Accelerator effect - Revision history

en>Wiki13565 at 13:18, 20 December 2014

2014-12-20T13:18:34Z

← Older revision		Revision as of 15:18, 20 December 2014
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	~~[[Image:Helicoid~~.~~svg\|right\|thumb\|350px\|A helicoid with α=1,~~ -~~1≤ρ≤1 and -π≤θ≤π~~.]]		Detta eventuellt låter enbart fason att få dig att protokollföra dig samt det . ni använder det intelligenta det avgiftsfri klöver pro någon nit pro casino ni har någonsin märkt en annons pro något on-line på linje casino har observerat erbjuder mestadels stort på kurs casinobonus.<br><br>

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	~~==Description==~~
	~~It was described by [[Euler]] in 1774 and by [[Jean Baptiste Meusnier]] in 1776~~. ~~Its [[Nomenclature\|name]] derives from its similarity to the [[helix]]: for every [[Point~~ (~~geometry~~)~~\|point]] on the helicoid~~, ~~there is a helix contained in the helicoid which passes through that point~~. Since it is considered that the planar range extends through negative and positive infinity, close observation shows the appearance of two parallel or mirror planes in the sense that if the slope of one plane is traced, the co-~~plane can be seen to be bypassed or skipped~~, ~~though in actuality the co-plane is also traced from the opposite perspective~~.

	~~The helicoid is also a [[ruled surface]] (and a [[right conoid]]), meaning that it is a trace of a line~~. ~~Alternatively, for any point on the surface, there is a line on the surface passing through it~~. ~~Indeed~~, ~~[[Eugène Charles Catalan\|Catalan]] proved in 1842 that the helicoid and the plane were the only ruled [[minimal surface]]s~~.<~~ref~~>~~''Elements of the Geometry and Topology of Minimal Surfaces in Three~~-~~dimensional Space''~~
	~~By [[A~~. ~~T. Fomenko]]~~, A. A. ~~Tuzhilin~~
	~~Contributor A~~. A. ~~Tuzhilin~~
	~~Published by AMS Bookstore~~, ~~1991~~
	~~ISBN 0-8218-4552-7, ISBN 978-0-8218-4552-3, p~~. 33<~~/ref~~>

	~~The helicoid and the [[catenoid]] are parts of a family of helicoid~~-~~catenoid minimal surfaces~~.

	~~The helicoid is shaped like [[Archimedes' screw]], but extends infinitely in all directions~~. ~~It can be described by the following [[parametric equation]]s in [[Cartesian coordinates]]:~~
	:<~~math~~> ~~x = \rho \cos (\alpha \theta)~~, \ <~~/math~~>
	:<~~math~~> ~~y = \rho \sin (\alpha \theta)~~, \ <~~/math~~>
	:<~~math~~> ~~z = \theta~~, ~~\ </math>~~
	~~where ''ρ'' and ''θ'' range from negative [[infinity]] to [[positive number\|positive]] infinity~~, ~~while ''α'' is a constant~~. ~~If ''α'' is positive~~, ~~then the helicoid is right~~-~~handed as shown in the figure; if negative then left~~-~~handed~~.

	~~The helicoid has [[principal curvature]]s~~ <~~math~~>~~\pm 1/(1+ \rho ^2) \~~ <~~/math~~>. ~~The sum of these quantities gives the [[mean curvature]] (zero since the helicoid is a minimal surface) and the product gives the [[Gaussian curvature]]~~.

	~~The helicoid is [[homeomorphism\|homeomorphic]] to the plane~~ <~~math~~> ~~\mathbb{R}^2~~ <~~/math~~>. ~~To see this~~, ~~let alpha decrease [[continuous function\|continuous]]ly from its given value down to [[0 (number)\|zero]].~~ ~~Each intermediate value of ''α'' will describe a different helicoid~~, ~~until ''α = 0'' is reached and the helicoid becomes a vertical [[plane~~ (~~mathematics~~)~~\|plane]].~~

	~~Conversely~~, ~~a plane can be turned into a helicoid by choosing a line~~, ~~or ''axis''~~, ~~on the plane~~, ~~then twisting the plane around that axis.~~

	~~For example~~, ~~if one takes ''h'' as the maximum value at z and R the radius~~, ~~the area of the surface is~~ <~~math~~>~~\pi [R \sqrt (R^2+h^2)+h2* \ln (R + \sqrt (R^2+h^2)/h)]~~<~~/math~~>.

	~~==Helicoid and catenoid==~~
	[~~[File~~:~~Helicatenoid~~.~~gif\|thumb\|256px\|Animation showing the transformation of a helicoid into a catenoid~~.]]
	~~The helicoid and the [[catenoid]] are locally isometric surfaces, see discussion there.~~

	==~~See also==~~
	*[[Dini's surface]]
	*[[Right conoid]]
	*[[Ruled surface]]

	~~==Notes==~~
	<~~references /~~>

	~~==External links==~~
	~~{{commonscat\|Helicoids}}~~
	* [http://~~chamicewicz~~.com/~~p5/helicoid~~/ ~~Interactive 3D Helicoid plotter using Processing (with code)]~~
	* {{springer\|title=Helicoid\|id=p/~~h046880}}~~
	* [http://~~www.princeton~~.~~edu/~rvdb/WebGL/helicoid.html WebGL-based Interactive 3D Helicoid]~~

	~~[[Category:Geometric shapes]]~~
	~~[[Category:Minimal surfaces]]~~
	~~[[Category:Surfaces]]~~

	~~[[de:Minimalfläche#Die Wendelfläche]]~~

en>Bender235: /* Accelerator models */

2013-07-31T15:44:42Z

Accelerator models

← Older revision		Revision as of 17:44, 31 July 2013
Line 1:		Line 1:
			[[Image:Helicoid.svg\|right\|thumb\|350px\|A helicoid with α=1, -1≤ρ≤1 and -π≤θ≤π.]]

