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''projective''
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A [[Projective transformation|projective transformation]] (projective isomorphism) of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232101.png" /> that is representable as the product of a finite number of perspectivities (cf. [[Perspective|Perspective]]); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232102.png" /> is a projective collineation, then for any subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232103.png" /> there exists a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232104.png" /> of not more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232105.png" /> perspectivities such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232106.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232107.png" />. For example, a projective transformation that leaves each point of some straight line fixed is a collineation, this is, a homology (in the narrow sense).
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232108.png" /> be interpreted as the collection of subspaces of the linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c0232109.png" /> over a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321010.png" />. Then in order that a projective transformation be a projective collineation, it is necessary and sufficient that it be induced by a linear transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321011.png" />. The collection of all projective collineations forms a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321012.png" /> of the group of projective transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321013.png" /> which is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321014.png" />.
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''projective''
  
The projective collineations exhaust all the projective transformations if and only if every automorphism of the skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321015.png" /> is inner. A field possesses this property if and only if any of its automorphisms is the identity, such as, for example, the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321016.png" />. The complex field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321017.png" /> does not possess this property, whereas every automorphism of the skew-field of quaternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321018.png" /> is inner.
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A [[Projective transformation|projective transformation]] (projective isomorphism) of a projective space  $  \Pi _ {n} $
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that is representable as the product of a finite number of perspectivities (cf. [[Perspective|Perspective]]); if  $  \nu $
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is a projective collineation, then for any subspace  $  S _ {q} $
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there exists a product  $  \pi $
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of not more than  $  q - 1 $
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perspectivities such that  $  \nu ( S _ {p} ) = \pi ( S _ {p} ) $
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for any  $  S _ {p} \subset  S _ {q} $.
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For example, a projective transformation that leaves each point of some straight line fixed is a collineation, this is, a homology (in the narrow sense).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321019.png" /> is a non-commutative skew-field, then there exists a non-trivial projective collineation that leaves every point of a given simplex fixed. Each simplex can be mapped onto any other simplex by precisely one projective collineation, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321020.png" /> is a field (the second fundamental theorem of projective geometry).
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Let  $  \Pi _ {n} $
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be interpreted as the collection of subspaces of the linear space  $  A _ {n + 1 }  ^ {e} ( K) $
 +
over a skew-field $  K $.
 +
Then in order that a projective transformation be a projective collineation, it is necessary and sufficient that it be induced by a linear transformation of  $  A _ {n + 1 }  ^ {e} ( K) $.  
 +
The collection of all projective collineations forms a subgroup  $  G _ {0} $
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of the group of projective transformations  $  G $
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which is a normal subgroup of  $  G $.
  
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The projective collineations exhaust all the projective transformations if and only if every automorphism of the skew-field  $  K $
 +
is inner. A field possesses this property if and only if any of its automorphisms is the identity, such as, for example, the field of real numbers  $  \mathbf R $.
 +
The complex field  $  \mathbf C $
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does not possess this property, whereas every automorphism of the skew-field of quaternions  $  \mathbf H $
 +
is inner.
  
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If  $  K $
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is a non-commutative skew-field, then there exists a non-trivial projective collineation that leaves every point of a given simplex fixed. Each simplex can be mapped onto any other simplex by precisely one projective collineation, if and only if  $  K $
 +
is a field (the second fundamental theorem of projective geometry).
  
 
====Comments====
 
====Comments====
There is a large amount of confusion in the literature on projective geometry about the terminology for the different kinds of transformations. The transformation defined above, a projective collineation, sometimes also called a projectivity, is usually defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023210/c02321021.png" /> transformation that is the product of a finite number of perspectivities. Some authors use the term projectivity, however, only for mappings between lines that are the product of a finite number of perspectivities. A projective transformation, usually called a projective isomorphism, or in older literature simply a collineation, is a bijection between projective spaces that preserves incidence.
+
There is a large amount of confusion in the literature on projective geometry about the terminology for the different kinds of transformations. The transformation defined above, a projective collineation, sometimes also called a projectivity, is usually defined as $  any $
 +
transformation that is the product of a finite number of perspectivities. Some authors use the term projectivity, however, only for mappings between lines that are the product of a finite number of perspectivities. A projective transformation, usually called a projective isomorphism, or in older literature simply a collineation, is a bijection between projective spaces that preserves incidence.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


projective

A projective transformation (projective isomorphism) of a projective space $ \Pi _ {n} $ that is representable as the product of a finite number of perspectivities (cf. Perspective); if $ \nu $ is a projective collineation, then for any subspace $ S _ {q} $ there exists a product $ \pi $ of not more than $ q - 1 $ perspectivities such that $ \nu ( S _ {p} ) = \pi ( S _ {p} ) $ for any $ S _ {p} \subset S _ {q} $. For example, a projective transformation that leaves each point of some straight line fixed is a collineation, this is, a homology (in the narrow sense).

Let $ \Pi _ {n} $ be interpreted as the collection of subspaces of the linear space $ A _ {n + 1 } ^ {e} ( K) $ over a skew-field $ K $. Then in order that a projective transformation be a projective collineation, it is necessary and sufficient that it be induced by a linear transformation of $ A _ {n + 1 } ^ {e} ( K) $. The collection of all projective collineations forms a subgroup $ G _ {0} $ of the group of projective transformations $ G $ which is a normal subgroup of $ G $.

The projective collineations exhaust all the projective transformations if and only if every automorphism of the skew-field $ K $ is inner. A field possesses this property if and only if any of its automorphisms is the identity, such as, for example, the field of real numbers $ \mathbf R $. The complex field $ \mathbf C $ does not possess this property, whereas every automorphism of the skew-field of quaternions $ \mathbf H $ is inner.

If $ K $ is a non-commutative skew-field, then there exists a non-trivial projective collineation that leaves every point of a given simplex fixed. Each simplex can be mapped onto any other simplex by precisely one projective collineation, if and only if $ K $ is a field (the second fundamental theorem of projective geometry).

Comments

There is a large amount of confusion in the literature on projective geometry about the terminology for the different kinds of transformations. The transformation defined above, a projective collineation, sometimes also called a projectivity, is usually defined as $ any $ transformation that is the product of a finite number of perspectivities. Some authors use the term projectivity, however, only for mappings between lines that are the product of a finite number of perspectivities. A projective transformation, usually called a projective isomorphism, or in older literature simply a collineation, is a bijection between projective spaces that preserves incidence.

References

[a1] M. Berger, "Geometry" , I , Springer (1987)
How to Cite This Entry:
Collineation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Collineation&oldid=46398
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article