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Affine cubic functions

III. The real plane

Published online by Cambridge University Press:  24 October 2008

C. T. C. Wall
Affiliation:
University of Liverpool

Extract

The classification of affine cubic functions in the real case is a fairly easy corollary of that in the complex case (9). However as the results can be easily interpreted by diagrams, one can obtain a much richer understanding. For example, the question of which types of cubic curve occur as level curves of which types of function is now much less trivial. This will lead us first to re-examine the classification of cubic curves going back to Newton (4). Next the ‘dynamic’ approach of considering these curves as members of families leads to the diagrams associated with the umbilic catastrophes of Thorn (8). However the consideration of functions rather than of curves gives a 1-dimensional foliation of these diagrams which we describe next. We conclude by placing the results back in a protective setting.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Cayley, A.On the classification of cubic curves. Camb. Phil. Trans. 11 (1866), 81128. (Collected works, vol. v, 354–400.)Google Scholar
(2) De Gua De Malves, J. P.Usage de l'analyse de Descartes pour decouvrir sans le secours du calcul différentiel…etc. 1740.Google Scholar
(3)Murdoch, P.Genesis curvarum per umbras. 1746.Google Scholar
(4)Newton, I.Enumeratio linearum tertii Ordinis, appendix to Treatise on Optics, 1706. English translation by Talbot, C. R. M. (with notes), London 1861.Google Scholar
(5)Pl'cker, J.System der Analytischen Geometrie. 1835.Google Scholar
(6)Porteous, I. R.The normal singularities of a submanifold, J. Diff. Geom. 5 (1971), 543564.Google Scholar
(7)Stirling, J.Illustratio tractatus D. Newtoni de enumeratione linearum tertii ordinis, 1717.Google Scholar
(8)Thom, R.Structural stability and morphogenesis, transl. by Fowler, D. (New York, Benjamin, 1975).Google Scholar
(9)Wall, C. T. C.Affine cubic functions I. The complex plane. Math. Proc. Cambridge Philos. Soc. 85 (1979), 387401.CrossRefGoogle Scholar
(10)Woodcock, A. E. R. and Poston, T.A geometrical study of the elementary catastrophes. Springer lecture notes 373 (1974).CrossRefGoogle Scholar
(11)Zeeman, E. C.The umbilic bracelet and the double-cusp catastrophe, pp. 328366 in Springer lecture notes 525 (1976).Google Scholar