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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 - Computer vision - ApplicationsIn the related fields machine vision and medical imaging, systems using computer vision techniques are sold in markets worth billions of US dollars per year.
One interesting application of computer vision, commonly used in the creation of visual effects for cinema and broadcast, is camera tracking or matchmoving. Computer vision also finds its applications in medicine, military industry, security and surveillance, quality inspection, robo ...
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 - Applications |
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| | | | | surfaces: Encyclopedia II - Computer vision - Related Fields Advanced systems are often borrowing from many different fields like pattern recognition, statistical learning, projective geometry, image processing, graph theory and other.
Cognitive computer vision is strongly related to cognitive psychology and biological computation.
Computer vision - A University Video Communication on Model-Based Computer Vision.
Joseph Mundy in a University Video Communication on Model-Based Computer Vision (1987):
"What do students need to l ...
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 - Related Fields |
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| | | | | surfaces: Encyclopedia II - Carl Friedrich Gauss - Commemorations From 1989 until the end of 2001, his portrait and a normal distribution curve were featured on the German ten-mark banknote. Germany has issued three stamps honouring Gauss, as well. The stamp pictured above, no. 725, was issued in 1955 on the hundredth anniversary of his death; two other stamps, no. 1246 and 1811, were issued in 1977, the 200th anniversary of his birth.
G. Waldo Dunnington was a lifelong student of Gauss. He wrote many articles, and a biography: Carl Frederick Gauss: Titan of Science. This book was reissued in 20 ...
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 - Commemorations |
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| | | | | surfaces: Encyclopedia II - Differential geometry and topology - Intrinsic versus extrinsic Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outsid ...
See also:Differential geometry and topology, Differential geometry and topology - Intrinsic versus extrinsic, Differential geometry and topology - Technical requirements, Differential geometry and topology - Differential topology, Differential geometry and topology - Branches of differential geometry, Differential geometry and topology - Contact geometry, Differential geometry and topology - Finsler geometry, Differential geometry and topology - Riemannian geometry, Differential geometry and topology - Symplectic topology, Differential geometry and topology - Reference books Read more here: » Differential geometry and topology: Encyclopedia II - Differential geometry and topology - Intrinsic versus extrinsic |
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| | | | | surfaces: Encyclopedia II - Manifold - Motivational example: the circle The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1 with its centre at the origin. If x and y are the coordinates of a point on the circle, then we have x² + y² = 1.
Locally, the circle resembles a line, which is one-dimensional. In other words, we need only one coordinate to describe the circle locally. Consider, for instance, the top part of the circle, for which the y-coordinate is positive (this is the yellow ...
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 - Motivational example: the circle |
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| | | | | surfaces: Encyclopedia II - Manifold - Introduction A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or three dimensional spaces: a line, a plane, or the three-dimensional space in which we live. Or, it may be an abstract space of some higher dimension or even of infinite dimension. Some authors allow manifolds to have separate pieces of different dimensions, but all authors require all pieces of a connected manifold to have the same dimension. A manifold with all pieces of dimension n is called an n-manifold. By contrast, gluing a one-dimensional string to three dimensional ball makes an ob ...
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 - Introduction |
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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 - 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
For more details on this topic, see topological manifold.
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. Th ...
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 - 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 - 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 - Differentiable manifolds
For more details on this topic, see differentiable manifold.
It is easy to define topological manifolds, but it is very hard to work with them. For most applications a special kind of topological manifold, a differentiable manifold, works better. If the local charts on a manifold are compatible in a certain sense, one can talk about directions, tangent spaces, and differentiable functions on that manifold. In particular it is possi ...
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 - Differentiable manifolds |
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| | | | | | surfaces: Encyclopedia II - Orientability - Examples in low dimensions Surfaces we normally encounter in every day life are orientable. For example, sphere, plane, torus. Example of non-orientable surfaces are Möbius strip, real projective plane, Klein bottle. These surfaces as visualized in 3-dimensions all have just one-side. Note that locally an embedded surface always has two sides, so a near-sighted ant crawling on a one-sided surface would think there is an "other side". The essence of one-sidedness is that the ant can crawl from one side of the surface to the "other" without going through the surface or ...
