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TetrahedronTetrahedronPickingDistribution

在一個單位體積的四面體內隨機選擇頂點的四面體平均四面體體積由下式給出

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)。

這提供了對該問題的解決方案是有理數的猜想(Croft等人1991,第 54 頁曾提出 1/57)的反駁,並使 Solomon 的宣告“顯然尚未成功計算出四面體、平行六面體等非球形區域中隨機點的顯式值”(Solomon 1978,第 124 頁)過時。

此外,Buchta 和 Reitzner (2001) 給出了在三維單純形中隨機選擇的 n 個點的凸包的期望體積的顯式公式,適用於任意 n


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球體 四面體 拾取, 立方體 四面體 拾取, 八面體 四面體 拾取, 單純形 單純形 拾取, 球體 四面體 拾取, 四面體 線 拾取, 四面體 三角形 拾取

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參考文獻

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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四面體 四面體 拾取

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Weisstein, Eric W. "Tetrahedron Tetrahedron Picking." 來自 Web 資源。 https://mathworld.tw/TetrahedronTetrahedronPicking.html

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