Envelopes and characteristics

Bruce, J.W. (1986) Envelopes and characteristics. Mathematical Proceedings of the Cambridge Philosophical Society, 100 (3). pp. 475-492. ISSN 0305-0041 DOI https://doi.org/10.1017/S0305004100066214

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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.

Item Type: Article
Subjects: Q Science > QA Mathematics
Date Deposited: 12 Jan 2011 10:18
URI: http://repository.edgehill.ac.uk/id/eprint/2189

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