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A quadratic mapping with invariant cubic curve

Published online by Cambridge University Press:  24 October 2008

Roger Penrose  and
Cedric A. B. Smith
Roger Penrose
Affiliation:
University Of Oxford
Cedric A. B. Smith
Affiliation:
University College London
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Abstract

In the projective plane, if H is a harmonic homology (linear transformation with H2 = I), and G a general inversion (quadratic transformation projectively equivalent to an inversion), then under a certain condition there is a pencil of cubics each of which is invariant under G, H separately. These are related to transformations discovered by Mandel, Todd and Lyness. As a near converse, we find that, given a Pascal configuration, there is a quadratic Cremona transformation under which each cubic passing through the vertices of the configuration is invariant. As a by-product, parametric expressions are found for elliptic functions of a fifth of a period.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 1 , January 1981 , pp. 89 - 105
Copyright
Copyright © Cambridge Philosophical Society 1981

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