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On the approximation of convex, closed plane curves by multifocal ellipses

Published online by Cambridge University Press:  14 July 2016

P. Erdös I. Vincze
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Abstract

The question whether a convex closed curve can be approximated by ellipses having a large number of foci is considered. It is shown that the limiting, convex figure of multifocal ellipses may have only one single straight segment. This happens only in the case, when the foci tend partly to infinity and partly to points of the line through the straight segment. The approximations of certain ‘distance integrals' are treated; the characterization of approximability remains an open problem.

Keywords

CONVEX CURVESMULTIFOCAL ELLIPSESAPPROXIMATION OF CONVEX CLOSED CURVES
Type
Part 2 — Geometry and Geometrical Probability
Information
Journal of Applied Probability , Volume 19 , Issue A , 1982 , pp. 89 - 96
Copyright
Copyright © 1982 Applied Probability Trust 

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References

[1] Erdös, P. and Vincze, I. (1958) On the approximation of convex, closed plane curves. Math. Lapok 9, 1–2.Google Scholar
[2] Weiszfeld, E. (1937) Sur le point pour lequel la somme des distances de n points donnés est minimum. Tohoku Math. J. 43, 355–386, and personal communication.Google Scholar
[3] Vincze, St. (1938) Über die Schwerpunkte der konvexen Kurven bei speziellen Belegungen. Acta Lit. Acad. Sci. Szeged 9, 52–59.Google Scholar