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

0,1

Comments

The corresponding value in degrees is 37.0071439021...

Links

Table of n, a(n) for n=0..104.

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

Cf. A086118 (mean), A336083 (median).

Sequence in context: A372995 A199385 A177159 * A356479 A317866 A309977

Adjacent sequences: A361602 A361603 A361604 * A361606 A361607 A361608

Keyword

nonn,cons

Author

Amiram Eldar, Mar 17 2023

Status

approved