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A one-to-one mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753801.png" /> of a [[Projective space|projective space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753802.png" /> onto itself preserving the order relation in the partially ordered (by inclusion) set of all subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753803.png" />, that is, a mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753804.png" /> onto itself such that:
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1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753805.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753806.png" />;
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{{TEX|done}}
  
2) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753807.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753808.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p0753809.png" />;
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A one-to-one mapping 
 +
of a [[Projective space|projective space]]    \Pi _ {n}
 +
onto itself preserving the order relation in the partially ordered (by inclusion) set of all subspaces of    \Pi _ {n} ,
 +
that is, a mapping of    \Pi _ {n}
 +
onto itself such that:
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538010.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538011.png" />.
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1) if    S _ {p} \subset  S _ {q} ,
 +
then    F ( S _ {p} ) \subset  F ( S _ {q} ) ;
 +
 
 +
2) for every    \widetilde{S}  _ {p}
 +
there is an    S _ {p}
 +
such that    F ( S _ {p} ) = \widetilde{S}  _ {p} ;
 +
 
 +
3) $  S _ {p} = S _ {q} $
 +
if and only if $  F ( S _ {p} ) = F ( S _ {q} ) $.
  
 
Under a projective transformation the sum and intersection of subspaces are preserved, points are mapped to points, and independence of points is preserved. The projective transformations constitute a group, called the projective group. Examples of projective transformations are: a [[Collineation|collineation]], a [[Perspective|perspective]] and a [[Homology|homology]].
 
Under a projective transformation the sum and intersection of subspaces are preserved, points are mapped to points, and independence of points is preserved. The projective transformations constitute a group, called the projective group. Examples of projective transformations are: a [[Collineation|collineation]], a [[Perspective|perspective]] and a [[Homology|homology]].
  
Let the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538012.png" /> be interpreted as the collection of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538013.png" /> of the left vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538014.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538015.png" />. A semi-linear transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538016.png" /> into itself is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538017.png" /> consisting of an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538018.png" /> of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538019.png" /> and an automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538020.png" /> of the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538021.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538023.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538024.png" /> holds. In particular, a semi-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538025.png" /> is linear if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538026.png" />. A semi-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538027.png" /> induces a projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538028.png" />. The converse assertion is the first fundamental theorem of projective geometry: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538029.png" />, then every projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538030.png" /> is induced by some semi-linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538031.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538032.png" />.
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Let the space   \Pi _ {n}
 +
be interpreted as the collection of subspaces   P _ {n} ( K )
 +
of the left vector space   A _ {n+} 1 ( K )
 +
over a skew-field   K .  
 +
A semi-linear transformation of   A _ {n+} 1
 +
into itself is a pair   ( \overline{F}\; , \phi )
 +
consisting of an automorphism   \overline{F}\;
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of the additive group   A _ {n+} 1
 +
and an automorphism   \phi
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of the skew-field   K
 +
such that for any   a \in A _ {n+} 1
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and   k \in K
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the equality $  \overline{F}\; ( ka ) = \phi ( k ) \overline{F}\; ( a ) $
 +
holds. In particular, a semi-linear transformation   ( \overline{F}\; , \phi )
 +
is linear if   \phi ( k) \equiv k .  
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A semi-linear transformation   ( \overline{F}\; , \phi )
 +
induces a projective transformation   F .  
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The converse assertion is the first fundamental theorem of projective geometry: If   n \geq  2 ,  
 +
then every projective transformation   F
 +
is induced by some semi-linear transformation   ( \overline{F}\; , \phi )
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of the space   A _ {n+} 1 ( K ) .
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Baer,   "Linear algebra and projective geometry" , Acad. Press (1952)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge,   D. Pedoe,   "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press (1947)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952) {{MR|0052795}} {{ZBL|0049.38103}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press (1947) {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
A projective transformation can also be defined as a bijection of the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075380/p07538033.png" /> preserving collinearity in both directions.
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A projective transformation can also be defined as a bijection of the points of   \Pi _ {n}
 +
preserving collinearity in both directions.
  
 
Other names used for a projective transformation are: projectivity, collineation. See also [[Collineation|Collineation]] for terminology.
 
Other names used for a projective transformation are: projectivity, collineation. See also [[Collineation|Collineation]] for terminology.

Latest revision as of 08:08, 6 June 2020


A one-to-one mapping F of a projective space \Pi _ {n} onto itself preserving the order relation in the partially ordered (by inclusion) set of all subspaces of \Pi _ {n} , that is, a mapping of \Pi _ {n} onto itself such that:

1) if S _ {p} \subset S _ {q} , then F ( S _ {p} ) \subset F ( S _ {q} ) ;

2) for every \widetilde{S} _ {p} there is an S _ {p} such that F ( S _ {p} ) = \widetilde{S} _ {p} ;

3) S _ {p} = S _ {q} if and only if F ( S _ {p} ) = F ( S _ {q} ) .

Under a projective transformation the sum and intersection of subspaces are preserved, points are mapped to points, and independence of points is preserved. The projective transformations constitute a group, called the projective group. Examples of projective transformations are: a collineation, a perspective and a homology.

Let the space \Pi _ {n} be interpreted as the collection of subspaces P _ {n} ( K ) of the left vector space A _ {n+} 1 ( K ) over a skew-field K . A semi-linear transformation of A _ {n+} 1 into itself is a pair ( \overline{F}\; , \phi ) consisting of an automorphism \overline{F}\; of the additive group A _ {n+} 1 and an automorphism \phi of the skew-field K such that for any a \in A _ {n+} 1 and k \in K the equality \overline{F}\; ( ka ) = \phi ( k ) \overline{F}\; ( a ) holds. In particular, a semi-linear transformation ( \overline{F}\; , \phi ) is linear if \phi ( k) \equiv k . A semi-linear transformation ( \overline{F}\; , \phi ) induces a projective transformation F . The converse assertion is the first fundamental theorem of projective geometry: If n \geq 2 , then every projective transformation F is induced by some semi-linear transformation ( \overline{F}\; , \phi ) of the space A _ {n+} 1 ( K ) .

References

[1] R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952) MR0052795 Zbl 0049.38103
[2] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 1 , Cambridge Univ. Press (1947) MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502

Comments

A projective transformation can also be defined as a bijection of the points of \Pi _ {n} preserving collinearity in both directions.

Other names used for a projective transformation are: projectivity, collineation. See also Collineation for terminology.

How to Cite This Entry:
Projective transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_transformation&oldid=12453
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article