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四面体 四面体 拾取


TetrahedronTetrahedronPickingDistribution

在一个单位体积的四面体内随机选择顶点的四面体平均四面体体积由下式给出

V^_=(13)/(720)-(pi^2)/(15015)
(1)
=0.017398...
(2)

(OEIS A093525; Buchta 和 Reitzner 1992; Mannion 1994; Schneider 1997, p. 170; Buchta 和 Reitzner 2001; Zinani 2003)。

这提供了对该问题的解决方案是有理数的猜想(Croft等人1991,第 54 页曾提出 1/57)的反驳,并使 Solomon 的声明“显然尚未成功计算出四面体、平行六面体等非球形区域中随机点的显式值”(Solomon 1978,第 124 页)过时。

此外,Buchta 和 Reitzner (2001) 给出了在三维单纯形中随机选择的 n 个点的凸包的期望体积的显式公式,适用于任意 n


另请参阅

球体 四面体 拾取, 立方体 四面体 拾取, 八面体 四面体 拾取, 单纯形 单纯形 拾取, 球体 四面体 拾取, 四面体 线 拾取, 四面体 三角形 拾取

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参考文献

Blaschke, W. "Über affine Geometrie XI: Lösung des 'Vierpunktproblems' von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten." Ber. Verh. Sachs. Akad. Wiss. Leipzig Math.-Phys. Kl. 69, 436-453, 1917.Buchta, C. and Reitzner, M. "What Is the Expected Volume of a Tetrahedron whose Vertices are Chosen at Random from a Given Tetrahedron." Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 129, 63-68, 1992.Buchta, C. and Reitzner, M. "The Convex Hull of Random Points in a Tetrahedron: Solution of Blaschke's Problem and More General Results." J. reine angew. Math. 536, 1-29, 2001.Croft, H. T.; Falconer, K. J.; and Guy, R. K. "Random Polygons and Polyhedra." §B5 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 54-57, 1991.Do, K.-A. and Solomon, H. "A Simulation Study of Sylvester's Problem in Three Dimensions." J. Appl. Prob. 23, 509-513, 1986.Klee, V. "What is the Expected Volume of a Simplex Whose Vertices are Chosen at Random from a Given Convex Body." Amer. Math. Monthly 76, 286-288, 1969.Mannion, D. "The Volume of a Tetrahedron Whose Vertices Are Chosen at Random in the Interior of a Parent Tetrahedron." Adv. Appl. Prob. 26, 577-596, 1994.Schneider, R. "Discrete Aspects of Stochastic Geometry." Ch. 9 in Handbook of Discrete and Computational Geometry (Ed. J. E. Goodman and J. O'Rourke). Boca Raton, FL: CRC Press, pp. 167-184, 1997.Sloane, N. J. A. Sequence A093525 in "The On-Line Encyclopedia of Integer Sequences."Solomon, H. Geometric Probability. Philadelphia, PA: SIAM, p. 124, 1978.Zinani, A. "The Expected Volume of a Tetrahedron Whose Vertices are Chosen at Random in the Interior of a Cube." Monatshefte Math. 139, 341-348, 2003.

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四面体 四面体 拾取

引用为

Weisstein, Eric W. "Tetrahedron Tetrahedron Picking." 来自 Web 资源。 https://mathworld.net.cn/TetrahedronTetrahedronPicking.html

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