Complex squaring map - Revision history

212.235.89.151: /* See also */

2014-03-21T13:11:18Z

See also

← Older revision		Revision as of 15:11, 21 March 2014
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	~~In [[geometry]] and in its applications to [[drawing]], a~~ '~~''perspectivity''' is the formation of an image in~~ a ~~[[picture plane]] of a scene viewed from a fixed point.~~		I'm a 33 years old and working at the high school (International Relations).<br>In my spare time I try to learn French. I have been twicethere and look forward to returning sometime in the future. I like to read, preferably on my kindle. I like to watch NCIS and The Vampire Diaries as well as documentaries about nature. I like Model Aircraft Hobbies.<br><br>Have a look at my blog ... [http://city-wiz.com/node/429955 для себя здесь]

	~~==Graphics==~~
	~~The science of [[graphical perspective]] uses perspectivities to make realistic images in proper proportion. According to [[Kirsti Andersen]],~~ the ~~first author to describe perspectivity was [[Leon Alberti]] in his ''De Pictura'' (1435).<ref>[[Kirsti Andersen]]~~ (~~2007~~) The Geometry of an Art, page 1,Springer ISBN 0-387-25967-9</ref> In English, [[Brook Taylor]] presented his ''Linear Perspective'' in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry".<ref>{{harvnb\|Andersen\|1992\|loc=p. ~~75}}~~<~~/ref~~> In ~~a second book, ''New Principles of Linear Perspective'' (1719), Taylor wrote~~
	~~:When Lines drawn according~~ to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the ''Projection'' of the other Figure. ~~The Lines producing that Projection, taken all together, are called the ''System of Rays''. And when those Rays all pass thro’ one~~ and ~~same Point, they are called~~ the ~~''Cone of Rays''~~. ~~And when that Point is consider’d as the Eye of a Spectator~~, ~~that System of Rays is called the ''Optic Cone''<ref>{{harvnb\|Andersen\|1992\|loc=p. 163}}</ref>~~

	~~==Projective Geometry==~~
	~~[[Image:Birapport et projection.png\|thumb\|300px\|A perspectivity:<p><math>ABCD \doublebarwedge A'B'C'D',</math>]]~~

	~~In [[projective geometry]] the points of a line are called a [[projective range]], and the set of lines in a plane~~ on ~~a point is called a [[pencil (mathematics)\|pencil]]~~.

	Given two [[line (geometry)\|lines]] <math>\ell</math> and <math>m</math> in a [[plane (geometry)\|plane]] and a point ''P'' of that plane on neither line, the [[bijection\|bijective mapping]] between the points of the range of <math>\ell</math> and the range of <math>m</math> determined by the lines of the pencil on ''P'' is called a '''perspectivity''' (or more precisely, a ''central perspectivity'' with center ''P'').<ref>{{harvnb\|Coxeter\|1969\|loc=p. 242}}</ref> A special symbol has been used to ~~show that points ''X''~~ and ''Y'' are related by a perspectivity; <math>X \doublebarwedge Y .</math> In this notation, to show that the center of perspectivity is ''P'', write <math>X \ \overset {P}{\doublebarwedge} \ Y.</math> Using the language of functions, a central perspectivity with center ''P'' is a function <math>f_P \colon [\ell] \mapsto [m]</math> (where the square brackets indicate the projective range of the line) defined by <math>f_P (X) = Y \text{ whenever } P \in XY</math>.<ref>{{harvnb\|Pedoe\|1988\|loc=p. 281}}</ref> This map is a [[Projection (mathematics)\|projection]] since it is an [[involution (mathematics)\|involution]], that is, <math>f_P (f_P (X)) = X \text{ for all }X \in [\ell]</math>.

	The existence of a perspectivity means that corresponding points are in [[perspective (geometry)\|perspective]]. The [[duality (projective geometry)\|dual]] concept, ''axial perspectivity'', is the correspondence between the lines of two pencils determined by a projective range.

