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 . We investigate some of it’s properties, and begin the study of non-convex cases.
@article{bwmeta1.element.bwnjournal-article-bcpv50z1p91bwm,
author = {Giblin, Peter and Holtom, Paul},
title = {The centre symmetry set},
journal = {Banach Center Publications},
volume = {50},
year = {1999},
pages = {91-105},
zbl = {0957.53003},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p91bwm}
}
Giblin, Peter; Holtom, Paul. The centre symmetry set. Banach Center Publications, Tome 50 (1999) pp. 91-105. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p91bwm/
[000] [1] V. I. Arnol’d, Critical points of functions on a manifold with boundary, the simple Lie groups , , and 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.
[001] [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. | Zbl 0656.58022
[002] [4] J. W. Bruce and P. J. Giblin, Projections of surfaces with boundary, Proc. London Math. Soc. (3) 60 (1990), 392-416. | Zbl 0667.58002
[003] [2] J. W. Bruce, P. J. Giblin and C. G. Gibson, Symmetry sets, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), 163-186. | Zbl 0593.58012
[004] [6] P. J. Giblin and S. A. Brassett, Local symmetry of plane curves, Amer. Math. Monthly 92 (1985), 689-707. | Zbl 0604.53001
[005] [7] P. J. Giblin and G. Sapiro, Affine-invariant distances, envelopes and symmetry sets, Geom. Dedicata 71 (1998), 237-261. | Zbl 0902.53001
[006] [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.
[007] [9] P. Holtom, Local Central Symmetry for Euclidean Plane Curves, M.Sc. Dissertation, University of Liverpool, Sept. 1997.
[008] [10] S. Janeczko, Bifurcations of the center of symmetry, Geom. Dedicata 60 (1996), 9-16. | Zbl 0868.58015
[009] [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.
[010] [5] Buchin Su, Affine Differential Geometry, Science Press, Beijing; Gordon and Breach, New York, 1983. | Zbl 0539.53002
[011] [12] V. M. Zakalyukin, Envelopes of families of wave fronts and control theory, Proc. Steklov Inst. Math. 209 (1995), 114-123. | Zbl 0883.93008