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''with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724101.png" />''
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''with centre $S$''
  
A mapping of a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724102.png" /> into a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724103.png" /> under which each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724104.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724105.png" /> is put into correspondence with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724106.png" /> of intersection of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724107.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724108.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p0724109.png" /> is not parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241010.png" />, see Fig.).
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A mapping of a plane $\pi$ into a plane $\pi_1$ under which each point $M$ in $\pi$ is put into correspondence with the point $M_1$ of intersection of the straight line $SM$ with $\pi_1$ (if $SM$ is not parallel to $\pi_1$, see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072410a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072410a.gif" />
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Figure: p072410a
 
Figure: p072410a
  
More generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241012.png" /> be proper subspaces of identical dimension in a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241013.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241014.png" /> be a subspace of maximal dimension not having points in common with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241015.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241016.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241017.png" /> be a subspace contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241018.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241019.png" /> be the subspace of minimal dimension containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241021.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241022.png" /> be the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241024.png" />.
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More generally, let $V$ and $V_1$ be proper subspaces of identical dimension in a projective space $\Omega$ and let $T$ be a subspace of maximal dimension not having points in common with $V$ or $V_1$. Let $U$ be a subspace contained in $V$, let $W$ be the subspace of minimal dimension containing $U$ and $T$ and let $U_1$ be the intersection of $W$ and $V_1$.
  
The correspondence by which each subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241025.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241026.png" /> is put into correspondence with the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241027.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241028.png" /> is called a perspective mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241029.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241030.png" /> with perspective centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241031.png" />.
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The correspondence by which each subspace $U$ contained in $V$ is put into correspondence with the subspace $U_1$ contained in $V_1$ is called a perspective mapping from $V$ into $V_1$ with perspective centre $T$.
  
A perspective is a [[Collineation|collineation]]. If the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241033.png" /> intersect, then each point in the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241034.png" /> corresponds to itself.
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A perspective is a [[Collineation|collineation]]. If the subspaces $V$ and $V_1$ intersect, then each point in the subspace $V\cap V_1$ corresponds to itself.
  
If projective coordinates are introduced into the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072410/p07241036.png" />, then a perspective correspondence can be specified by a linear mapping.
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If projective coordinates are introduced into the spaces $V$ and $V_1$, then a perspective correspondence can be specified by a linear mapping.
  
 
====References====
 
====References====

Latest revision as of 22:17, 11 April 2014

with centre $S$

A mapping of a plane $\pi$ into a plane $\pi_1$ under which each point $M$ in $\pi$ is put into correspondence with the point $M_1$ of intersection of the straight line $SM$ with $\pi_1$ (if $SM$ is not parallel to $\pi_1$, see Fig.).

Figure: p072410a

More generally, let $V$ and $V_1$ be proper subspaces of identical dimension in a projective space $\Omega$ and let $T$ be a subspace of maximal dimension not having points in common with $V$ or $V_1$. Let $U$ be a subspace contained in $V$, let $W$ be the subspace of minimal dimension containing $U$ and $T$ and let $U_1$ be the intersection of $W$ and $V_1$.

The correspondence by which each subspace $U$ contained in $V$ is put into correspondence with the subspace $U_1$ contained in $V_1$ is called a perspective mapping from $V$ into $V_1$ with perspective centre $T$.

A perspective is a collineation. If the subspaces $V$ and $V_1$ intersect, then each point in the subspace $V\cap V_1$ corresponds to itself.

If projective coordinates are introduced into the spaces $V$ and $V_1$, then a perspective correspondence can be specified by a linear mapping.

References

[1] E. Artin, "Geometric algebra" , Interscience (1957)
[2] N.A. Glagolev, "Projective geometry" , Moscow (1963) (In Russian)


Comments

Perspective mappings are also called central projections or perspectivities.

References

[a1] M. Berger, "Geometry" , I , Springer (1987)
[a2] H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)
[a3] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a4] H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964)
[a5] H.S.M. Coxeter, "The real projective plane" , McGraw-Hill (1949)
How to Cite This Entry:
Perspective. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perspective&oldid=31589
This article was adapted from an original article by P.S. Modenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article