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[[Image:Desargues theorem alt.svg|thumb|right|300px|Two perspective triangles, with their perspective axis and center]] | |||
Two figures in a [[plane (geometry)|plane]] are '''perspective from a [[point (geometry)|point]]''' ''O'' if the lines joining corresponding points of the figures all meet at ''O''. [[Duality (projective geometry)|Dually]], the figures are said to be '''perspective from a line''' if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in [[projective geometry]] where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions. | |||
==Terminology== | |||
The line which goes through the points where the figure's corresponding sides intersect is known as the '''axis of perspectivity''', '''perspective axis''', '''homology axis''', or archaically, '''perspectrix'''. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the '''center of perspectivity''', '''perspective center''', '''homology center''', '''pole''', or archaically '''perspector'''. The figures are said to be perspective from this center.<ref>{{harvnb|Young|1930|loc=p. 28}}</ref> | |||
==Perspectivity== | |||
If each of the perspective figures consists of all the points on a line (a [[projective range|range]]) then transformation of the points of one range to the other is called a ''central perspectivity''. A dual transformation, taking all the lines through a point (a [[pencil (mathematics)|pencil]]) to another pencil by means of an axis of perspectivity is called an ''axial perspectivity''.<ref>{{harvnb|Young|1930|loc=p. 29}}</ref> | |||
==Triangles== | |||
An important special case occurs when the figures are [[triangle]]s. Two triangles that are perspective from a point are called a ''central couple'' and two triangles that are perspective from a line are called an ''axial couple''.<ref>{{harvnb|Dembowski|1968|loc=p. 26}}</ref> | |||
==Notation== | |||
[[Karl von Staudt]] introduced the notation <math>ABC \doublebarwedge abc</math> to indicate that triangles ABC and abc are perspective.<ref>[[H. S. M. Coxeter]] (1942) ''Non-Euclidean Geometry'', [[University of Toronto Press]], reissued 1998 by [[Mathematical Association of America]], ISBN 0-88385-522-4 . 21,2.</ref> | |||
==Related theorems and configurations== | |||
[[Desargues' theorem]] states that, a central couple of triangles is axial. The converse statement, an axial couple of triangles is central, is equivalent (either can be used to prove the other). Desargues' theorem can be proved in the [[real projective plane]], and with suitable modifications for special cases, in the [[Euclidean plane]]. [[Projective plane]]s in which this result can be proved are called ''Desarguesian planes''. | |||
There are ten points associated with these two kinds of perspective: six on the two triangles, three on the axis of perspectivity, and one at the center of perspectivity. [[Projective duality|Dually]], there are also ten lines associated with two perspective triangles: three sides of the triangles, three lines through the center of perspectivity, and the axis of perspectivity. These ten points and ten lines form an instance of the [[Desargues configuration]]. | |||
[[File:Pappus hexagon.svg|thumb|240px|Two triply-perspective triangles ''BbY'' and ''CcX'']] | |||
If two triangles are a central couple in at least two different ways (with two different associations of corresponding vertices, and two different centers of perspectivity) then they are perspective in three ways. This is one of the equivalent forms of [[Pappus's hexagon theorem|Pappus's (hexagon) theorem]].<ref>{{harvnb|Coxeter|1969|loc=p. 233 exercise 2}} </ref> When this happens, the nine associated points (six triangle vertices and three centers) and nine associated lines (three through each perspective center) form an instance of the [[Pappus configuration]]. | |||
The [[Reye configuration]] is formed by four quadruply-perspective tetrahedra in an analogous way to the Pappus configuration. | |||
== Notes == | |||
{{reflist}} | |||
==References== | |||
* {{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 | last1=Dembowski | first1=Peter | title=Finite geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8}} | |||
* {{citation|first=John Wesley|last=Young|title=Projective Geometry|year=1930|publisher=Mathematical Association of America|series=The Carus Mathematical Monographs (#4)}} | |||
[[Category:Triangles]] | |||
[[Category:Projective geometry]] | |||
Revision as of 22:54, 31 January 2014
Two figures in a plane are perspective from a point O if the lines joining corresponding points of the figures all meet at O. Dually, the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in projective geometry where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions.
Terminology
The line which goes through the points where the figure's corresponding sides intersect is known as the axis of perspectivity, perspective axis, homology axis, or archaically, perspectrix. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the center of perspectivity, perspective center, homology center, pole, or archaically perspector. The figures are said to be perspective from this center.[1]
Perspectivity
If each of the perspective figures consists of all the points on a line (a range) then transformation of the points of one range to the other is called a central perspectivity. A dual transformation, taking all the lines through a point (a pencil) to another pencil by means of an axis of perspectivity is called an axial perspectivity.[2]
Triangles
An important special case occurs when the figures are triangles. Two triangles that are perspective from a point are called a central couple and two triangles that are perspective from a line are called an axial couple.[3]
Notation
Karl von Staudt introduced the notation to indicate that triangles ABC and abc are perspective.[4]
Related theorems and configurations
Desargues' theorem states that, a central couple of triangles is axial. The converse statement, an axial couple of triangles is central, is equivalent (either can be used to prove the other). Desargues' theorem can be proved in the real projective plane, and with suitable modifications for special cases, in the Euclidean plane. Projective planes in which this result can be proved are called Desarguesian planes.
There are ten points associated with these two kinds of perspective: six on the two triangles, three on the axis of perspectivity, and one at the center of perspectivity. Dually, there are also ten lines associated with two perspective triangles: three sides of the triangles, three lines through the center of perspectivity, and the axis of perspectivity. These ten points and ten lines form an instance of the Desargues configuration.
If two triangles are a central couple in at least two different ways (with two different associations of corresponding vertices, and two different centers of perspectivity) then they are perspective in three ways. This is one of the equivalent forms of Pappus's (hexagon) theorem.[5] When this happens, the nine associated points (six triangle vertices and three centers) and nine associated lines (three through each perspective center) form an instance of the Pappus configuration.
The Reye configuration is formed by four quadruply-perspective tetrahedra in an analogous way to the Pappus configuration.
Notes
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References
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ H. S. M. Coxeter (1942) Non-Euclidean Geometry, University of Toronto Press, reissued 1998 by Mathematical Association of America, ISBN 0-88385-522-4 . 21,2.
- ↑ Template:Harvnb