature: Encyclopedia II - Curvature - Curvature of surfaces in 3-space |
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| | | | | surfaces: Encyclopedia II - Manifold - Charts, atlases and transition maps Charts
A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and we are interested in the topological structure. This structure is preserved by homeomo ...
See also:Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts, atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History Read more here: » Manifold: Encyclopedia II - Manifold - Charts, atlases and transition maps |
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| | | | | surfaces: Encyclopedia II - Unifying theories in mathematics - Recent developments in relation with modular theory A well-known example is the Taniyama-Shimura conjecture, now the Taniyama-Shimura theorem, which proposed that each elliptic curve over the rational numbers can be translated into a modular form (in such a way as to preserve the associated L-function). There are difficulties in identifying this with an isomorphism, in any strict sense of the word. Certain curves had been known to be both elliptic curves (of genus 1) and modular curves, before the conjecture was formulated (about 1955). The surprising part of the conjecture was the extension ...
See also:Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory Read more here: » Unifying theories in mathematics: Encyclopedia II - Unifying theories in mathematics - Recent developments in relation with modular theory |
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| | | | | surfaces: Encyclopedia II - Sandpaper - Types of sandpaper There are countless varieties of sandpaper, with variations in the paper or backing, the material used for the grit, grit size, and the bond.
Sandpaper - Backing.
In addition to paper, backing for sandpaper includes cloth (cotton, polyester, rayon), polyester film (Mylar), and "Fibre". Cloth backing is used for sanding discs and belts, while mylar is used with extremely fine grits. Fibre or vulcanized fibre is a strong backing material consisting of many layers of impregnated paper made from rags. The weig ...
See also:Sandpaper, Sandpaper - Types of sandpaper, Sandpaper - Backing, Sandpaper - Material, Sandpaper - Bonds, Sandpaper - Shapes, Sandpaper - Grit sizes, Sandpaper - Grit size table, Sandpaper - History Read more here: » Sandpaper: Encyclopedia II - Sandpaper - Types of sandpaper |
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| | | | | | surfaces: Encyclopedia II - Unifying theories in mathematics - Geometrical theories A well-known example was the development of analytic geometry, which in the hands of mathematicians such as Descartes and Fermat showed that many theorems about curves and surfaces of special types could be stated in algebraic language (then new), each of which could then be proved using the same techniques (that is, the theorems were very similar algebraically, even if the geometrical interpretations were distinct.)
At the end of the 19th century, Felix Klein noted that the many branches of geometry which had been developed during th ...
See also:Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory Read more here: » Unifying theories in mathematics: Encyclopedia II - Unifying theories in mathematics - Geometrical theories |
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| | | | | surfaces: Encyclopedia II - Manifold - Charts atlases and transition maps Charts
A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and we are interested in the topological structure. This structure is preserved by homeomor ...
See also:Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History Read more here: » Manifold: Encyclopedia II - Manifold - Charts atlases and transition maps |
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| | | | | surfaces: Encyclopedia II - Manifold - Topological manifolds The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rn. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rn. These homeomorphisms are the charts of the manifold.
Usually additional t ...
See also:Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History Read more here: » Manifold: Encyclopedia II - Manifold - Topological manifolds |
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| | | | | surfaces: Encyclopedia II - Manifold - Orientability Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to itself acquire a sense of ordering, which we can think of as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, we can choose charts so that overlapping regions agree on their "handedness"; these are orientable manifo ...
See also:Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History Read more here: » Manifold: Encyclopedia II - Manifold - Orientability |
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| | | | | | surfaces: Encyclopedia II - Manifold - History The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intrinsic differential geometry with his theorema egregium.
Bernhard Riemann was the first to do extensive work that required a generalization of manifolds to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translates as "manifoldness". In his Göttingen inaugural lecture, Riemann states the possible values a p ...
See also:Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History Read more here: » Manifold: Encyclopedia II - Manifold - History |
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