	~~===Projectivity===~~
	The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a '''projectivity''' (''projective transformation'', ''projective collineation'' and ''[[homography]]'' are [[synonym]]s).

	~~There are several results concerning projectivities and perspectivities which hold in any [[pappus's hexagon theorem\|pappian]] projective plane:<ref>{{harvnb\|Fishback\|1969\|loc=pp. 65-66}}</ref>~~

	~~Theorem: Any projectivity between two distinct projective ranges can be written~~ as ~~the composition of no more than two perspectivities.~~

	~~Theorem: Any projectivity from a projective range to itself can be written~~ as ~~the composition of three perspectivities.~~

	~~Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity~~.

	~~===Higher dimensional perspectivities===~~
	~~The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher dimensional analogues which will also be called perspectivities~~.

	~~Let ''S''<sub>''m''~~<~~/sub~~> ~~and ''T''~~<~~sub~~>''m''</sub> be two distinct ''m''-dimensional projective spaces contained in an ''n''-dimensional projective space ''R''<sub>''n''</sub>. Let ''P''<sub>''n''-''m''-1</sub> be an (''n''-''m''-1) - dimensional subspace of ''R''<sub>''n''</sub> with no points in common with either ''S''<sub>''m''</sub> or ''T''<sub>''m''</sub>. For each point ''X'' of ''S''<sub>''m''</sub>, the space ''L'' spanned by ''X'' and ''P''<sub>''n''-''m''-1</sub> meets ''T''<sub>''m''</sub> in a ~~point ''Y'' = f<sub>''P''</sub>(''X''). This correspondence f<sub>''P''</sub> is also called a perspectivity.<ref>{{harvnb\|Pedoe\|1988\|loc=pp~~. ~~282-3}}</ref> The central perspectivity described above is the case with ''n'' = 2 and ''m'' = 1~~.

	~~===Perspective collineations===~~
	~~Let ''S''<sub>2</sub> and ''T''<sub>2</sub> be two distinct projective planes in a projective 3-space ''R''<sub>3</sub>~~. With ''O'' and ''O''* being points of ''R''<sub>3</sub> in neither plane, use the construction of the last section to project ''S''<sub>2</sub> onto ''T''<sub>2</sub> by the perspectivity with center ''O'' followed by the projection of ''T''<sub>2</sub> back onto ''S''<sub>2</sub> with the perspectivity with center ''O''. This composition is a [[bijection\|bijective map]] of the points of ''S''<sub>2</sub> onto itself which preserves [[collinear]] points and is called a ''perspective collineation'' (''central collineation'' in more modern terminology).<ref>{{harvnb\|Young\|1930\|loc=p. 116}}</ref> Let φ be a perspective collineation of ''S''<sub>2</sub>. Each point of the line of intersection of ''S''<sub>2</sub> and ''T''<sub>2</sub> will be fixed by φ and this line is called the ''axis'' of φ. Let point ''P'' be the intersection of line ''OO'' with the plane ''S''<sub>2</sub>. ''P'' is also fixed by φ and every line of ''S''<sub>2</sub> that passes through ''P'' is stabilized by φ (fixed, but not necessarily pointwise fixed). ''P'' is called the ''center'' of φ. The restriction of φ to any line of ''S''<sub>2</sub> not passing through ''P'' is the central perspectivity in ''S''<sub>2</sub> with center ''P'' between that line and the line which is its image under φ.

