Envelopes and characteristics

J.W. Bruce

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    Abstract

    Envelopes of hypersurfaces arise in many different geometric situations. For example, the canal surfaces associated with a space curve can be obtained as the envelope of spheres of a fixed radius centred on that curve. (Other examples can be found in [11], chapter II and [4], chapter 5). In what follows we shall be entirely concerned with the local structure of such envelopes. So essentially we have a smooth map germ with 0 a regular value of the restriction of F to . For near 0 we have a smooth hypersurface near , where . The envelope of this family of hypersurfaces is obtained by eliminating t from the equations . Moreover, it can be identified with the discriminant of F viewed as an unfolding, by the x variables, of the function germ . This observation, together with standard results on versal unfoldings of functions, allows one to determine the local structure of many ‘generic’ geometric envelopes. See [2] for details.
    Original languageEnglish
    Pages (from-to)475-492
    JournalMathematical Proceedings of the Cambridge Philosophical Society
    Volume100
    Issue number3
    DOIs
    Publication statusPublished - 1986

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    Envelope
    Hypersurface
    Local Structure
    Unfolding
    Space Curve
    Discriminant
    Radius
    Restriction
    Curve

    Cite this

    @article{5f70c20492734a02aea758795722c6f3,
    title = "Envelopes and characteristics",
    abstract = "Envelopes of hypersurfaces arise in many different geometric situations. For example, the canal surfaces associated with a space curve can be obtained as the envelope of spheres of a fixed radius centred on that curve. (Other examples can be found in [11], chapter II and [4], chapter 5). In what follows we shall be entirely concerned with the local structure of such envelopes. So essentially we have a smooth map germ with 0 a regular value of the restriction of F to . For near 0 we have a smooth hypersurface near , where . The envelope of this family of hypersurfaces is obtained by eliminating t from the equations . Moreover, it can be identified with the discriminant of F viewed as an unfolding, by the x variables, of the function germ . This observation, together with standard results on versal unfoldings of functions, allows one to determine the local structure of many ‘generic’ geometric envelopes. See [2] for details.",
    author = "J.W. Bruce",
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    doi = "10.1017/S0305004100066214",
    language = "English",
    volume = "100",
    pages = "475--492",
    journal = "Mathematical Proceedings of the Cambridge Philosophical Society",
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    Envelopes and characteristics. / Bruce, J.W.

    In: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 100, No. 3, 1986, p. 475-492.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Envelopes and characteristics

    AU - Bruce, J.W.

    PY - 1986

    Y1 - 1986

    N2 - Envelopes of hypersurfaces arise in many different geometric situations. For example, the canal surfaces associated with a space curve can be obtained as the envelope of spheres of a fixed radius centred on that curve. (Other examples can be found in [11], chapter II and [4], chapter 5). In what follows we shall be entirely concerned with the local structure of such envelopes. So essentially we have a smooth map germ with 0 a regular value of the restriction of F to . For near 0 we have a smooth hypersurface near , where . The envelope of this family of hypersurfaces is obtained by eliminating t from the equations . Moreover, it can be identified with the discriminant of F viewed as an unfolding, by the x variables, of the function germ . This observation, together with standard results on versal unfoldings of functions, allows one to determine the local structure of many ‘generic’ geometric envelopes. See [2] for details.

    AB - Envelopes of hypersurfaces arise in many different geometric situations. For example, the canal surfaces associated with a space curve can be obtained as the envelope of spheres of a fixed radius centred on that curve. (Other examples can be found in [11], chapter II and [4], chapter 5). In what follows we shall be entirely concerned with the local structure of such envelopes. So essentially we have a smooth map germ with 0 a regular value of the restriction of F to . For near 0 we have a smooth hypersurface near , where . The envelope of this family of hypersurfaces is obtained by eliminating t from the equations . Moreover, it can be identified with the discriminant of F viewed as an unfolding, by the x variables, of the function germ . This observation, together with standard results on versal unfoldings of functions, allows one to determine the local structure of many ‘generic’ geometric envelopes. See [2] for details.

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