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On the density of lines and Santalo’s formula for computing geometric size measures

Author

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  • El Khaldi Khaldoun

    (Department of Computer Science, Notre Dame University-Louaize, Zouk Mosbeh, Lebanon)

  • Saleeby Elias G.

    (Mount Lebanon, Lebanon)

Abstract

Methods from integral geometry and geometric probability allow us to estimate geometric size measures indirectly. In this article, a Monte Carlo algorithm for simultaneous estimation of hyper-volumes and hyper-surface areas of a class of compact sets in Euclidean space is developed. The algorithm is based on Santalo’s formula and the Hadwiger formula from integral geometry, and employs a comparison principle to assign geometric probabilities. An essential component of the method is to be able to generate uniform sets of random lines on the sphere. We utilize an empirically established method to generate these random chords, and we describe a geometric randomness model associated with it. We verify our results by computing measures for hyper-ellipsoids and certain non-convex sets.

Suggested Citation

  • El Khaldi Khaldoun & Saleeby Elias G., 2020. "On the density of lines and Santalo’s formula for computing geometric size measures," Monte Carlo Methods and Applications, De Gruyter, vol. 26(4), pages 315-323, December.
    • Handle: RePEc:bpj:mcmeap:v:26:y:2020:i:4:p:315-323:n:2
    DOI: 10.1515/mcma-2020-2071
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    References listed on IDEAS

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    1. El Khaldi Khaldoun & Saleeby Elias G., 2017. "On the tangent model for the density of lines and a Monte Carlo method for computing hypersurface area," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 13-20, March.
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