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立方体线段选取

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在单位立方体内部随机选择两点之间的平均距离（n=3 情况下的超立方体线段选取），有时被称为罗宾斯常数，是

Delta(3)=1/(105)[4+17sqrt(2)-6sqrt(3)+21ln(1+sqrt(2))+42ln(2+sqrt(3))-7pi]
(1)
=1/(105)[4+17sqrt(2)-6sqrt(3)+21sinh^(-1)1+42ln(2+sqrt(3))-7pi]
(2)
=0.66170...
(3)

(OEIS A073012; 罗宾斯 1978, 勒利奥内 1983, 贝克 2023)。

此值在 Wolfram 语言中实现为PolyhedronData["Cube", "MeanInteriorLineSegmentLength"].

CubeLinePickingDistribution

概率函数作为线段长度的函数，如上所示，由 Mathai 等人 (1999) 以（几乎）闭合形式找到。经过简化、更正错别字和完成积分，给出了闭合形式

P(l)={-l^2[(l-8)l^2+pi(6l-4)] for 0<=l<=1; 2l[(l^2-8sqrt(l^2-1)+3)l^2-4sqrt(l^2-1)+12l^2sec^(-1)l+pi(3-4l)-1/2] for 1<l<=sqrt(2); l[(1+l^2)(6pi+8sqrt(l^2-2)-5-l^2)-16lcsc^(-1)(sqrt(2-2l^(-2)))+16ltan^(-1)(lsqrt(l^2-2))-24(l^2+1)tan^(-1)(sqrt(l^2-2))] for sqrt(2)<l<=sqrt(3).
(4)

前几个偶数阶原点矩 mu_n^' 对于 n=0, 2, ... 是 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 和 A160694)。

在立方体上选取 n 个点，并尽可能将它们分隔开。下表给出了任意两点之间最小直线线距离的最佳已知值。

nd(n)
51.1180339887498
61.0606601482100
71
81
90.86602540378463
100.74999998333331
110.70961617562351
120.70710678118660
130.70710678118660
140.70710678118660
150.625

另请参阅

立方体线段选取--面与面, 立方体线段选取--面与内部, 立方体点选取, 立方体三角形选取, 差异定理, 超立方体线段选取, 线段线段选取, 点选取, 罗宾斯常数, 正方形线段选取

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参考文献

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006b.Beck, D. "Mean Distance in Polyhedra." 22 Sep 2023. https://arxiv.org/abs/2309.13177.Bolis, T. S. Solution to Problem E2629. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 277-278, 1978.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Finch, S. R. "Geometric Probability Constants." §8.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 479-484, 2003.Ghosh, B. "Random Distances within a Rectangle and between Two Rectangles." Bull. Calcutta Math. Soc. 43, 17-24, 1951.Holshouser, A. L.; King, L. R.; and Klein, B. G. Solution to Problem E3217, "Minimum Average Distance between Points in a Rectangle." Amer. Math. Monthly 96, 64-65, 1989.Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983.Mathai, A. M.; Moschopoulos, P.; and Pederzoli, G. "Distance between Random Points in a Cube." J. Statistica 59, 61-81, 1999.Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978.Santaló, L. A. Integral Geometry and Geometric Probability. Reading, MA: Addison-Wesley, 1976.Schroeppel, R. (results due to R. H. Hardin and N. J. A. Sloane) "points in a cube." [email protected] posting, May 30, 1996.Sloane, N. J. A. Sequences A073012, A160693, and A160694 in "The On-Line Encyclopedia of Integer Sequences."

在 Wolfram|Alpha 中被引用

立方体线段选取

请引用为

韦斯坦因，埃里克·W. "立方体线段选取。" 来自 MathWorld——Wolfram Web 资源。 https://mathworld.net.cn/CubeLinePicking.html

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