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Generic isotopies of space curves

Published online by Cambridge University Press:  18 May 2009

J. W. Bruce  and
P. J. Giblin
J. W. Bruce
Affiliation:
Department of Mathematics, The University Newcastle-upon-Tyne, NE17RUEngland
P. J. Giblin
Affiliation:
Department of Pure Mathematics, The University Liverpool, L69 3BXEngland
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For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 29 , Issue 1 , January 1987 , pp. 41 - 63
Copyright
Copyright © Glasgow Mathematical Journal Trust 1987

References

REFERENCES

1.Arnold, V. I., Wavefront evolution and equivariant Morse lemma, Comm. Pure Appl. Math. 29 (1976), 557–582.CrossRefGoogle Scholar
2.Arnold, V. I., Catastrophe theory (Springer-Verlag, 1984).CrossRefGoogle Scholar
3.Bruce, J. W., Wavefronts and parallels in Euclidean space, Math. Proc. Cambridge Philos. Soc. 93 (1983), 323–333.CrossRefGoogle Scholar
4.Bruce, J. W., Isotopies of generic plane curves Glasgow Math. J. 24 (1983), 195–206.CrossRefGoogle Scholar
5.Bruce, J. W., Self-intersections of wavefront evolution, Proc. Roy. Irish Acad. Sect. A. 83 (1983), 225–229.Google Scholar
6.Bruce, J. W. and Giblin, P. J., Curves and singularities (Cambridge University Press, 1984).Google Scholar
7.Bruce, J. W. and Giblin, P. J., Outlines and their duals, Proc. London Math. Soc (3) 50 (1985), 552–570.CrossRefGoogle Scholar
8.Shcherbak, O. P., Projectively dual space curves and Legendre singularities (Russian), Trudy Tbiliss. Univ. 232/233 (1982), 280–336.Google Scholar