TOPICS
Search

Cube Point Picking


Cube point picking is the three-dimensional case of hypercube point picking.

The average distance from a point picked at random inside a unit cube to the center is given by

d^__(center)=int_(-1/2)^(1/2)int_(-1/2)^(1/2)int_(-1/2)^(1/2)sqrt(x^2+y^2+z^2)dxdydz
(1)
=8int_0^(1/2)int_0^(1/2)int_0^(1/2)sqrt(x^2+y^2+z^2)dxdydz
(2)
=1/8sqrt(3)-1/(48)pi-1/4ln2+1/2ln(1+sqrt(3))
(3)
=0.48029597....
(4)

Similarly, the average distance from a point picked at random to a fixed corner is given by

d^__(corner)=B_3(1)
(5)
=2d^__(center)
(6)
=int_0^1int_0^1int_0^1sqrt(x^2+y^2+z^2)dxdydz
(7)
=1/4sqrt(3)-1/(24)pi-1/2ln2+ln(1+sqrt(3))
(8)
=0.9605919...,
(9)

where B_n(s) is the B-box integral.

The average distance from the center of a unit cube to a given face is

d^_=int_(-1/2)^(1/2)int_(-1/2)^(1/2)sqrt(x^2+y^2+1/4)dxdy
(10)
=8int_0^(1/2)int_0^ysqrt(x^2+y^2+1/4)dxdy
(11)
=1/3ln(2+sqrt(3))+1/6sqrt(3)-1/(36)pi
(12)
=0.640394...
(13)

(OEIS A097047).


See also

Ball Point Picking, Cube Line Picking, Hypercube Point Picking, Sphere Point Picking, Square Point Picking, Unit Cube, Unit Square Integral

Explore with Wolfram|Alpha

References

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Box Integrals." J. Comput. Appl. Math. 206, 196-208, 2007.Sloane, N. J. A. Sequence A097047 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Cube Point Picking

Cite this as:

Weisstein, Eric W. "Cube Point Picking." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CubePointPicking.html

Subject classifications