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

El Khaldi Khaldoun () and Saleeby Elias G. ()
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El Khaldi Khaldoun: Department of Computer Science, Notre Dame University-Louaize, Zouk Mosbeh, Lebanon
Saleeby Elias G.: Mount Lebanon, Lebanon

Monte Carlo Methods and Applications, 2020, vol. 26, issue 4, 315-323

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.

Keywords: Density of lines; randomness model; Monte Carlo method; geometric size measures (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1515/mcma-2020-2071

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