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~
	* {{citation\|last=Andersen\|first=Kirsti\|title=Brook Taylor's Work on Linear Perspective\|publisher=Springer\|year=1992\|isbn=0-387-97486-5}}
	* {{Citation \| last1=Coxeter \| first1=Harold Scott MacDonald \| author1-link=Harold Scott MacDonald Coxeter \| title=Introduction to Geometry \| publisher=[[John Wiley & Sons]] \| location=New York \| edition=2nd \| isbn=978-0-471-50458-0 \| mr=123930 \| year=1969}}
	* {{citation\|last=Fishback\|first=W.T.\|title=Projective and Euclidean Geometry\|year=1969\|publisher=John Wiley & Sons}}
	* {{citation\|last=Pedoe\|first=Dan\|title=Geometry/A Comprehensive Course\|year=1988\|publisher=Dover\|isbn=0-486-65812-0}}
	* {{citation\|first=John Wesley\|last=Young\|title=Projective Geometry\|year=1930\|publisher=Mathematical Association of America\|series=The Carus Mathematical Monographs (#4)}}

	~~==External links==~~
	* Christopher Cooper [http://~~web.science~~.~~mq.edu.au/~chris/geometry/CHAP05%20Perspectivities%20and%20Projectivities.pdf Perspectivities and Projectivities] from [[Macquarie University]].~~
	* James C. Morehead Jr. (1911) [http://scholarship.rice.edu/bitstream/handle/1911/~~62737~~/~~article_RIP421_part1.pdf?sequence=1 Perspective and Projective Geometries: A Comparison] from [[Rice University]].~~
	* John Taylor [http://www.cmis.brighton.ac.uk/staff/jt40/EM225/EM225_Projective_geometry_2.pdf Projective Geometry] from [[University of Brighton]].

	~~[[Category:Projective geometry]]~~
	~~[[Category:Perspective projection]]~~
	~~[[Category:Technical drawing]]~~
	~~[[Category:Functions and mappings]]~~
	~~[[Category:Composition in visual art]~~]

en>Funandtrvl: nvbx

2013-06-28T22:59:25Z

nvbx

← Older revision		Revision as of 00:59, 29 June 2013
Line 1:		Line 1:
			In [[geometry]] and in its applications to [[drawing]], a '''perspectivity''' is the formation of an image in a [[picture plane]] of a scene viewed from a fixed point.

			==Graphics==
			The science of [[graphical perspective]] uses perspectivities to make realistic images in proper proportion. According to [[Kirsti Andersen]], the first author to describe perspectivity was [[Leon Alberti]] in his ''De Pictura'' (1435).<ref>[[Kirsti Andersen]] (2007) The Geometry of an Art, page 1,Springer ISBN 0-387-25967-9</ref> In English, [[Brook Taylor]] presented his ''Linear Perspective'' in 1715, where he explained "Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry".<ref>{{harvnb\|Andersen\|1992\|loc=p. 75}}</ref> In a second book, ''New Principles of Linear Perspective'' (1719), Taylor wrote
			:When Lines drawn according to a certain Law from the several Parts of any Figure, cut a Plane, and by that Cutting or Intersection describe a figure on that Plane, that Figure so described is called the ''Projection'' of the other Figure. The Lines producing that Projection, taken all together, are called the ''System of Rays''. And when those Rays all pass thro’ one and same Point, they are called the ''Cone of Rays''. And when that Point is consider’d as the Eye of a Spectator, that System of Rays is called the ''Optic Cone''<ref>{{harvnb\|Andersen\|1992\|loc=p. 163}}</ref>

