surfaces

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve). Curve - Definitions. In ma ...

Including:

• Curve - Definitions
• Curve - Conventions and terminology
• Curve - Lengths of curves
• Curve - Differential geometry
• Curve - Algebraic curve
• Curve - History

Read more here: » Curve: Encyclopedia - Curve

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds. There are close connections to other fields, such as 4-manifolds, surfaces, knot theory, topological quantum field theory, and gauge theory. 3-manifold theory is a part o ...

Including:

• 3-manifold - Some famous examples of 3-manifolds
• 3-manifold - Some important classes of 3-manifolds
• 3-manifold - Some important structures on 3-manifolds
• 3-manifold - Foundational results
• 3-manifold - Some famous conjectures

Read more here: » 3-manifold: Encyclopedia - 3-manifold

Curvature is the amount by which a geometric object deviates from being flat. The word flat might have very different meanings depending on the objects considered (for curves it is a straight line and for surfaces it is a Euclidean plane). For example, curvature of a circle is the inverse of its radius. Smaller circles bend more sharply, and hence have higher curvature. Further, curvature of a smooth curve is defined as the curvature of its osculating circle at each point. In a plane, this is a scalar quantity, bu ...

Including:

• 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

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Cubism was probably the most important and influential art movement since the Italian Renaissance; it was an avant-garde art movement that revolutionized European painting and sculpture in the early 20th century. In cubist artworks, objects are broken up, analyzed, and re-assembled in an abstracted form — instead of rendering objects from a single fixed angle, the artist depicts the subject from multiple angles simultaneously as an attempt to present the subject in the most complete manner. Often the surfaces of the facets, o ...

Including:

• Cubism - History
• Cubism - Analytical cubism
• Cubism - Synthetic cubism
• Cubism - Well-known cubists

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In dynamical systems, an attractor is a set to which the system evolves after a long enough time. For the set to be an attractor, trajectories that get close enough to the attractor must remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with fractal structures known as a strange attractor. Describing the attractors of chaotic dynamical systems h ...

Including:

• 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

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A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, di ...

Including:

• 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 - Manifold

Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. Sometimes known as "the prince of mathematicians", Gauss had a remarkable influence in many fields of mathematics and science and is ranked beside Euler, Newton ...

Including:

• 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 - Carl Friedrich Gauss

Computer vision is the study and application of methods which allow computers to "understand" image content or content of multidimensional data in general. The term "understand" means here that specific information is being extracted from the image data for a specific purpose: either for presenting it to a human operator (e. g., if cancerous cells have been detected in a microscopy image), or for controlling some process (e. g., an industry robot or an autonomous vehicle). The image data that is fed into a computer vision system is of ...

Including:

• 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 - Computer vision

For a finite CW-complex and in particular for a finite simplicial complex, the Euler characteristic can be defined as the alternating sum where ki denotes the number of cells of dimension i. Then, one can define the Euler characteristic of a manifold as the Euler characteristic of a simplicial complex homeomorphic to it. For example, the circle and torus have ...

See also:

Euler characteristic, Euler characteristic - Definitions and properties, Euler characteristic - Partially ordered set, Euler characteristic - Proof

Read more here: » Euler characteristic: Encyclopedia II - Euler characteristic - Definitions and properties

Mechanically, a fan can be any revolving vane or vanes used for producing currents of air. Fans produce air flows with high volume and low pressure, as opposed to a gas compressor which produces high pressures at a comparatively low volume. Fans are useful for moving large quantities of air, which is suited for applications such as winnowing grain or blowing a fire, cooling and ventilation purposes, and in conjunction with a heat source for heating and drying. A fan blade will often rotate when exposed to an air stream, and devices that take advantage of this, such as anemometers and wind t ...

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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 - Mechanical devices

The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives,integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives. These all relate to multivariable calculus; but for the geometric applications must be developed in a way that makes good sense without a preferred coordinate system. The distinctive concepts of differential geometry can be said to be those that embody the geo ...

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 - Technical requirements

The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to product of the arc length s of C and the distance d1 traveled by its centroid. For example, the surface area of the torus with minor radius r and major radius R is ...

See also:

Pappus's centroid theorem, Pappus's centroid theorem - The first theorem, Pappus's centroid theorem - The second theorem

Read more here: » Pappus's centroid theorem: Encyclopedia II - Pappus's centroid theorem - The first theorem

One natural generalization in differential geometry is hyperbolic n-space Hn, the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. In this terminology, the upper half-plane is H2 since it has real dimension 2. In number theory, the theory of Hilbert modular forms is concerned with the study of certain functions on the direct product Hn of n copies of the upper half-plane. Yet another space interesting to number theorists is the Siegel upper half-space Hn ...

See also:

Upper half-plane, Upper half-plane - Generalizations

Read more here: » Upper half-plane: Encyclopedia II - Upper half-plane - Generalizations

Carl Friedrich Gauss - Early years. Gauss was born in Braunschweig, in the Duchy of Braunschweig-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 t ...

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

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

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 - Construction

Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge. We can orient each triangle, by choosing a direction for each edge (think of this as drawing an arrow on each edge) so that the arrows go from head to tail as we go around the boundary of the triangle. If we can do this so that in addition triangles sharing an edge have arrows on that edge ...

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 - Orientation by a triangulation

His contributions include a proof of the Smale conjecture and important results in the theory of surfaces and 3-manifolds. Allen Hatcher - 3-manifolds. Perhaps among his most recognized results in 3-manifolds concern the classification of incompressible surfaces in certain 3-manifolds and their boundary slopes. Bill Floyd and Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. Bill Thurston and Hatcher classified the incompressible surfaces in 2-bridge knot comple ...

See also:

Allen Hatcher, Allen Hatcher - Mathematical contributions, Allen Hatcher - 3-manifolds, Allen Hatcher - Surfaces, Allen Hatcher - Selected publications, Allen Hatcher - Papers, Allen Hatcher - Books

Read more here: » Allen Hatcher: Encyclopedia II - Allen Hatcher - Mathematical contributions

Grit size refers to the size of the particles of abrading materials embedded in the sandpaper. A number of different standards have been established for grit size. These standards establish not only the average grit size, but also the allowable variation from the average. The two most common are the United States CAMI (Coated Abrasive Manufacturers Institute, now part of the Unified Abrasives Manufacturers' Association) and the European FEPA (Federation of European Producers of Abrasives) "P" grade. The FEPA system is the same as the ISO 634 ...

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 - Grit sizes

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

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 - Construction

In mathematics, a (topological) curve is defined as follows. Let I be an interval of real numbers (i.e. a non-empty connected subset of ). Then a curve is a continuous mapping , where X is a topological space. The curve is said to be simple if it is injective, i.e. if for all x, y in I, we have . If I is a closed ...

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 - Definitions