See also:Orientability, Orientability - Examples in low dimensions, Orientability - Orientation by a triangulation, Orientability - Orientation by top-dimensional forms, Orientability - Orientation and vector bundles Read more here: » Orientability: Encyclopedia II - Orientability - Examples in low dimensions |
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| | | | | surfaces: Encyclopedia II - Differential geometry and topology - Branches of differential geometry
Differential geometry and topology - Contact geometry.
Contact geometry is an analog of symplectic geometry which works for certain manifolds of odd dimension. Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a hyperplane field that is nowhere integrable. This is equivalent to the hyperplane field being defined by a 1-form α such that does not vanish anywh ...
See also:Differential geometry and topology, Differential geometry and topology - Intrinsic versus extrinsic, Differential geometry and topology - Technical requirements, Differential geometry and topology - Differential topology, Differential geometry and topology - Branches of differential geometry, Differential geometry and topology - Contact geometry, Differential geometry and topology - Finsler geometry, Differential geometry and topology - Riemannian geometry, Differential geometry and topology - Symplectic topology, Differential geometry and topology - Reference books Read more here: » Differential geometry and topology: Encyclopedia II - Differential geometry and topology - Branches of differential geometry |
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| | | | | surfaces: Encyclopedia II - Differential geometry and topology - Differential topology Differential topology per se considers the properties and structures that require only a smooth structure on a manifold to define (such as those in the previous section). Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but ...
See also:Differential geometry and topology, Differential geometry and topology - Intrinsic versus extrinsic, Differential geometry and topology - Technical requirements, Differential geometry and topology - Differential topology, Differential geometry and topology - Branches of differential geometry, Differential geometry and topology - Contact geometry, Differential geometry and topology - Finsler geometry, Differential geometry and topology - Riemannian geometry, Differential geometry and topology - Symplectic topology, Differential geometry and topology - Reference books Read more here: » Differential geometry and topology: Encyclopedia II - Differential geometry and topology - Differential topology |
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| | | | | surfaces: Encyclopedia II - Fan implement - History
Fan implement - Etymology.
Old English fann referred to a basket or shovel for winnowing. It was a loan from Latin vannus, with the same meaning, derived from ventus "wind" or a related root (cf. vates). In the sense of "device for moving air" the word is first attested 1390, the hand-held version is first recorded in 1555.
Fan implement - Ancient.
Fan history stretches back thousands of years. Since antiquity, fans have possessed a dual function – a stat ...
See also:Fan implement, Fan implement - History, Fan implement - Etymology, Fan implement - Ancient, Fan implement - Asia, Fan implement - Europe, Fan implement - Mechanical development, Fan implement - Mechanical devices, Fan implement - Types, Fan implement - Table fan, Fan implement - Ceiling fan, Fan implement - Solar powered fan, Fan implement - Gas turbine fan, Fan implement - Aft fan, Fan implement - Supersonic fan, Fan implement - Supersonic through-flow fan, Fan implement - Variable pitch fan, Fan implement - Variable geometry fan, Fan implement - Propfan, Fan implement - Overhung fan, Fan implement - Snubbered fan, Fan implement - Wide chord fan, Fan implement - Swept fan, Fan implement - Other meanings, Fan implement - Books Read more here: » Fan implement: Encyclopedia II - Fan implement - History |
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| | | | | surfaces: Encyclopedia II - Laplace operator - Definition The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient:
Equivalently, the Laplacian is the sum of all the unmixed second partial derivatives:
Here, it is understood that the xi are Cartesian coordinates on the space; the equation takes a different form in spherical coordinates and cylindrical coordinates, as shown below.
In the three-dimensional space the Laplac ...
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 - Definition |
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| | | | | surfaces: Encyclopedia II - Michael Polanyi - Physical chemistry Polanyi's scientific interests were diverse, embracing chemical kinetics, x-ray diffraction and the absorption of gases at solid surfaces.
In 1934, Polanyi, roughly contemporarily with G. I. Taylor and Egon Orowan realised that the plastic deformation of ductile materials could be explained in terms of the theory of dislocations developed by Vito Volterra in 1905. The insight was critical in developing the modern science of solid mechanics.
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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 - Physical chemistry |
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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 - 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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