surfaces - Article Index

Encyclopedia - Curve: Encyclopedia - Curve
In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple...   » Read the article

Encyclopedia - 3-manifold: Encyclopedia - 3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in ...   » Read the article

Encyclopedia - Curvature: Encyclopedia - Curvature
Curvature is the amount by which a geometric object deviates from being flat. The word flat might have very different meanings depending ...   » Read the article

Encyclopedia - Cubism: Encyclopedia - Cubism
Cubism was probably the most important and influential art movement since the Italian Renaissance; it was an avant-garde art movement tha...   » Read the article

Encyclopedia - Attractor: Encyclopedia - Attractor
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, traject...   » Read the article

Encyclopedia - Manifold: Encyclopedia - Manifold
A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For ex...   » Read the article

Encyclopedia - Carl Friedrich Gauss: Encyclopedia - Carl Friedrich Gauss
Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist of profound genius who contr...   » Read the article

Encyclopedia - Computer Vision: Encyclopedia - Computer Vision
Computer vision is the study and application of methods which allow computers to "understand" image content or content of multidimensiona...   » Read the article

Encyclopedia - Euler Characteristic: Encyclopedia Ii - Euler Characteristic - Definitions And Properties
For a finite CW-complex and in particular for a finite simplicial complex, the Euler characteristic can be defined as the alternating sum...   » Read the article

Encyclopedia - Fan Implement: Encyclopedia Ii - Fan Implement - Mechanical Devices
Mechanically, a fan can be any revolving vane or vanes used for producing currents of air. Fans produce air flows with high volume and lo...   » Read the article

Encyclopedia - Differential Geometry And Topology: Encyclopedia Ii - Differential Geometry And Topology - Technical Requirements
The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent...   » Read the article

Encyclopedia - Pappus's Centroid Theorem: Encyclopedia Ii - Pappus's Centroid Theorem - The First Theorem
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 ...   » Read the article

Encyclopedia - Upper Half-plane: Encyclopedia Ii - Upper Half-plane - Generalizations
One natural generalization in differential geometry is hyperbolic n-space Hn, the maximally symmetric, simply connected, n-dimensional Ri...   » Read the article

Encyclopedia - Carl Friedrich Gauss: Encyclopedia Ii - Carl Friedrich Gauss - Biography
Carl Friedrich Gauss - Early years. Gauss was born in Braunschweig, in the Duchy of Braunschweig-Lüneburg (now part of Lower Saxony, G...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Construction
A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly ...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Orientation By A Triangulation
Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has...   » Read the article

Encyclopedia - Allen Hatcher: Encyclopedia Ii - Allen Hatcher - Mathematical Contributions
His contributions include a proof of the Smale conjecture and important results in the theory of surfaces and 3-manifolds. Allen Hatcher...   » Read the article

Encyclopedia - Sandpaper: Encyclopedia Ii - Sandpaper - Grit Sizes
Grit size refers to the size of the particles of abrading materials embedded in the sandpaper. A number of different standards have been ...   » Read the article

Encyclopedia - Manifold: Encyclopedia Ii - Manifold - Construction
A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly ...   » Read the article

Encyclopedia - Curve: Encyclopedia Ii - Curve - Definitions
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 )...   » Read the article

Encyclopedia - Carl Friedrich Gauss: 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...   » Read the article

Encyclopedia - Michael Polanyi: 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 ent...   » Read the article

Encyclopedia - Cubism: Encyclopedia Ii - Cubism - History
Cubism began in 1887 with Georges Braque and Pablo Picasso, who lived in the Montmartre quarter of Paris, France. They met in 1907, and w...   » Read the article

Encyclopedia - Laplace Operator: 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 mo...   » Read the article

Encyclopedia - Mapping Class Group: Encyclopedia Ii - Mapping Class Group - Examples
An easy exercise is to show that: For manifolds of dimension two or higher the mapping class group is often infinite. Generalizing the...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: 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 o...   » Read the article

Encyclopedia - Computer Vision: 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 197...   » Read the article

Encyclopedia - Cubism: Encyclopedia Ii - Cubism - History
Cubism began in 1906 with Georges Braque and Pablo Picasso, who lived in the Montmartre quarter of Paris, France. They met in 1907, and w...   » Read the article

Encyclopedia - Curvature: 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 ci...   » Read the article

Encyclopedia - Attractor: 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 ...   » Read the article

Encyclopedia - Curvature: 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...   » Read the article

Encyclopedia - Manifold: 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 ...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: 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 t...   » Read the article

