Mean line segment length

For a line segment of length d, the average distance between two points is 1/3d.[1]

For a triangle with side lengths a, b, and c, the average distance between two points in its interior is given by the formula[2]

${\displaystyle {\frac {4ss_{a}s_{b}s_{c}}{15}}\left[{\frac {1}{a^{3}}}\ln \left({\frac {s}{s_{a}}}\right)+{\frac {1}{b^{3}}}\ln \left({\frac {s}{s_{b}}}\right)+{\frac {1}{c^{3}}}\ln \left({\frac {s}{s_{c}}}\right)\right]+{\frac {a+b+c}{15}}+{\frac {(b+c)(b-c)^{2}}{30a^{2}}}+{\frac {(a+c)(a-c)^{2}}{30b^{2}}}+{\frac {(a+b)(a-b)^{2}}{30c^{2}}},}$

where ${\displaystyle s=(a+b+c)/2}$ is the semiperimeter, and ${\displaystyle s_{i}}$ denotes ${\displaystyle s-i}$.

For an equilateral triangle with side length a, this is equal to

${\displaystyle \left({\frac {4+3\ln 3}{20}}\right)a\approx 0.364791843\ldots a.}$

The average distance between two points inside a square with side length s is[3]

${\displaystyle \left({\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{15}}\right)s\approx 0.521405433\ldots s.}$

More generally, the mean line segment length of a rectangle with side lengths l and w is[1]

${\displaystyle {\frac {1}{15}}\left[{\frac {l^{3}}{w^{2}}}+{\frac {w^{3}}{l^{2}}}+d\left(3-{\frac {l^{2}}{w^{2}}}-{\frac {w^{2}}{l^{2}}}\right)+{\frac {5}{2}}\left({\frac {w^{2}}{l}}\ln \left({\frac {l+d}{w}}\right)+{\frac {l^{2}}{w}}\ln \left({\frac {w+d}{l}}\right)\right)\right]}$

where ${\displaystyle d={\sqrt {l^{2}+w^{2}}}}$ is the length of the rectangle's diagonal.

If the two points are instead chosen to be on different sides of the square, the average distance is given by[3][4]

${\displaystyle \left({\frac {2+{\sqrt {2}}+5\ln(1+{\sqrt {2}})}{9}}\right)s\approx 0.869009\ldots s.}$

The average distance between points inside an n-dimensional unit hypercube is denoted as Δ(n), and is given as[5]

${\displaystyle \Delta (n)=\underbrace {\int _{0}^{1}\cdots \int _{0}^{1}} _{2n}{\sqrt {(x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}+\cdots +(x_{n}-y_{n})^{2}}}\,dx_{1}\cdots \,dx_{n}\,dy_{1}\cdots \,dy_{n}}$

The first two values, Δ(1) and Δ(2), refer to the unit line segment and unit square respectively.

For the three-dimensional case, the mean line segment length of a unit cube is also known as Robbins constant, named after David P. Robbins. This constant has a closed form,[6]

${\displaystyle \Delta (3)={\frac {4+17{\sqrt {2}}-6{\sqrt {3}}-7\pi }{105}}+{\frac {\ln(1+{\sqrt {2}})}{5}}+{\frac {2\ln(2+{\sqrt {3}})}{5}}.}$

Its numerical value is approximately 0.661707182... (sequence A073012 in the OEIS)

Andersson et. al. (1976) showed that Δ(n) satisfies the bounds[7]

${\displaystyle {\tfrac {1}{3}}n^{1/2}\leq \Delta (n)\leq ({\tfrac {1}{6}}n)^{1/2}{\sqrt {{\frac {1}{3}}\left[1+2\left(1-{\frac {3}{5n}}\right)^{1/2}\right]}}.}$

Choosing points from two different faces of the unit cube also gives a result with a closed form, given by,[4]

${\displaystyle {\frac {4+17{\sqrt {2}}-6{\sqrt {3}}-7\pi }{75}}+{\frac {7\ln {(1+{\sqrt {2}})}}{25}}+{\frac {14\ln {(2+{\sqrt {3}})}}{25}}.}$

The average chord length between points on the circumference of a circle of radius r is[8]

${\displaystyle {\frac {4}{\pi }}r\approx 1.273239544\ldots r}$

And picking points on the surface of a sphere with radius r is [9]

${\displaystyle {\frac {4}{3}}r}$

The average distance between points inside a disk of radius r is[10]

${\displaystyle {\frac {128}{45\pi }}r\approx 0.905414787\ldots r.}$

The values for a half disk and quarter disk are also known.[11]

For a half disk of radius 1:

${\displaystyle {\frac {64}{135}}{\frac {12\pi -23}{\pi ^{2}}}\approx 0.706053409\ldots }$

For a quarter disk of radius 1:

${\displaystyle {\frac {32}{135\pi ^{2}}}(6\ln {(2{\sqrt {2}}-2)}-94{\sqrt {2}}+48\pi +3)\approx 0.473877262\ldots }$

For a three-dimensional ball, this is

${\displaystyle {\frac {36}{35}}r\approx 1.028571428\ldots r.}$

More generally, the mean line segment length of an n-ball is[1]

${\displaystyle {\frac {2n}{2n+1}}\beta _{n}r}$

where β_n depends on the parity of n,

${\displaystyle \beta _{n}={\begin{cases}{\dfrac {2^{3n+1}\,(n/2)!^{2}\,n!}{(n+1)\,(2n)!\,\pi }}&({\text{for even }}n)\\{\dfrac {2^{n+1}\,n!^{3}}{(n+1)\,((n-1)/2)!^{2}\,(2n)!}}&({\text{for odd }}n)\end{cases}}}$