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

Fingerprint

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",
year = "1986",
doi = "10.1017/S0305004100066214",
language = "English",
volume = "100",
pages = "475--492",
journal = "Mathematical Proceedings of the Cambridge Philosophical Society",
issn = "0305-0041",
publisher = "Cambridge University Press",
number = "3",

}

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.

U2 - 10.1017/S0305004100066214

DO - 10.1017/S0305004100066214

M3 - Article

VL - 100

SP - 475

EP - 492

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 3

ER -