	~~Surely the second option would be more beneficial for any website. Should you have just about any questions with regards to wherever along with how you can make use of~~ [~~http~~:~~//go1~~.in/~~backup_plugin_588842 wordpress dropbox backup~~]~~, it is possible to email us on~~ the ~~website. The next step is to visit your Word - Press blog dashboard. I thought about what would happen by placing~~ a ~~text widget in the sidebar beneath my banner ad, and so it went. s ultimately easy to implement and virtually maintenance free. All this is very simple~~, and the ~~best thing~~ is ~~that it is totally free, and you don't need~~ a ~~domain name or web hosting~~. <br><br>~~Most Word - Press web developers can provide quality CMS website solutions~~ and ~~they price their services at reasonable rates. Best~~ of ~~all~~, ~~you can still have all~~ the ~~functionality that you desire when you use~~ the ~~Word - Press platform. This may possibly also permit it currently being seriously straightforward to modify~~ the ~~hues within your Ad - Sense code so~~ the ~~ads blend nicely with~~ the ~~many term broad internet word wide web web page in case you can come to your conclusion to run~~ the ~~adverts. Apart from these, you are also required to give some backlinks~~ on ~~other sites as well. But in case you want some theme~~ or ~~plugin in sync~~ with ~~your business needs, it is advisable that you must seek some professional help~~. <br><br>~~It is very easy~~ to ~~install Word - Press blog or website~~. ~~Word - Press has ensured the users of~~ this ~~open source blogging platform do not have to troubleshoot on their own~~, ~~or seek outside help. Those who cannot conceive with donor eggs due~~ to ~~some problems can also opt for surrogacy option using~~ the ~~services~~ of ~~surrogate mother~~. ~~To turn~~ the ~~Word - Press Plugin on~~, ~~click Activate on~~ the ~~far right side~~ of the ~~list~~. So, ~~if you~~ are ~~looking online to hire dedicated Wordpress developers~~, ~~India PHP Expert can give~~ a ~~hand you~~ in ~~each~~ and ~~every best possible way~~. <br><br>~~It has become a more prevalent cause~~ of ~~infertility and the fertility clinic are having~~ more ~~and more couples with infertility problems~~. ~~But the Joomla was created~~ as the ~~CMS over years~~ of ~~hard work~~. ~~Normally, the Word - Press developers make~~ a ~~thorough research~~ on ~~your website goals~~ and ~~then ingrain the most suitable graphical design elements to your website~~. ~~A whole lot worse, your site will likely~~ be ~~useless as well as your merchandise won~~'~~t sell if~~ no ~~one has~~ the ~~endurance to wait for the web pages to load~~. The ~~popularity of Word - Press has increased~~ the ~~demand for Word - Press themes~~ and ~~these themes sells like hot cake on the internet~~. <br><br>~~Every single module contains published data~~ and ~~guidelines, usually~~ a ~~lot more than 1 video,~~ and ~~when pertinent~~, ~~incentive links and PDF files~~ to ~~assist you out~~. ~~An ease~~ of ~~use~~ which ~~pertains to both internet site back-end~~ and ~~front-end users alike~~. It'~~s not a secret that a lion share~~ of ~~activity on~~ the ~~internet~~ is ~~takes place on~~ the ~~Facebook~~. ~~If this~~ is not ~~possible you still have~~ the ~~choice~~ of the ~~default theme~~ that is ~~Word~~ - ~~Press 3~~. ~~Definitely when you wake up~~ from ~~the slumber, you can be sure that you will be lagging behind~~ and ~~getting on track would be a tall order~~.		==Projective Geometry==
			[[Image:Birapport et projection.png\|thumb\|300px\|A perspectivity:<p><math>ABCD \doublebarwedge A'B'C'D',</math>]]

			In [[projective geometry]] the points of a line are called a [[projective range]], and the set of lines in a plane on a point is called a [[pencil (mathematics)\|pencil]].

			Given two [[line (geometry)\|lines]] <math>\ell</math> and <math>m</math> in a [[plane (geometry)\|plane]] and a point ''P'' of that plane on neither line, the [[bijection\|bijective mapping]] between the points of the range of <math>\ell</math> and the range of <math>m</math> determined by the lines of the pencil on ''P'' is called a '''perspectivity''' (or more precisely, a ''central perspectivity'' with center ''P'').<ref>{{harvnb\|Coxeter\|1969\|loc=p. 242}}</ref> A special symbol has been used to show that points ''X'' and ''Y'' are related by a perspectivity; <math>X \doublebarwedge Y .</math> In this notation, to show that the center of perspectivity is ''P'', write <math>X \ \overset {P}{\doublebarwedge} \ Y.</math> Using the language of functions, a central perspectivity with center ''P'' is a function <math>f_P \colon [\ell] \mapsto [m]</math> (where the square brackets indicate the projective range of the line) defined by <math>f_P (X) = Y \text{ whenever } P \in XY</math>.<ref>{{harvnb\|Pedoe\|1988\|loc=p. 281}}</ref> This map is a [[Projection (mathematics)\|projection]] since it is an [[involution (mathematics)\|involution]], that is, <math>f_P (f_P (X)) = X \text{ for all }X \in [\ell]</math>.

