OFFSET
0,1
COMMENTS
The corresponding value in degrees is 37.0071439021...
LINKS
Math Stackexchange, Mean value of the rotation angle is 126.5 degrees, 2013.
Marc B. Reynolds, Volume element of SO(3) and average uniform random rotation angle, 2017.
Hansklaus Rummler, On the distribution of rotation angles how great is the mean rotation angle of a random rotation?, The Mathematical Intelligencer, Vol. 24, No. 4 (2002), pp. 6-11; alternative link.
Eugene Salamin, Application of quaternions to computations with rotations, Working Paper, Stanford AI Lab, 1979.
FORMULA
Equals sqrt(<t^2> - <t>^2), where <t^k> = Integral_{t=0..Pi} t^k * P(t) dt, and P(t) = (1 - cos(t))/Pi is the probability distribution function of the angles in radians.
Equals sqrt((Pi^4 - 48)/3)/(2*Pi).
EXAMPLE
0.64589650785149948235874138426552716216750326306111...
MATHEMATICA
RealDigits[Sqrt[(Pi^4 - 48)/3]/(2*Pi), 10, 100][[1]]
PROG
(PARI) sqrt((Pi^4 - 48)/3)/(2*Pi)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Mar 17 2023
STATUS
approved