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Extrema of principal curvature and symmetry

Published online by Cambridge University Press:  20 January 2009

J. W. Bruce  and
F. Tari
J. W. Bruce
Affiliation:
Department of Pure Mathematics The University P.O. Box 147 Liverpool L69 3BX, U.K.
F. Tari
Affiliation:
Department of Pure Mathematics The University P.O. Box 147 Liverpool L69 3BX, U.K.
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Abstract

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In this paper we show that away from umbilic points certain measures of the local reflectional symmetry of a surface in Euclidean 3-space are detected by the extrema of the sectional curvatures along lines of curvature. There are two types of reflectional symmetry, with one detected by the contact between the surface and spheres, and in this case the result is due to Porteous and is 20 years old. We show that an analogous result remains true for the second type of symmetry.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 2 , June 1996 , pp. 397 - 402
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Bruce, J. W., Generic reflections and projections, Math. Scand. 54 (1984), 262–278.CrossRefGoogle Scholar
2. Bruce, J. W. and Wilkinson, T. C., Folding maps and focal sets, in Proceedings of Warwick Symposium on Singularities (Springer Lecture Notes in Math., 1462 Springer-Verlag, Berlin and New York, 1991), 63–72.Google Scholar
3. Gourdon, A. P., Face recognition from depth maps and surface curvature, in Proceedings of SPIE Conference on Geometric Methods in Computer Vision (San Diego, CA, 07 1991).Google Scholar
4. Guéziec, A., Large deformable splines, crest lines and matching, INRIA, 11 1992, preprint.Google Scholar
5. Lipschutz, M. M. Differential Geometry (Schaum's Outline Guide Series, McGraw-Hill, New York, 1969).Google Scholar
6. Mond, D. M. Q., On the classification of germs of maps from R2 to R3 Proc. London Math. Soc. 50 (1985), 333–369.CrossRefGoogle Scholar
7. Morris, R. J. Symmetry of curves and the geometry of surfaces: two explorations with the aid of computer graphics (Thesis, University of Liverpool, 1991).Google Scholar
8. Porteous, I. R., The normal singularities of a submanifold, J. Differential Geom. 5 (1971), 543–564.CrossRefGoogle Scholar
9. Porteous, I. R., The normal singularities of surfaces in R3, in Proceedings of Symposia in Pure Mathematics (Volume 40, Part 2, American Mathematical Society, 1983, Providence), 379–394.Google Scholar
10. Porteous, I. R. Geometric Differentiation (Cambridge University Press, Cambridge, 1994).Google Scholar
11. Thirion, J. -P. and Gourdon, A., The 3D marching lines algorithm, INRIA, preprint.Google Scholar
12. Thirion, J. -P. and Gourdon, A., The marching lines algorithm: new results and proofs (INRIA research report No. 1881, March 1993).Google Scholar
13. Wall, C. T. C., Geometric properties of generic differentiable manifolds, in Geometry and Topology, Rio de Janeiro 1976 (Lecture Notes in Mathematics 597, 707–774, Springer-Verlag, New York), 707–775.Google Scholar
14. Wilkinson, T. C., The geometry of folding maps (Thesis, University of Newcastle-upon-Tyne, 1991).Google Scholar