Complex projective space - Revision history

en>Mark viking: /* top */ Added wl

2014-09-29T01:23:56Z

top: Added wl

← Older revision		Revision as of 03:23, 29 September 2014
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	~~In the [[mathematical]] field~~ of ~~[[projective geometry]], a '''projective frame''' is an ordered collection~~ of ~~points~~ in ~~[[projective space]] which can be used as reference points to describe any other point~~ in ~~that space~~. ~~For example:~~
	* Given three distinct points on a ~~[[projective line]]~~, ~~any other point can be described by its [[cross-ratio]] with these three points.~~
	* In a [[projective plane]], ~~a projective frame consists of four points~~, ~~no three of which lie on a projective line~~.

	~~In general, let '''K'''''P''~~<~~sup~~>~~''n''~~<~~/sup~~> ~~denote ''n''-dimensional projective space over an arbitrary field '''K'''. This is the projectivization of the vector space '''K'''~~<~~sup~~>~~''n''+1~~<~~/sup~~>~~. Then~~ a ~~projective frame is an (''n''+2)-[[tuple]]~~ of ~~points in [[general position]] in~~
	~~'''K'''''P''<sup>''n''</sup>~~. ~~Here ''general position'' means~~ that ~~no subset~~ of ~~''n''+1 of these points lies in a [[hyperplane]] (a projective subspace of dimension ''n''&minus~~;1).

	~~Sometimes it is convenient to describe a projective frame by ''n''+2 representative vectors ''v''<sub>0</sub>~~, ~~''v''<sub>1</sub>~~, ..~~., ''v''~~<~~sub~~>~~''n''+1~~<~~/sub~~> in ~~'''K'''<sup>n+1</sup>~~. ~~Such a tuple~~ of ~~vectors defines a projective frame if any subset~~ of ~~''n''+1~~ of ~~these vectors~~ is a ~~basis for '''K'''<sup>''n''+1</sup>~~. ~~The full set of ''n''+2 vectors must satisfy [[linear dependence]] relation~~
	~~:<math> \lambda_0 v_0 + \lambda_1 v_1 + \cdots +\lambda_n v_n + \lambda_{n+1} v_{n+1} = 0~~. ~~</math>~~
	~~However~~, ~~because~~ the ~~subsets~~ of ~~''n''+1 vectors are linearly independent~~, the ~~scalars ''λ''~~<~~sub~~>~~''j''~~<~~/sub~~> ~~must~~ all ~~be nonzero~~. ~~It follows that~~ the ~~representative vectors can be rescaled so that ''λ''<sub>''j''</sub>=1 for all ''j''=0,1~~,...,'~~'n''+1~~. ~~This fixes~~ the ~~representative vectors up~~ to ~~an overall scalar multiple. Hence~~ a ~~projective frame~~ is ~~sometimes defined~~ to ~~be a (''n''+ 2)-tuple~~ of ~~vectors which span '''K'''~~<~~sup~~>~~''n''+1~~<~~/sup~~> and ~~sum to zero~~. ~~Using such a frame~~, ~~any point ''p''~~ in '~~''K'''''P''<sup>''n''</sup> may~~ be ~~described by a projective version of ''[[barycentric coordinates (mathematics)\|barycentric coordinates]]'': a collection of ''n''+2 scalars ''μ''<sub>''j''</sub> which sum~~ to ~~zero, such that ''p'' is represented by~~ the ~~vector~~
	~~:<math> \mu_0 v_0 + \mu_1 v_1 + \cdots +\mu_n v_n + \mu_{n+1} v_{n+1}~~. ~~</math>~~

	~~==References==~~
	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Projective Frame}}~~
	~~[[Category:Projective geometry]]~~

en>Michaelfalk: /* Line bundles */

2014-01-07T23:59:18Z

Line bundles

← Older revision		Revision as of 01:59, 8 January 2014
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	~~39 year~~-~~old Mechanical Executive Draftsperson Smolenski from Igloolik~~, ~~likes tarot~~, ~~diet and warhammer~~. ~~Finds encouragement through travel and just spent 6 days at Al~~-~~Hijr Archaeological Site~~ (~~Madâin Sâlih~~).<br><br>~~Feel free~~ to ~~visit my page~~ [~~http~~:/~~/tinyurl~~.~~com~~/~~50o61vunjm27 dash diet~~]		{{Unreferenced\|date=December 2009}}
			In the [[mathematical]] field of [[projective geometry]], a '''projective frame''' is an ordered collection of points in [[projective space]] which can be used as reference points to describe any other point in that space. For example:
			* Given three distinct points on a [[projective line]], any other point can be described by its [[cross-ratio]] with these three points.
			* In a [[projective plane]], a projective frame consists of four points, no three of which lie on a projective line.

			In general, let '''K'''''P''<sup>''n''</sup> denote ''n''-dimensional projective space over an arbitrary field '''K'''. This is the projectivization of the vector space '''K'''<sup>''n''+1</sup>. Then a projective frame is an (''n''+2)-[[tuple]] of points in [[general position]] in
			'''K'''''P''<sup>''n''</sup>. Here ''general position'' means that no subset of ''n''+1 of these points lies in a [[hyperplane]] (a projective subspace of dimension ''n''−1).

			Sometimes it is convenient to describe a projective frame by ''n''+2 representative vectors ''v''<sub>0</sub>, ''v''<sub>1</sub>, ..., ''v''<sub>''n''+1</sub> in '''K'''<sup>n+1</sup>. Such a tuple of vectors defines a projective frame if any subset of ''n''+1 of these vectors is a basis for '''K'''<sup>''n''+1</sup>. The full set of ''n''+2 vectors must satisfy [[linear dependence]] relation
			:<math> \lambda_0 v_0 + \lambda_1 v_1 + \cdots +\lambda_n v_n + \lambda_{n+1} v_{n+1} = 0. </math>
			However, because the subsets of ''n''+1 vectors are linearly independent, the scalars ''λ''<sub>''j''</sub> must all be nonzero. It follows that the representative vectors can be rescaled so that ''λ''<sub>''j''</sub>=1 for all ''j''=0,1,...,''n''+1. This fixes the representative vectors up to an overall scalar multiple. Hence a projective frame is sometimes defined to be a (''n''+ 2)-tuple of vectors which span '''K'''<sup>''n''+1</sup> and sum to zero. Using such a frame, any point ''p'' in '''K'''''P''<sup>''n''</sup> may be described by a projective version of ''[[barycentric coordinates (mathematics)\|barycentric coordinates]]'': a collection of ''n''+2 scalars ''μ''<sub>''j''</sub> which sum to zero, such that ''p'' is represented by the vector
			:<math> \mu_0 v_0 + \mu_1 v_1 + \cdots +\mu_n v_n + \mu_{n+1} v_{n+1}. </math>

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Projective Frame}}
			[[Category:Projective geometry]]

en>TakuyaMurata: /* Differentiall geometry */ misspell

2012-08-26T23:13:52Z

Differentiall geometry: misspell

New page

39 year-old Mechanical Executive Draftsperson Smolenski from Igloolik, likes tarot, diet and warhammer. Finds encouragement through travel and just spent 6 days at Al-Hijr Archaeological Site (Madâin Sâlih).<br><br>Feel free to visit my page [http://tinyurl.com/50o61vunjm27 dash diet]