Quadrangles inscribed in a closed curve. (English. Russian original) Zbl 0862.57007
Math. Notes 57, No. 1, 91-93 (1995); translation from Mat. Zametki 57, No. 1, 129-132 (1995).
The article investigates when four points of a quadrangle in the plane can be moved by a similarity such that they lie on a given Jordan curve. The main result states that for any planar \(C^2\) smooth starlike Jordan curve and any four points on a circle there exists a similarity that maps the four points to points of the Jordan curve, provided the curve intersects the circle in at most four points. The author shows that this problem is related to a certain problem on the sphere. This problem then can be solved with rather elementary methods.
Reviewer: A.Schroth (Braunschweig)
MSC:
| 57N05 | Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) |
References:
| [1] | L. G. Shnirel’man, Usp. Mat. Nauk,10, 34–44 (1944). |
| [2] | H. B. Griffiths, Proc. London Math. Soc.,62, No. 3, 647–672 (1991). · Zbl 0696.55003 · doi:10.1112/plms/s3-62.3.647 |
| [3] | V. V. Makeev, Mat. Sb.,3, 424–431 (1989). |
| [4] | H. Yamabe and Z. Yujobo, Osaka Math. J., No. 2, 19–22 (1950). |
| [5] | C. Pucci, Atti Acad. Naz. Rend. C1. Sci. Fis. Mat. Nat.,21, 61–65 (1956). |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
