### Abstract

Original language | English |
---|---|

Pages (from-to) | 243-268 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 125 |

Issue number | 2 |

Publication status | Published - 1999 |

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### Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*,

*125*(2), 243-268.

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*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 125, no. 2, pp. 243-268.

**Families of surfaces: focal sets, ridges and umbilics.** / Bruce, J.W.; Giblin, P.J.; Tari, F.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Families of surfaces: focal sets, ridges and umbilics

AU - Bruce, J.W.

AU - Giblin, P.J.

AU - Tari, F.

PY - 1999

Y1 - 1999

N2 - The contact between a surface in Euclidean space and the family of spheres carries a good deal of useful geometrical information about the surface. The centres of those spheres having degenerate tangency with the surface (in Arnold's notation of type A2) form the focal set, and the set of points on the surface where there is a contact of type A3, the so-called ridge curve, is of some interest in the field of computer vision, as a robust feature of the surface. These ridges come together at umbilics, where the contact between the surface and the unique sphere of curvature is of type D. A dual notion, that of a subparabolic curve is also of some interest. In this paper we describe the way in which the ridge and subparabolic curves can evolve in a generic 1-parameter family of surfaces.

AB - The contact between a surface in Euclidean space and the family of spheres carries a good deal of useful geometrical information about the surface. The centres of those spheres having degenerate tangency with the surface (in Arnold's notation of type A2) form the focal set, and the set of points on the surface where there is a contact of type A3, the so-called ridge curve, is of some interest in the field of computer vision, as a robust feature of the surface. These ridges come together at umbilics, where the contact between the surface and the unique sphere of curvature is of type D. A dual notion, that of a subparabolic curve is also of some interest. In this paper we describe the way in which the ridge and subparabolic curves can evolve in a generic 1-parameter family of surfaces.

M3 - Article

VL - 125

SP - 243

EP - 268

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 2

ER -