Quadrangles Inscribed in a Closed Curve and the Vertices of a Curve
 V. V. Makeev
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Let ABCDE be a pentagon inscribed in a circle. It is proved that if \(\mathcal{O}\) is a C^{4}generic smooth convex planar oval with four vertices (stationary points of curvature), then there are two similarities φ such that the quadrangle φ(ABCD) is inscribed in \(\mathcal{O}\) and the point φ(E)lies inside \(\mathcal{O}\) , as well as two similarities ψ such that the quadrangle ψ(ABCD) is inscribed in \(\mathcal{O}\) and ψ(E)lies outside \(\mathcal{O}\) . Itisalsoprovedthatif n is odd, then any smoothly embedded circle γ ↪ ℝ^{n} contains the vertices of an equilateral (n + 1)link polygonal line lying in a hyperplane of ℝ^{n}. Bibliography: 7 titles.
 Mukhopadhayaya, S. (1909) New methods in the geometry of planar arcI, cyclic, and sextactic points. Bull. Calcutta Math. Soc. 1: pp. 3137
 Musin, O. R. (2001) Extrema of the curvature and the fourvertex theorem for polygons and polyhedra. Zap. Nauchn. Semin. POMI 280: pp. 251271
 Makeev, V. V. (1995) On quadrangles inscribed in a closed curve. Mat. Zametki 57: pp. 129132
 Shnirel'man, L. G. (1944) On certain geometric properties of closed curves. Usp. Mat. Nauk 10: pp. 3444
 Bose, R. C. (1932) On the number of circles of curvature perfectly enclosing or perfectly enclosed by a closed oval. Math. Ann. 35: pp. 1624
 Makeev, V. V. (1994) Inscribed and circumscribed polyhedra for a convex body. Mat. Zametki 55: pp. 128130
 Griffits, H. (1991) The topology of square pegs in round holes. Proc. London Math. Soc. 363: pp. 647672
 Title
 Quadrangles Inscribed in a Closed Curve and the Vertices of a Curve
 Journal

Journal of Mathematical Sciences
Volume 131, Issue 1 , pp 53955400
 Cover Date
 20051101
 DOI
 10.1007/s1095800504128
 Print ISSN
 10723374
 Online ISSN
 15738795
 Publisher
 Kluwer Academic PublishersConsultants Bureau
 Additional Links
 Topics
 Industry Sectors
 Authors

 V. V. Makeev ^{(1)}
 Author Affiliations

 1. St. Petersburg State University, St. Petersburg, Russia