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F. Piefke, “Beziehungen zwischen der Sehnenlängenverteilung und der Verteilung des Abstandes zweier zufälliger Punkte im Eikörper”, Probability Theory and Related Fields , 43(2): 129–134, 1978
1978
Earlier work this paper cites.
Y. Zhuang and J. Pan, “Random distances associated with rhombuses”, ArXiv Technical Report, arXiv: 1106.1257, 2011
2011
Earlier work this paper cites.

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Y. Zhuang and J. Pan, “Random distances associated with hexagons”, ArXiv Technical Report, arXiv: 1106.2200, 2011
2011
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U. Bäsel, “Random chords and point distances in regular polygons”, ArXiv Technical Report, arXiv: 1204.2707, 2012
2012
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2012

Random Distances Associated With Equilateral Triangles

Zhuang, Yanyan, Pan, Jianping

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In this report, the explicit probability density functions of the random Euclidean distances associated with equilateral triangles are given, when the two endpoints of a link are randomly distributed in 1) the same triangle, 2) two adjacent triangles sharing a side, 3) two parallel triangles sharing a vertex, and 4) two diagonal triangles sharing a vertex, respectively.

The density function for 1) is based on the most recent work by Basel [1].
2)-4) are based on 1) and our previous work in [2], [3].
Simulation results show the accuracy of the obtained closed-form distance distribution functions, which are important in the theory of geometrical probability.