A361605 Decimal expansion of the standard deviation of the probability distribution function of angles of random rotations in 3D space uniformly distributed with respect to Haar measure (in radians).
6, 4, 5, 8, 9, 6, 5, 0, 7, 8, 5, 1, 4, 9, 9, 4, 8, 2, 3, 5, 8, 7, 4, 1, 3, 8, 4, 2, 6, 5, 5, 2, 7, 1, 6, 2, 1, 6, 7, 5, 0, 3, 2, 6, 3, 0, 6, 1, 1, 1, 1, 7, 0, 2, 7, 3, 2, 9, 1, 2, 0, 4, 9, 9, 3, 8, 5, 5, 1, 4, 6, 1, 9, 3, 6, 7, 7, 7, 5, 7, 2, 1, 7, 1, 5, 2, 5, 9, 5, 1, 1, 4, 9, 1, 6, 6, 3, 5, 0, 5, 2, 1, 0, 8, 0
Offset: 0
Examples
0.64589650785149948235874138426552716216750326306111...
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.
Programs
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Mathematica
RealDigits[Sqrt[(Pi^4 - 48)/3]/(2*Pi), 10, 100][[1]]
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PARI
sqrt((Pi^4 - 48)/3)/(2*Pi)
Formula
Equals sqrt( - ^2), where = 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).
Comments