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Reconstruction of Three Dimensional Convex Bodies from the Curvatures of Their Shadows
Russian-Armenian (Slavonic) University, Yerevan, Armenia
Institute of Mathematics Armenian Academy of Sciences, Yerevan, Armenia
- 1 Russian-Armenian (Slavonic) University, Yerevan, Armenia
- 2 Institute of Mathematics Armenian Academy of Sciences, Yerevan, Armenia
American Journal of Computational Mathematics·Volume 05 (2015)·Pages 86–95·Published 13 May 2015·DOI10.4236/ajcm.2015.52007
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Abstract
In this article, we study necessary and sufficient conditions for a function, defined on the space of flags to be the projection curvature radius function for a convex body. This type of inverse problems has been studied by Christoffel, Minkwoski for the case of mean and Gauss curvatures. We suggest an algorithm of reconstruction of a convex body from its projection curvature radius function by finding a representation for the support function of the body. We lead the problem to a system of differential equations of second order on the sphere and solve it applying a consistency method suggested by the author of the article.
KeywordsIntegral GeometryConvex BodyProjection CurvatureSupport Function
- Minkowski, H. (1911) Theorie der konvexen Korper, insbesondere Begrundung ihresb Oberflachenbergriffs. Ges. Abh., 2, Leipzig, Teubner, 131-229.
- Blaschke, W. (1923) Vorlesungen uber Differentialgeometrie. II. Affine Differentialgeometrie, Springer-Verlag, Berlin.
- Pogorelov, A.V. (1969) Exterior Geometry of Convex Surfaces [in Russian]. Nauka, Moscow.
- Alexandrov, A.D. (1956) Uniqueness Theorems for Surfaces in the Large [in Russian]. Vesti Leningrad State University, 19, 25-40.
- Bakelman, I.Ya., Verner, A.L. and Kantor, B.E. (1973) Differential Geometry in the Large [in Russian]. Nauka, Moskow.
- Firey, W.J. (1970) Intermediate Christoffel-Minkowski Problems for Figures of Revolution. Israel Journal of Mathematics, 8, 384-390. http://dx.doi.org/10.1007/BF02798684
- Berg, C. (1969) Corps convexes et potentiels spheriques. Matematisk-fysiske Meddelelser Udgivet af. Det Kongelige Danske Videnskabernes Selska, 37, 64.
- Wiel, W. and Schneider, R. (1983) Zonoids and Related Topics. In: Gruber, P. and Wills, J., Eds., Convexity and Its Applications, Birkhauser, Basel, 296-317.
- Gardner R.J. and Milanfar, P. (2003) Reconstruction of Convex Bodies from Brightness Functions. Discrete & Computational Geometry, 29, 279-303. http://dx.doi.org/10.1007/s00454-002-0759-2
- Ryabogin, D. and Zvavich, A. (2004) Reconstruction of Convex Bodies of Revolution from the Areas of Their Shadows. Archiv der Mathematik, 5, 450-460. http://dx.doi.org/10.1007/s00454-002-0759-2
- Leichtweiz, K. (1980) Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin. http://dx.doi.org/10.1007/978-3-642-95335-4
- Ambartzumian, R.V. (1990) Factorization Calculus and Geometrical Probability. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9781139086561
- Aramyan, R.H. (2001) An Approach to Generalized Funk Equations I [in Russian]. Izvestiya Akademii Nauk Armenii. Matematika [English Translation: Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)], 36, 47-58.
- Aramyan, R.H. (2010) Generalized Radon Transform on the Sphere. Analysis International Mathematical Journal of Analysis and Its Applications, 30, 271-284.
- Aramyan, R.H. (2010) Solution of an Integral Equation by Consistency Method. Lithuanian Mathematical Journal, 50, 133-139.