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[[Image:Desargues theorem alt.svg|thumb|right|300px|Two perspective triangles, with their perspective axis and center]]
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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]]

Latest revision as of 14:19, 13 December 2014

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