			The '''helicoid''', after the [[Plane (geometry)\|plane]] and the [[catenoid]], is the third [[minimal surface]] to be known.

	~~Glen Wendler~~ is the ~~title people use~~ to ~~call me~~ and ~~my spouse doesn~~'t like ~~it at~~ all. ~~Some time ago she selected~~ to ~~live~~ in ~~Delaware but her spouse desires them to move~~. ~~Dispatching~~ has ~~been my profession for some time~~. ~~Going~~ to [~~http:~~//~~browse~~.~~deviantart~~.~~com~~/~~?qh~~=~~&section~~=~~&global~~=~~1&q~~=~~karaoke karaoke] is something that I've done for years. Go to my web site to find out much more:~~ http://~~conex360~~.com/~~members~~/~~armantillery~~/~~activity~~/~~139214/<br><br>Here is my web blog ...~~ [http://~~conex360~~.~~com~~/~~members~~/~~armantillery~~/~~activity/139214/ nya online svenska casino på nätet~~]		==Description==
			It was described by [[Euler]] in 1774 and by [[Jean Baptiste Meusnier]] in 1776. Its [[Nomenclature\|name]] derives from its similarity to the [[helix]]: for every [[Point (geometry)\|point]] on the helicoid, there is a helix contained in the helicoid which passes through that point. Since it is considered that the planar range extends through negative and positive infinity, close observation shows the appearance of two parallel or mirror planes in the sense that if the slope of one plane is traced, the co-plane can be seen to be bypassed or skipped, though in actuality the co-plane is also traced from the opposite perspective.

			The helicoid is also a [[ruled surface]] (and a [[right conoid]]), meaning that it is a trace of a line. Alternatively, for any point on the surface, there is a line on the surface passing through it. Indeed, [[Eugène Charles Catalan\|Catalan]] proved in 1842 that the helicoid and the plane were the only ruled [[minimal surface]]s.<ref>''Elements of the Geometry and Topology of Minimal Surfaces in Three-dimensional Space''
			By [[A. T. Fomenko]], A. A. Tuzhilin
			Contributor A. A. Tuzhilin
			Published by AMS Bookstore, 1991
			ISBN 0-8218-4552-7, ISBN 978-0-8218-4552-3, p. 33</ref>

			The helicoid and the [[catenoid]] are parts of a family of helicoid-catenoid minimal surfaces.

			The helicoid is shaped like [[Archimedes' screw]], but extends infinitely in all directions. It can be described by the following [[parametric equation]]s in [[Cartesian coordinates]]:
			:<math> x = \rho \cos (\alpha \theta), \ </math>
			:<math> y = \rho \sin (\alpha \theta), \ </math>
			:<math> z = \theta, \ </math>
			where ''ρ'' and ''θ'' range from negative [[infinity]] to [[positive number\|positive]] infinity, while ''α'' is a constant. If ''α'' is positive, then the helicoid is right-handed as shown in the figure; if negative then left-handed.

			The helicoid has [[principal curvature]]s <math>\pm 1/(1+ \rho ^2) \ </math>. The sum of these quantities gives the [[mean curvature]] (zero since the helicoid is a minimal surface) and the product gives the [[Gaussian curvature]].

			The helicoid is [[homeomorphism\|homeomorphic]] to the plane <math> \mathbb{R}^2 </math>. To see this, let alpha decrease [[continuous function\|continuous]]ly from its given value down to [[0 (number)\|zero]]. Each intermediate value of ''α'' will describe a different helicoid, until ''α = 0'' is reached and the helicoid becomes a vertical [[plane (mathematics)\|plane]].

			Conversely, a plane can be turned into a helicoid by choosing a line, or ''axis'', on the plane, then twisting the plane around that axis.

			For example, if one takes ''h'' as the maximum value at z and R the radius, the area of the surface is <math>\pi [R \sqrt (R^2+h^2)+h2* \ln (R + \sqrt (R^2+h^2)/h)]</math>.

			==Helicoid and catenoid==
			[[File:Helicatenoid.gif\|thumb\|256px\|Animation showing the transformation of a helicoid into a catenoid.]]
			The helicoid and the [[catenoid]] are locally isometric surfaces, see discussion there.

			==See also==
			*[[Dini's surface]]
			*[[Right conoid]]
			*[[Ruled surface]]

			==Notes==
			<references />

			==External links==
			{{commonscat\|Helicoids}}
			* [http://chamicewicz.com/p5/helicoid/ Interactive 3D Helicoid plotter using Processing (with code)]
			* {{springer\|title=Helicoid\|id=p/h046880}}
			* [http://www.princeton.edu/~rvdb/WebGL/helicoid.html WebGL-based Interactive 3D Helicoid]

			[[Category:Geometric shapes]]
			[[Category:Minimal surfaces]]
			[[Category:Surfaces]]

			[[de:Minimalfläche#Die Wendelfläche]]

en>Ehsanit: /* Business Cycles vs. Acceleration effect */

2012-01-27T14:31:26Z

Business Cycles vs. Acceleration effect

New page

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