“Cube Point Picking”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cube Point Picking

释义

Cube Point Picking

N.B. A detailed on-line essay by S. Finchwas the starting point for this entry.

Let two points be picked randomly from a unit -D Hypercube. The expected distance between the points is then

The function

satisfies

(1)

(Anderssen et al. 1976).

Pick points , ..., randomly in a unit -cube. Let be the Convex Hull, so

(2)

Let

be the expected

-D Volume (the Content) of

be the expected

-D SurfaceArea of

, and

the expected number of Vertices on the Polygonalboundary of

. Then

(3)

(4)

and

(5)

(Rényi and Sulanke 1963, 1964). The average Distance between two points chosen at random inside a unit cube is

(6)

(Robbins 1978, Le Lionnais 1983).

Pick points on a Cube, and space them as far apart as possible. The best value known for the minimum straight Line distance between any two points is given in the following table.


5	1.1180339887498
6	1.0606601482100
7	1
8	1
9	0.86602540378463
10	0.74999998333331
11	0.70961617562351
12	0.70710678118660
13	0.70710678118660
14	0.70710678118660
15	0.625

See also Cube Triangle Picking, Discrepancy Theorem, Point Picking

References

Anderssen, R. S.; Brent, R. P.; Daley, D. J.; and Moran, A. P. ``Concerning and a Taylor Series Method.'' SIAM J. Appl. Math. 30, 22-30, 1976.

Bolis, T. S. Solution to Problem E2629. ``Average Distance Between Two Points in a Box.'' Amer. Math. Monthly 85, 277-278, 1978.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/geom/geom.html

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.

Rényi, A. and Sulanke, R. ``Über die konvexe Hülle von zufällig gewählten Punkten, I.'' Z. Wahrscheinlichkeits 2, 75-84, 1963.

Rényi, A. and Sulanke, R. ``Über die konvexe Hülle von zufällig gewählten Punkten, II.'' Z. Wahrscheinlichkeits 3, 138-147, 1964.

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.

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