The centre symmetry set

Peter Giblin; Paul Holtom

Banach Center Publications (1999)

  • Volume: 50, Issue: 1, page 91-105
  • ISSN: 0137-6934

Abstract

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A centrally symmetric plane curve has a point called it’s centre of symmetry. We define (following Janeczko) a set which measures the central symmetry of an arbitrary strictly convex plane curve, or surface in R 3 . We investigate some of it’s properties, and begin the study of non-convex cases.

How to cite

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Giblin, Peter, and Holtom, Paul. "The centre symmetry set." Banach Center Publications 50.1 (1999): 91-105. <http://eudml.org/doc/209020>.

@article{Giblin1999,
abstract = {A centrally symmetric plane curve has a point called it’s centre of symmetry. We define (following Janeczko) a set which measures the central symmetry of an arbitrary strictly convex plane curve, or surface in $R^3$. We investigate some of it’s properties, and begin the study of non-convex cases.},
author = {Giblin, Peter, Holtom, Paul},
journal = {Banach Center Publications},
keywords = {generalized central symmetry; oval; envelope of lines; central symmetry set},
language = {eng},
number = {1},
pages = {91-105},
title = {The centre symmetry set},
url = {http://eudml.org/doc/209020},
volume = {50},
year = {1999},
}

TY - JOUR
AU - Giblin, Peter
AU - Holtom, Paul
TI - The centre symmetry set
JO - Banach Center Publications
PY - 1999
VL - 50
IS - 1
SP - 91
EP - 105
AB - A centrally symmetric plane curve has a point called it’s centre of symmetry. We define (following Janeczko) a set which measures the central symmetry of an arbitrary strictly convex plane curve, or surface in $R^3$. We investigate some of it’s properties, and begin the study of non-convex cases.
LA - eng
KW - generalized central symmetry; oval; envelope of lines; central symmetry set
UR - http://eudml.org/doc/209020
ER -

References

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  1. [1] V. I. Arnol’d, Critical points of functions on a manifold with boundary, the simple Lie groups B k , C k , and F 4 and singularities of evolutes (in Russian), Uspekhi Mat. Nauk 33 no. 5 (1978), 91-105, 237; English transl.: Russian Math. Surveys 33 no. 5 (1978), 99-116. 
  2. [3] J. W. Bruce and P. J. Giblin, Growth, motion and 1-parameter families of symmetry sets, Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 179-204. Zbl0656.58022
  3. [4] J. W. Bruce and P. J. Giblin, Projections of surfaces with boundary, Proc. London Math. Soc. (3) 60 (1990), 392-416. Zbl0667.58002
  4. [2] J. W. Bruce, P. J. Giblin and C. G. Gibson, Symmetry sets, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 163-186. Zbl0593.58012
  5. [6] P. J. Giblin and S. A. Brassett, Local symmetry of plane curves, Amer. Math. Monthly 92 (1985), 689-707. Zbl0604.53001
  6. [7] P. J. Giblin and G. Sapiro, Affine-invariant distances, envelopes and symmetry sets, Geom. Dedicata 71 (1998), 237-261. Zbl0902.53001
  7. [8] V. V. Goryunov, Projections of generic surfaces with boundary, in: Theory of Singularities and its Applications, V. I. Arnol'd (ed.), Adv. Soviet Math. 1, Amer. Math. Soc., Providence, 1990, 157-200. 
  8. [9] P. Holtom, Local Central Symmetry for Euclidean Plane Curves, M.Sc. Dissertation, University of Liverpool, Sept. 1997. 
  9. [10] S. Janeczko, Bifurcations of the center of symmetry, Geom. Dedicata 60 (1996), 9-16. Zbl0868.58015
  10. [11] Liverpool Surface Modelling Package, written by Richard Morris for Silicon Graphics and X Windows. See R. J. Morris, The use of computer graphics for solving problems in singularity theory, in: Visualization in Mathematics, H.-C. Hege and K. Polthier (eds.), Springer, Heidelberg, 1997, 53-66. 
  11. [5] Buchin Su, Affine Differential Geometry, Science Press, Beijing; Gordon and Breach, New York, 1983. Zbl0539.53002
  12. [12] V. M. Zakalyukin, Envelopes of families of wave fronts and control theory, Proc. Steklov Inst. Math. 209 (1995), 114-123. Zbl0883.93008

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