			The existence of a perspectivity means that corresponding points are in [[perspective (geometry)\|perspective]]. The [[duality (projective geometry)\|dual]] concept, ''axial perspectivity'', is the correspondence between the lines of two pencils determined by a projective range.

			===Projectivity===
			The composition of two perspectivities is, in general, not a perspectivity. A perspectivity or a composition of two or more perspectivities is called a '''projectivity''' (''projective transformation'', ''projective collineation'' and ''[[homography]]'' are [[synonym]]s).

			There are several results concerning projectivities and perspectivities which hold in any [[pappus's hexagon theorem\|pappian]] projective plane:<ref>{{harvnb\|Fishback\|1969\|loc=pp. 65-66}}</ref>

			Theorem: Any projectivity between two distinct projective ranges can be written as the composition of no more than two perspectivities.

			Theorem: Any projectivity from a projective range to itself can be written as the composition of three perspectivities.

			Theorem: A projectivity between two distinct projective ranges which fixes a point is a perspectivity.

			===Higher dimensional perspectivities===
			The bijective correspondence between points on two lines in a plane determined by a point of that plane not on either line has higher dimensional analogues which will also be called perspectivities.

			Let ''S''<sub>''m''</sub> and ''T''<sub>''m''</sub> be two distinct ''m''-dimensional projective spaces contained in an ''n''-dimensional projective space ''R''<sub>''n''</sub>. Let ''P''<sub>''n''-''m''-1</sub> be an (''n''-''m''-1) - dimensional subspace of ''R''<sub>''n''</sub> with no points in common with either ''S''<sub>''m''</sub> or ''T''<sub>''m''</sub>. For each point ''X'' of ''S''<sub>''m''</sub>, the space ''L'' spanned by ''X'' and ''P''<sub>''n''-''m''-1</sub> meets ''T''<sub>''m''</sub> in a point ''Y'' = f<sub>''P''</sub>(''X''). This correspondence f<sub>''P''</sub> is also called a perspectivity.<ref>{{harvnb\|Pedoe\|1988\|loc=pp. 282-3}}</ref> The central perspectivity described above is the case with ''n'' = 2 and ''m'' = 1.

			===Perspective collineations===
			Let ''S''<sub>2</sub> and ''T''<sub>2</sub> be two distinct projective planes in a projective 3-space ''R''<sub>3</sub>. With ''O'' and ''O''* being points of ''R''<sub>3</sub> in neither plane, use the construction of the last section to project ''S''<sub>2</sub> onto ''T''<sub>2</sub> by the perspectivity with center ''O'' followed by the projection of ''T''<sub>2</sub> back onto ''S''<sub>2</sub> with the perspectivity with center ''O''. This composition is a [[bijection\|bijective map]] of the points of ''S''<sub>2</sub> onto itself which preserves [[collinear]] points and is called a ''perspective collineation'' (''central collineation'' in more modern terminology).<ref>{{harvnb\|Young\|1930\|loc=p. 116}}</ref> Let φ be a perspective collineation of ''S''<sub>2</sub>. Each point of the line of intersection of ''S''<sub>2</sub> and ''T''<sub>2</sub> will be fixed by φ and this line is called the ''axis'' of φ. Let point ''P'' be the intersection of line ''OO'' with the plane ''S''<sub>2</sub>. ''P'' is also fixed by φ and every line of ''S''<sub>2</sub> that passes through ''P'' is stabilized by φ (fixed, but not necessarily pointwise fixed). ''P'' is called the ''center'' of φ. The restriction of φ to any line of ''S''<sub>2</sub> not passing through ''P'' is the central perspectivity in ''S''<sub>2</sub> with center ''P'' between that line and the line which is its image under φ.

