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Affine geometry

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The branch of geometry whose subject are the properties of figures that are invariant with respect to affine transformations (cf. Affine transformation). Examples are the simple relation for three points to lie on a straight line, or the parallelity of straight lines (planes). A.F. Möbius, in the first half of the 19th century, was the first to study the properties of geometric images that go over into each other as a result of affine transformations; however, the concept of "affine geometry" only arose following the appearance in 1872 of the Erlangen program, according to which each group of transformations has its own geometry, the subject of which are the properties of the figures that are invariant with respect to the transformations of this group. The group of affine transformations contains various subgroups; as a result, in addition to general affine geometry, subordinated geometries — equi-affine geometry, centro-affine geometry, etc. — arose, corresponding to these groups. Affine geometry also deals with problems of differential geometry corresponding to specific transformation subgroups (cf. Affine differential geometry).

References

[1] P.S. Aleksandrov, "Lectures on analytical geometry" , Moscow (1968) (In Russian)
[2] N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)


Comments

Two English-language references are [a1], [a2].

References

[a1] K. Borsuk, "Multidimensional analytic geometry" , PWN (1969)
[a2] B.E. Meserve, "Fundamental concepts of geometry" , Addison-Wesley (1955)
How to Cite This Entry:
Affine geometry. E.V. Shikin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Affine_geometry&oldid=18996
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098