Difference between revisions of "Projective transformation"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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.
Projective transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_transformation&oldid=23939