A336083 Decimal expansion of the arclength on the unit circle such that the corresponding chord separates the interior into segments having 3 = ratio of segment areas; see Comments.

2, 3, 0, 9, 8, 8, 1, 4, 6, 0, 0, 1, 0, 0, 5, 7, 2, 6, 0, 8, 8, 6, 6, 3, 3, 7, 7, 9, 3, 1, 3, 6, 2, 4, 8, 4, 6, 1, 1, 1, 9, 9, 6, 4, 5, 8, 5, 8, 8, 3, 1, 0, 3, 7, 5, 4, 5, 3, 1, 5, 2, 9, 3, 1, 9, 2, 7, 1, 9, 2, 8, 5, 8, 0, 2, 6, 6, 5, 2, 0, 9, 3, 9, 1, 3, 3

Offset: 1

Author

Clark Kimberling, Jul 11 2020

Keywords

• nonn
• cons

Comments

Suppose that s in (0,Pi) is the length of an arc of the unit circle. The associated chord separates the interior into two segments. Let A1 be the area of the larger and A2 the area of the smaller. The term "ratio of segment areas" means A1/A2. See A336073 for a guide to related sequences.

Equals the median of the probability distribution function of angles of random rotations in 3D space uniformly distributed with respect to the Haar measure, i.e., the solution x to Integral_{t=0..x} ((1 - cos(t))/Pi) dt = 1/2 (see Reynolds, 2017; cf. A086118, A361605). - Amiram Eldar, Mar 17 2023

Examples

			arclength = 2.3098814600100572608866337793136248461119964...
		

Links

• Marc B. Reynolds, Volume element of SO(3) and average uniform random rotation angle, 2017.

Crossrefs

Cf. A336073, A003957, A019669.

Cf. A086118, A361605.

Programs

Mathematica

k = 3; s = s /. FindRoot[(2 Pi - s + Sin[s])/(s - Sin[s]) == k, {s, 2}, WorkingPrecision -> 200]
RealDigits[s][[1]]

PARI

d=solve(x=0,1,cos(x)-x); d+Pi/2 \\ Gleb Koloskov, Feb 21 2021

Formula

Equals d+Pi/2 = A003957 + A019669, where d is the Dottie number. - Gleb Koloskov, Feb 21 2021