Lines, circles, focal and symmetry sets

J.W. Bruce

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    Abstract

    Let X be a surface in Euclidean 3-space, hereafter denoted by 3. In the paper [13] Montaldi considered the contact of the surface X with circles, and obtained some very attractive results. In this piece of work we want to address some more detailed questions concerning such contact. In keeping with a general theme within singularity theory we shall bundle the circles up into fibres of certain maps and consider the restriction of these mappings to our surface X. In other words we shall be interested in the simultaneous contact of the surface X with special families of circles. The particular families we shall consider are parameterized by the set K of all lines in 3; associated to such a line we have the family of all circles lying in planes orthogonal to the line, and centred on the line. The line will be referred to as the axis of the circle. Suppose, for example, the line in question is given by x1 = x2 = 0. We can consider the map 3 → 2 given by . The fibres of this mapping are clearly the set of circles with the properties described above together, of course, with single points on the line itself. So the family of oriented lines parameterizes a family of mappings 3 → 2, and by restriction a family of mappings X → 2. It is of interest to relate the singularities of this mapping to the differential geometry of X. The key geometric invariant of any smooth family is its bifurcation set, that is the set of parameter values for which the corresponding map fails to be stable. We shall see that for the family of circle maps the bifurcation set is of some interest.
    Original languageEnglish
    Pages (from-to)411-436
    JournalMathematical Proceedings of the Cambridge Philosophical Society
    Volume118
    Issue number3
    DOIs
    Publication statusPublished - 1995

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