Projective transformation

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.