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The intuitive idea of flatness is important in several fields. Image File history File links Broom_icon. ...
[edit] Flatness in mathematics The flatness of a surface is the degree to which it approximates a mathematical plane. The term is generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. See curvature. An open surface with X-, Y-, and Z-contours shown. ...
Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
Curvature refers to a number of loosely related concepts in different areas of geometry. ...
Flatness in homological algebra and algebraic geometry means, of an object A in an abelian category, that  is an exact functor. See flat module or, for more generality, flat morphism. Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. ...
In homological algebra, an exact functor is one which preserves exact sequences. ...
In abstract algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. ...
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i. ...
[edit] Flatness in systems theory Flatness is a property of nonlinear dynamic systems. It extends the notion of controllability from linear time-invariant systems to nonlinear systems. Flatness is closely related to Feedback linearization by dynamic state feedback. Flatness in systems theory is a system property that extends the notion of Controllability from linear systems to nonlinear dynamical systems. ...
In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. ...
The Lorenz attractor is an example of a non-linear dynamical system. ...
Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. ...
In electrical engineering, specifically in signal processing and control theory, LTI system theory investigates the response of a linear, time-invariant system to an arbitrary input signal. ...
Feedback linearization is a common approach used in controlling nonlinear systems. ...
[edit] Flatness in cosmology In cosmology, the concept of "curvature of space" is considered. A space without curvature is called a "flat space" or Euclidean space. Physical cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
A question often asked is "is the Universe flat"? According to Albert Einstein's theory of relativity, it probably is curved and warped due to gravity. The Universe is defined as the summation of all particles and energy that exist and the space-time in which all events occur. ...
Two-dimensional analogy of space-time curvature described in General Relativity. ...
Gravity is a force of attraction that acts between bodies that have mass. ...
[edit] Flatness in mechanical engineering Joseph Whitworth popularized the first practical method of making accurate flat surfaces during the 1830s, using engineer's blue and scraping techniques on three trial surfaces. By testing all three pairs against each other, it is ensured that the surfaces become flat. Using two surfaces would result in a concave surface and a convex surface. Eventually a point is reached when many points of contact are visible within each square inch, at which time the three surfaces are uniformly flat to a very close tolerance.[1] Sir Joseph Whitworth Sir Joseph Whitworth, Baronet (December 21, 1803 - January 22, 1887) was an English engineer and entrepreneur. ...
Prussian blue is a blue pigment used in paints and formerly in blueprints. ...
Up until his introduction of the scraping technique, the same three plate method was employed using polishing techniques, giving less accurate results. This led to an explosion of development of precision instruments using these flat surface generation techniques as a basis for further construction of precise shapes. Captain Nemo and Professor Aronnax contemplating measuring instruments in Twenty Thousand Leagues Under the Sea In physics and engineering, measurement is the activity of comparing physical quantities of real-world objects and events. ...
[edit] Flatness in precision manufacturing In the manufacture of precision parts and assemblies, especially where parts will be required to be connected across a surface area in an air-tight or liquid-tight manner, flatness is a critical quality of the manufactured surfaces. such surfaces are usually machined or ground to achieve the required degree of flatness. High-definition metrology, such as digital holographic interferometry, of such a surface to confirm and ensure that the required degree of flatness has been achieved is a key step in such manufacturing processes. Flatness may be defined in terms of least squares fit to a plane ("statistical flatness"), worst-case or overall flatness (the distance between the two closest parallel planes within which the surface barely will fit, or other mathematical definitions that fit the intended use of the manufactured part. A lathe is a common tool used in machining. ...
The word ground has several meanings: The surface of the Earth Soil, a mixture of sand and organic material present on the surface of the Earth Ground (electricity), in electrical engineering, something that is connected to the Earth or at the voltage defined as zero (in the US, called ground...
High-definition metrology refers to measurment of dimensional or other attributes of a surface or an object in which measurements are made densely across the observable extent of that surface or object, so that the measured attribute of the surface or object can be portrayed (displayed) with high-definition. ...
[edit] Flatness in art In art criticism of the 1960s and 1970s, flatness described the smoothness and absence of curvature or surface detail of a two-dimensional work of art. Critic Clement Greenberg believed that flatness, or two-dimensionality, was an essential and desirable quality in painting, a criterion which implies rejection of painterliness and impasto. The valorization of flatness led to a number of art movements, including minimalism and post-painterly abstractionism.[1][2] Art criticism is the study and evaluation of art. ...
The 1960s decade refers to the years from January 1, 1960 to December 31, 1969, inclusive. ...
The 1970s decade refers to the years from 1970 to 1979. ...
Clement Greenberg (January 16, 1909 - May 7, 1994) was an influential American art critic closely associated with the abstract art movement in the United States. ...
For building painting, see painter and decorator. ...
Painterly is a translation of the German term malerisch, one of the opposed categories popularized by Swiss art historian Heinrich Wölfflin (1864 - 1945) in order to help focus, enrich and standardize the terms being used by art historians of his time to characterize works of art. ...
Image:Jane Frank Crgs And Crevices. ...
Minimalism describes movements in various forms of art and design, especially visual art and music, where the work is stripped down to its most fundamental features and core self expression. ...
Post-painterly Abstraction is a term created by art critic, Clement Greenberg as the title for an exhibit he curated for the Los Angeles County Museum of Art in 1964, which subsequently travelled to the Walker Art Center and the Art Gallery of Ontario in Toronto. ...
[edit] Flatness in liquids A carbonated beverage becomes flat when it loses enough of its carbon dioxide that there is no more "fizz" left, although this refers to the intrinsic properties of the substance, rather than the geometric properties of the liquid. Bubbles of carbon dioxide float to the surface of a soft drink. ...
Carbon dioxide is a chemical compound composed of one carbon and two oxygen atoms. ...
On planet earth, the flatness of a liquid is a function of the curvature of the earth, and from trigonometry, can be found to deviate from true flatness by approximately 19.6 nanometers over an area of 1 square meter. This is using the earths mean radius at sea level, however a liquid will be slightly flatter at the poles.. A nanometre (American spelling: nanometer) is 1. ...
Since the Earth, like all planets, is not a perfect sphere, the radius of Earth can refer to various values. ...
[edit] References - Wayne R. Moore, Foundations of Mechanical Accuracy, Moore Special Tool Company, Bridgeport, CT (1970)
- Joseph Whitworth, Plane Metallic Surfaces, Longman, Brown, and Co., London (1858)
[edit] External links
[edit] References - ^ Art-Lex, "Flat"
- ^ Tom Wolfe, The Painted Word (Bantam, 1975: ISBN 0-553-38065-6)
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