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surfaces | A Wisdom Archive on surfaces | | surfaces A selection of articles related to surfaces | |
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surfaces
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| ARTICLES RELATED TO surfaces | | | |  surfaces: Encyclopedia II - Manifold - Motivational example: the circleThe 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, only one coordinate is needed to describe the circle locally. Consider, for instance, the top part of the circle, for which the y-coordinate is positive (the yellow part i ...
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 - 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 which we inhabit; 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 - 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 - Introduction |
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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 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 - Unifying theories in mathematics - Uniting theories On a less grandiose scale, there are frequent instances in which it appears that sets of results in two different branches of mathematics are similar, and one might ask whether there is a unifying framework which clarifies the connections. We have already noted the example of analytic geometry, and more generally the field of algebraic geometry thoroughly develops the connections between geometric objects (varieties, or more generally schemes) and algebraic ones (ideals); the touchstone result here is Hilbert's Nullstellensatz which roughly speaking shows that there is ...
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 - Uniting theories |
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| | | | | surfaces: Encyclopedia II - Unifying theories in mathematics - Category theory as a rival An alternative (mostly complementary) to set theory but also serving to give a consistent approach to most of axiomatic mathematics is category theory, developed in the second half of the 20th century. A key theme from this point of view is that mathematics studies not only of certain kinds of objects (Lie groups, Banach spaces, etc.) but also of the mappings between them.
In particular, this clarifies exactly what it means for the mathematical objects to be considered to be the same. (For example, are all equilateral triangles ...
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 - Category theory as a rival |
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| | | | | surfaces: Encyclopedia II - Unifying theories in mathematics - Bourbaki The cause of axiomatic development was taken up in earnest by the Bourbaki group of mathematicians. Taken to its extreme this attitude demanded developing mathematics in its greatest generality, starting from the most general axioms and then specializing (e.g. introducing vector spaces over arbitrary fields and limiting to the real numbers only when absolutely necessary) even when the specializations were the theorems of primary interest.
In particular, this perspective placed little value on fields of mathematics (such as combinatori ...
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 - Bourbaki |
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| | | | | surfaces: Encyclopedia II - Unifying theories in mathematics - Through-axiomatisation Early in the 20th century, parallel to the development of mathematical logic as a stand-alone branch of mathematics, many parts of mathematics began to treated by delineating useful sets of axioms and then studying their consequences. Thus for example the studies of "hypercomplex numbers", popular at the turn of the century, were put onto an axiomatic footing as branches of ring theory (in this case, sp ...
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 - Through-axiomatisation |
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| | | | | | surfaces: Encyclopedia II - Michael Polanyi - Philosophy of science From the middle years of the Nineteen-Thirties Polanyi began to articulate his opposition to the prevailing positivist account of science, arguing that it failed to recognise the part played by tacit knowledge and the creative role played by the imagination. He viewed positivism as encouraging some to believe that scientific research ought to be directed by the State. He drew attention to what happened to genetics in the Soviet Union, once the doctrines of Trofim Lysenko gained political approval. Polanyi, like Friedrich Hayek, supplied r ...
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 - Philosophy of science |
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| | | | | surfaces: Encyclopedia II - Curve - Conventions and terminology The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading.
Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more ...
See also:Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History Read more here: » Curve: Encyclopedia II - Curve - Conventions and terminology |
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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 space curves, 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 - Curve - Differential geometry Main article: differential geometry of curves
While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general re ...
See also:Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History Read more here: » Curve: Encyclopedia II - Curve - Differential geometry |
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| | | | | surfaces: Encyclopedia II - Curve - History A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachroni ...
See also:Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History Read more here: » Curve: Encyclopedia II - Curve - History |
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| | | | | surfaces: Encyclopedia II - Curve - Algebraic curve Main article: Algebraic curve
In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces.
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See also:Curve, Curve - Definitions, Curve - Conventions and terminology, Curve - Lengths of curves, Curve - Differential geometry, Curve - Algebraic curve, Curve - History Read more here: » Curve: Encyclopedia II - Curve - Algebraic curve |
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| | | | | surfaces: Encyclopedia II - Cubism - Synthetic cubism The second phase of cubism, began in 1912, it is called "synthetic cubism".
These works of art are composed of distinct superimposed parts — painted or often pasted onto the canvas — and are characterized by brighter colours, something that they had previously tried to reintroduce, but were unsuccesful in doing so in a smooth transitory way. Unlike analytic cubism, which fragmented objects into its composing parts or facets, synthetic cubism attempted more to bring many diff ...
See also:Cubism, Cubism - History, Cubism - Analytical cubism, Cubism - Synthetic cubism, Cubism - Well-known cubists Read more here: » Cubism: Encyclopedia II - Cubism - Synthetic cubism |
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| | | | | | surfaces: Encyclopedia II - Computer vision - Examples of applications for computer vision Another way to describe computer vision is in terms of applications areas. One of the most prominent application fields is medical computer vision or medical image processing. This area is characterized by the extraction of information from image data for the purpose of making a medical diagnosis of a patient. Typically image data is in the form of microscopy images, X-ray images, angiography images, ultrasonic images, and tomography images. An example of information which can be extracted from such image data is detection of tumours, arteri ...
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 - Examples of applications for computer vision |
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| | | | | surfaces: Encyclopedia II - Attractor - Motivation and definition Dynamical systems are often described in terms of differential equations. These equations describe the behavior of the system for a short period of time. To determine the behavior for longer periods it is necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers. Dynamical systems that come from applications tend to be dissipative: if it were not for some driving force the motion would cease. (The dissipation may come from internal friction, thermodynamic losses, or loss of ma ...
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 - Motivation and definition |
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| | | | | surfaces: Encyclopedia II - Computer vision - Typical tasks of computer vision
Computer vision - Object Recognition.
Detecting the presence of known objects or living beings in an image, possibly together with estimating the pose of these objects.
Examples:
Searching in digital images for specific content (content-based image retrieval)
Recognizing human faces and their location in images.
Estimation of the three-dimensional pose of humans and their limbs
Detection of objects which are passing through a manufacturing process, e.g., on a conveyor belt, and estimation of their po ...
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 - Typical tasks of computer vision |
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| | | | | surfaces: Encyclopedia II - Computer vision - Computer Vision Systems A typical computer vision system can be divided in the following subsystems:
Computer vision - Image acquisition.
The image or image sequence is acquired with a imaging system (camera,radar,lidar,tomography system). Often the imaging system has to be calibrated before being used.
Computer vision - Preprocessing.
In the preprocessing step, the image is being treated with "low-level"-operations. The aim of this step is to do noise reduction on the image (i.e. to dissociate ...
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 - Computer Vision Systems |
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