			==Notes==
			{{Reflist}}

			==References==
			* {{citation\|last=Andersen\|first=Kirsti\|title=Brook Taylor's Work on Linear Perspective\|publisher=Springer\|year=1992\|isbn=0-387-97486-5}}
			* {{Citation \| last1=Coxeter \| first1=Harold Scott MacDonald \| author1-link=Harold Scott MacDonald Coxeter \| title=Introduction to Geometry \| publisher=[[John Wiley & Sons]] \| location=New York \| edition=2nd \| isbn=978-0-471-50458-0 \| mr=123930 \| year=1969}}
			* {{citation\|last=Fishback\|first=W.T.\|title=Projective and Euclidean Geometry\|year=1969\|publisher=John Wiley & Sons}}
			* {{citation\|last=Pedoe\|first=Dan\|title=Geometry/A Comprehensive Course\|year=1988\|publisher=Dover\|isbn=0-486-65812-0}}
			* {{citation\|first=John Wesley\|last=Young\|title=Projective Geometry\|year=1930\|publisher=Mathematical Association of America\|series=The Carus Mathematical Monographs (#4)}}

			==External links==
			* Christopher Cooper [http://web.science.mq.edu.au/~chris/geometry/CHAP05%20Perspectivities%20and%20Projectivities.pdf Perspectivities and Projectivities] from [[Macquarie University]].
			* James C. Morehead Jr. (1911) [http://scholarship.rice.edu/bitstream/handle/1911/62737/article_RIP421_part1.pdf?sequence=1 Perspective and Projective Geometries: A Comparison] from [[Rice University]].
			* John Taylor [http://www.cmis.brighton.ac.uk/staff/jt40/EM225/EM225_Projective_geometry_2.pdf Projective Geometry] from [[University of Brighton]].

			[[Category:Projective geometry]]
			[[Category:Perspective projection]]
			[[Category:Technical drawing]]
			[[Category:Functions and mappings]]
			[[Category:Composition in visual art]]

en>Adam majewski: /* Generalisations */ link

2012-06-07T11:31:07Z

Generalisations: link

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Surely the second option would be more beneficial for any website. Should you have just about any questions with regards to wherever along with how you can make use of [http://go1.in/backup_plugin_588842 wordpress dropbox backup], it is possible to email us on the website. The next step is to visit your Word - Press blog dashboard. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. s ultimately easy to implement and virtually maintenance free. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. Best of all, you can still have all the functionality that you desire when you use the Word - Press platform. This may possibly also permit it currently being seriously straightforward to modify the hues within your Ad - Sense code so the ads blend nicely with the many term broad internet word wide web web page in case you can come to your conclusion to run the adverts. Apart from these, you are also required to give some backlinks on other sites as well. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. It is very easy to install Word - Press blog or website. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. To turn the Word - Press Plugin on, click Activate on the far right side of the list. So, if you are looking online to hire dedicated Wordpress developers, India PHP Expert can give a hand you in each and every best possible way. It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. But the Joomla was created as the CMS over years of hard work. Normally, the Word - Press developers make a thorough research on your website goals and then ingrain the most suitable graphical design elements to your website. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. The popularity of Word - Press has increased the demand for Word - Press themes and these themes sells like hot cake on the internet. Every single module contains published data and guidelines, usually a lot more than 1 video, and when pertinent, incentive links and PDF files to assist you out. An ease of use which pertains to both internet site back-end and front-end users alike. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. If this is not possible you still have the choice of the default theme that is Word - Press 3. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order.