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Difference between revisions of "Projective transformation"

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<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>
  
  

Revision as of 21:55, 30 March 2012

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

1) if , then ;

2) for every there is an such that ;

3) if and only if .

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 be interpreted as the collection of subspaces of the left vector space over a skew-field . A semi-linear transformation of into itself is a pair consisting of an automorphism of the additive group and an automorphism of the skew-field such that for any and the equality holds. In particular, a semi-linear transformation is linear if . A semi-linear transformation induces a projective transformation . The converse assertion is the first fundamental theorem of projective geometry: If , then every projective transformation is induced by some semi-linear transformation of the space .

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 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=23939
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article