Parallelograms inscribed in a curve having a circle as π/2-isoptic

• Autorzy: Andrzej Miernowski
• Języki publikacji: EN

Abstrakty

EN

Jean-Marc Richard observed in [7] that maximal perimeter of a parallelogram inscribed in a given ellipse can be realized by a parallelogram with one vertex at any prescribed point of ellipse. Alain Connes and Don Zagier gave in [4] probably the most elementary proof of this property of ellipse. Another proof can be found in [1]. In this note we prove that closed, convex curves having circles as π/2-isoptics have the similar property.

Słowa kluczowe

EN

Convex curve; support function; curvature

• Wydawca: Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej
• Czasopismo: Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
• Rocznik: 2008
• Tom: 62
• Numer: 1
• Strony: 105-111

Daty

wydano

2008-01-01

online

2009-02-09

Twórcy

autor

Andrzej Miernowski

• Institute of Mathematics, M. Curie-Skłodowska University, pl. Marii Curie-Skłodowskiej 1 20-031 Lublin, Poland

Bibliografia

• Berger, M., Geometrie, Vol. 2, Nathan, Paris, 1990.
• Cieślak, W., Miernowski, A. and Mozgawa, W., Isoptics of a strictly convex curve, Global Differential Geometry and Global Analysis, 1990 (Berlin), Lecture Notes in Math., 1481, Springer, Berlin, 1991, 28-35.
• Cieślak, W., Miernowski, A. and Mozgawa, W., Isoptics of a strictly convex curve II, Rend. Sem. Mat. Univ. Padova 96 (1996), 37-49.
• Connes, A., Zagier, D., A property of parallelograms inscribed in ellipses, Amer. Math. Monthly 114 (2007), 909-914.
• Green, J. W., Sets subtending a constant angle on a circle, Duke Math. J. 17 (1950), 263-267.
• Matsuura, S., On nonconvex curves of constant angle, Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math., 1540, Springer, Berlin, 1993, 251-268.
• Richard, J-M., Safe domain and elementary geometry, Eur. J. Phys. 25 (2004), 835-844.[Crossref]
• Wunderlich, W., Kurven mit isoptischem Kreis, Aequationes Math. 6 (1971), 71-78.

Identyfikatory

DOI

10.2478/v10062-008-0012-4