Flatness
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The intuitive idea of '''flatness''' is important in several fields.
==Flatness in mathematics==
The '''flatness''' of a [[surface]] is the degree to which it approximates a [[Plane (mathematics)|mathematical plane]]. The term is generalized for higher-dimensional [[manifold]]s to describe the degree to which they approximate the [[Euclidean space]] of the same dimensionality. See [[curvature]].
'''Flatness''' in [[homological algebra]] and [[algebraic geometry]] means, of an object <math>A</math> in an [[abelian category]], that <math>- \otimes A</math> is an [[exact functor]]. See [[flat module]] or, for more generality, [[flat morphism]].
== Flatness in systems theory ==
'''[[Flatness (systems theory)|Flatness]]''' is a property of [[Nonlinearity|nonlinear]] [[Dynamical system|dynamic systems]]. It extends the notion of [[controllability]] from [[LTI system theory|linear time-invariant systems]] to nonlinear systems.
Flatness is closely related to [[Feedback linearization]] by dynamic state feedback.
==Flatness in cosmology==
In [[physical cosmology|cosmology]], the concept of "curvature of space" is considered. A space without curvature is called a "flat space" or [[Euclidean space]].
A question often asked is "is the [[Universe]] flat"? The geometry of spacetime has been measured by the [[WMAP]] probe to be nearly flat. The data are consistent with a flat geometry, with Ω = 1.02 +/- 0.02
==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.[http://journals.iranscience.net:800/www.newscientist.com/www.newscientist.com/lastword/article.jsp@id=lw801]
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 [[measuring instrument|instrument]]s using these flat surface generation techniques as a basis for further construction of precise shapes.
==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.
==Flatness in electrical engineering==
When measuring the flatness of a particular non-time-domain response, the measure of flatness defines the difference in a maximum and minimum value. For example, in a frequency response plot for an amplifier the flatness is defined as
: <math> \text{flatness} = \max \left ( P_\text{out} \right ) - \min \left ( P_\text{out} \right ) </math>
where each output power measurement is in [[decibel]]s.
==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]].<ref>[http://artlex.com/ArtLex/Fi.html Art-Lex], "Flat"</ref><ref>[[Tom Wolfe]], ''The Painted Word'' (Bantam, 1975: ISBN 0-553-38065-6)</ref>
==Flatness in liquids==
A [[carbonation|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.
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 [[nanometer]]s over an area of 1 square meter. This is using the [[earth radius|earths mean radius]] at sea level, however a liquid will be slightly flatter at the poles.
==See also==
*[[Flat Earth]]
==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)
==External links==
* http://archive.ncsa.uiuc.edu/Cyberia/Cosmos/FlatnessProblem.html
*[http://www.scienceandsociety.co.uk/results.asp?image=10311926&wwwflag=2&imagepos=1 Two surface plates made by Whitworth]
==References==
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[[Category:Geometry]]
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