en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2137303

2013-03-16T18:39:25Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2137303

← Older revision		Revision as of 20:39, 16 March 2013
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	The ~~writer is known as Wilber Pegues~~. ~~For a whilst I~~'~~ve been~~ in ~~Mississippi but now I~~'~~m contemplating other choices~~. ~~Invoicing~~ is ~~my occupation~~. ~~One~~ of the ~~very best things~~ in the ~~world~~ for ~~him~~ is ~~doing ballet~~ and ~~he'll be starting something else along~~ with it.<br><br>~~Here~~ is ~~my website~~ ... ~~real psychics~~ (~~[http:~~//~~cpacs~~.~~org/index~~.~~php?document_srl~~=~~90091&mid~~=~~board_zTGg26 http://cpacs~~.~~org/index~~.~~php?document_srl~~=~~90091&mid~~=~~board_zTGg26~~])		[[Image:Moulton plane.jpg\|thumb\|right\|300px\|The ''Moulton plane''. Lines sloping down and to the right are bent where they cross the ''y''-axis.]]
			In [[incidence geometry]], the '''Moulton plane''' is an example of an [[affine plane (incidence geometry)\|affine plane]] in which [[Desargues' theorem]] does not hold. It is named after the American astronomer [[Forest Ray Moulton]]. The points of the Moulton plane are simply the points in the real plane '''R'''<sup>2</sup> and the lines are the regular lines as well with the exception, that for lines with a negative [[slope]] the slope doubles when they pass the ''y''-axis.

			==Formal definition==
			The Moulton plane is an [[incidence structure]] <math>\mathfrak M=\langle P, G,\textrm I\rangle</math>, where <math>P</math> denotes the set of points, <math>G</math> the set of lines and <math>\textrm I</math> the incidence relation "lies on":

			:<math> P:=\mathbb R^2 \,</math>

			:<math> G:=(\mathbb R \cup \{\infty\}) \times \mathbb R,</math>

			<math>\infty</math> is just a formal symbol for an element <math>\not\in\mathbb R</math>. It is used to describe vertical lines, which you may think of as lines with an infinitely large slope.

			The incidence relation is defined as follows:

			For ''p'' = (''x'', ''y'') ∈ ''P'' and ''g'' = (''m'', ''b'') ∈ ''G'' we have

			:<math>
			p\,\textrm I\,g\iff\begin{cases}
			x=b&\text{if }m=\infty\\
			y=\frac{1}{2}mx+b&\text{if }m\leq 0, x\leq 0\\
			y=mx+b&\text{if }m\geq 0 \text{ or } x\geq 0.
			\end{cases}
			</math>

			==Application==
			The Moulton plane is an affine plane in which Desargues' theorem does not hold. The associated projective plane is consequently non-desarguesian as well. This means that there are projective planes not isomorphic to <math> PG(2,F) </math> for any (skew) field ''F''. Here <math> PG(2,F) </math> is the projective plane <math> P(F^3) </math> determined by a 3-dimensional vector space.

			==References==
			* Beutelspacher, A Rosenbaum, U. : ''Projektive Geometrie'': Vieweg (1992), s. 70-71 (German)
			*{{Citation \| last1=Moulton \| first1=Forest Ray \| title=A Simple Non-Desarguesian Plane Geometry \| jstor=1986419 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| year=1902 \| journal=[[Transactions of the American Mathematical Society]] \| issn=0002-9947 \| volume=3 \| issue=2 \| pages=192–195}}

			{{DEFAULTSORT:Moulton Plane}}
			[[Category:Incidence geometry]]

en>Brad7777: /* References */ chnaged category mathematical analysis to real analysis added types of functions

2011-11-17T16:51:33Z

References: chnaged category mathematical analysis to real analysis added types of functions

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The writer is known as Wilber Pegues. For a whilst I've been in Mississippi but now I'm contemplating other choices. Invoicing is my occupation. One of the very best things in the world for him is doing ballet and he'll be starting something else along with it.<br><br>Here is my website ... real psychics ([http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26 http://cpacs.org/index.php?document_srl=90091&mid=board_zTGg26])

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en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q2137303

en>Brad7777: /* References */ chnaged category mathematical analysis to real analysis added types of functions