Advertisement

Discrete & Computational Geometry

, Volume 62, Issue 4, pp 856–864 | Cite as

On Recognizing Shapes of Polytopes from Their Shadows

  • Sergii MyroshnychenkoEmail author
Article
  • 37 Downloads

Abstract

Let P and Q be two convex polytopes both contained in the interior of a Euclidean ball \(r\mathbf B ^{d}\). We prove that \(P=Q\) provided that their sight cones from any point on the sphere \(rS^{d-1}\) are congruent. We also prove an analogous result for spherical projections.

Keywords

Convex geometry Geometric tomography Projections of convex bodies Shadow picture Isoptic characterization 

Notes

Acknowledgements

The author is very grateful the anonymous referees for their suggestions that greatly improved the manuscript; and also to Vlad Yaskin and Dmitry Ryabogin for many fruitful and interesting discussions.

References

  1. 1.
    Bianchi, G., Gruber, P.M.: Characterizations of ellipsoids. Arch. Math. (Basel) 49(4), 344–350 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gardner, R.J.: Geometric Tomography. 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 58. Cambridge University Press, Cambridge (2006)Google Scholar
  3. 3.
    Green, J.W.: Sets subtending a constant angle on a circle. Duke Math. J. 17, 263–267 (1950)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kincses, J., Kurusa, Á.: Can you recognize the shape of a figure from its shadows? Beiträge Algebra Geom. 36(1), 25–35 (1995)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Klamkin, M.S.: Conjectured isoptic characterization of a circle. Amer. Math. Monthly 95(9), 845 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kleinschmidt, P., Pachner, U.: Shadow-boundaries and cuts of convex polytopes. Mathematika 27(1), 58–63 (1980)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kurusa, Á., Ódor, T.: Isoptic characterization of spheres. J. Geom 106(3), 63–73 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
  9. 9.
    Matsuura, S.: A problem in solid geometry. J. Math. Osaka City Univ. 12(1–2), 89–95 (1961)MathSciNetGoogle Scholar
  10. 10.
    Myroshnychenko, S., Ryabogin, D.: On polytopes with congruent projections or sections. Adv. Math. 325, 482–504 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Nitsche, J.C.C.: Isoptic characterization of a circle (proof of a conjecture of M.S. Klamkin). Amer. Math. Monthly 97(1), 45–47 (1990)Google Scholar
  12. 12.
    Rudin, W.: Functional Analysis, 2nd edn. International Series in Pure and Applied Mathematics. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  13. 13.
    Zhang, N.: On bodies with congruent sections or projections. J. Differential Equations 265(5), 2064–2075 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations