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Homology

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in projective geometry

An automorphism of the projective plane that leaves fixed all the points of a given straight line (the homology axis) and maps onto themselves all the lines through exactly one fixed point (the homology centre). If the homology centre does not lie on the homology axis, the homology is known as non-singular (or hyperbolic); if the homology centre lies on the homology axis, the homology is called singular (or parabolic). A homology is usually specified by a centre, an axis and a pair of points in correspondence under the homology. A homology of the affine plane with finite centre and axis at infinity is a homothety; one with a centre at infinity and finite axis not through the centre is an extension or contraction towards the axis; one with both centre and axis at infinity represents a parallel translation; a singular homology with finite axis and centre at infinity is a shift.

References

[1] R. Hartshorne, "Foundations of projective geometry" , Benjamin (1967)


Comments

A homology is also called a central collineation, a hyperbolic homology — a dilation, and a parabolic homology — an elation or transvection.

References

[a1] H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953)
[a2] H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964)
How to Cite This Entry:
Homology. A.B. Ivanov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Homology&oldid=17654
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098