主題
Search

立方體線段選取

DOWNLOAD Mathematica Notebook下載 Wolfram 筆記本

在單位立方體內部隨機選擇兩點之間的平均距離（n=3 情況下的超立方體線段選取），有時被稱為羅賓斯常數，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; 羅賓斯 1978, 勒利奧內 1983, 貝克 2023)。

此值在 Wolfram 語言中實現為PolyhedronData["Cube", "MeanInteriorLineSegmentLength"].

CubeLinePickingDistribution

機率函式作為線段長度的函式，如上所示，由 Mathai 等人 (1999) 以（幾乎）閉合形式找到。經過簡化、更正錯別字和完成積分，給出了閉合形式

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)

前幾個偶數階原點矩 mu_n^' 對於 n=0, 2, ... 是 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 和 A160694)。

在立方體上選取 n 個點，並儘可能將它們分隔開。下表給出了任意兩點之間最小直線線距離的最佳已知值。

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

另請參閱

立方體線段選取--面與面, 立方體線段選取--面與內部, 立方體點選取, 立方體三角形選取, 差異定理, 超立方體線段選取, 線段線段選取, 點選取, 羅賓斯常數, 正方形線段選取

使用 探索

參考文獻

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." math-fun@cs.arizona.edu posting, May 30, 1996.Sloane, N. J. A. Sequences A073012, A160693, and A160694 in "The On-Line Encyclopedia of Integer Sequences."

在 中被引用

立方體線段選取

請引用為

韋斯坦因，埃裡克·W. "立方體線段選取。" 來自 —— 資源。 https://mathworld.tw/CubeLinePicking.html

主題分類