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

Andrzej Miernowski

Annales UMCS, Mathematica (2008)

  • Volume: 62, Issue: 1, page 105-111
  • ISSN: 2083-7402

Abstract

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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.

How to cite

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Andrzej Miernowski. "Parallelograms inscribed in a curve having a circle as π/2-isoptic." Annales UMCS, Mathematica 62.1 (2008): 105-111. <http://eudml.org/doc/268006>.

@article{AndrzejMiernowski2008,
abstract = {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.},
author = {Andrzej Miernowski},
journal = {Annales UMCS, Mathematica},
keywords = {Convex curve; support function; curvature; convex curve},
language = {eng},
number = {1},
pages = {105-111},
title = {Parallelograms inscribed in a curve having a circle as π/2-isoptic},
url = {http://eudml.org/doc/268006},
volume = {62},
year = {2008},
}

TY - JOUR
AU - Andrzej Miernowski
TI - Parallelograms inscribed in a curve having a circle as π/2-isoptic
JO - Annales UMCS, Mathematica
PY - 2008
VL - 62
IS - 1
SP - 105
EP - 111
AB - 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.
LA - eng
KW - Convex curve; support function; curvature; convex curve
UR - http://eudml.org/doc/268006
ER -

References

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  1. Berger, M., Geometrie, Vol. 2, Nathan, Paris, 1990. 
  2. 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. Zbl0739.53001
  3. Cieślak, W., Miernowski, A. and Mozgawa, W., Isoptics of a strictly convex curve II, Rend. Sem. Mat. Univ. Padova 96 (1996), 37-49. Zbl0881.53003
  4. Connes, A., Zagier, D., A property of parallelograms inscribed in ellipses, Amer. Math. Monthly 114 (2007), 909-914. Zbl1140.51010
  5. Green, J. W., Sets subtending a constant angle on a circle, Duke Math. J. 17 (1950), 263-267. Zbl0039.18201
  6. 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. 
  7. Richard, J-M., Safe domain and elementary geometry, Eur. J. Phys. 25 (2004), 835-844.[Crossref] Zbl1162.70320
  8. Wunderlich, W., Kurven mit isoptischem Kreis, Aequationes Math. 6 (1971), 71-78. Zbl0215.50103

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