Encyclopedia - Sandpaper: 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 b...   » Read the article

Encyclopedia - Euler Characteristic: Encyclopedia Ii - Euler Characteristic - Proof
The first rigorous proof of Euler's formula, given by a 20-year-old Cauchy, is as follows: Remove one face of the polyhedron. By pulling ...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: 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 t...   » Read the article

Encyclopedia - Manifold: 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 ...   » Read the article

Encyclopedia - Manifold: 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. Formall...   » Read the article

Encyclopedia - Manifold: 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 its...   » Read the article

Encyclopedia - Curvature: Encyclopedia Ii - Curvature - Curvature Of Space
In cosmology, the concept of "curvature of space" is considered, which is the curvature of corresponding pseudo-Riemannian manifolds, see...   » Read the article

Encyclopedia - Manifold: 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 intr...   » Read the article

Encyclopedia - Manifold: 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...   » Read the article

Encyclopedia - Manifold: 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 t...   » Read the article

Encyclopedia - Curvature: 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 ci...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: 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...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: 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 ca...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: 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 de...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: 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 mathem...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: Encyclopedia Ii - Unifying Theories In Mathematics - Reference List Of Major Unifying Concepts
A short list of these theories might include: Cartesian geometry Calculus Complex analysis Galois theory Erlangen programme Lie group Se...   » Read the article

Encyclopedia - Michael Polanyi: 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...   » Read the article

Encyclopedia - Curve: 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 ...   » Read the article

Encyclopedia - Curvature: Encyclopedia Ii - Curvature - Curvature Of Space
In cosmology, the concept of "curvature of space" is considered, which is the curvature of corresponding pseudo-Riemannian manifolds, see...   » Read the article

Encyclopedia - Curvature: 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...   » Read the article

Encyclopedia - Curve: Encyclopedia Ii - Curve - Lengths Of Curves
If X is a metric space with metric d, then we can define the length of a curve by A rectifiable curve is a curve with finite length. ...   » Read the article

Encyclopedia - Curve: 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 everyd...   » Read the article

Encyclopedia - Curve: 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...   » Read the article

Encyclopedia - Curve: 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, f...   » Read the article

Encyclopedia - Cubism: 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 part...   » Read the article

Encyclopedia - Cubism: Encyclopedia Ii - Cubism - Analytical Cubism
Picasso and Braque worked alongside one another (1906-1909 pre-cubism) and then started to work hand-in-hand to further advance their con...   » Read the article

Encyclopedia - Computer Vision: 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 comput...   » Read the article

Encyclopedia - Attractor: 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...   » Read the article

Encyclopedia - Computer Vision: 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 esti...   » Read the article

Encyclopedia - Computer Vision: 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 se...   » Read the article

Encyclopedia - Computer Vision: Encyclopedia Ii - Computer Vision - Applications
In the related fields machine vision and medical imaging, systems using computer vision techniques are sold in markets worth billions of ...   » Read the article

Encyclopedia - Computer Vision: Encyclopedia Ii - Computer Vision - Related Fields
Advanced systems are often borrowing from many different fields like pattern recognition, statistical learning, projective geometry, imag...   » Read the article

Encyclopedia - Carl Friedrich Gauss: 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 ...   » Read the article

Encyclopedia - Differential Geometry And Topology: 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, sur...   » Read the article

Encyclopedia - Manifold: 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...   » Read the article

Encyclopedia - Manifold: 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 t...   » Read the article

Encyclopedia - Manifold: 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 ...   » Read the article

Encyclopedia - Manifold: 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 l...   » Read the article

Encyclopedia - Manifold: 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 its...   » Read the article

Encyclopedia - Manifold: 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 w...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Orientation By Top-dimensional Forms
Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in th...   » Read the article

Encyclopedia - Orientability: 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 ar...   » Read the article

Encyclopedia - Differential Geometry And Topology: 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 man...   » Read the article

Encyclopedia - Differential Geometry And Topology: 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 a...   » Read the article

Encyclopedia - Fan Implement: 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 sam...   » Read the article

Encyclopedia - Laplace Operator: 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 gradi...   » Read the article

Encyclopedia - Michael Polanyi: 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...   » Read the article

Encyclopedia - Laplace Operator: Encyclopedia Ii - Laplace Operator - Laplace-de Rham Operator
In the general case of differential geometry, one defines the Laplace-de Rham operator as the generalization of the Laplacian. It is a di...   » Read the article

Encyclopedia - Manifold: 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 intr...   » Read the article