A133741 Decimal expansion of offset at which two unit disks overlap by half each's area.
8, 0, 7, 9, 4, 5, 5, 0, 6, 5, 9, 9, 0, 3, 4, 4, 1, 8, 6, 3, 7, 9, 2, 3, 4, 8, 0, 1, 3, 2, 6, 3, 0, 8, 8, 5, 8, 0, 4, 4, 7, 1, 9, 2, 9, 1, 4, 8, 1, 9, 6, 8, 4, 4, 5, 0, 0, 1, 9, 5, 2, 0, 3, 4, 6, 7, 7, 4, 1, 0, 9, 9, 9, 4, 2, 5, 9, 0, 7, 0, 7, 0, 0, 2, 4, 8, 6, 7, 8, 0, 3, 3, 0, 4, 4, 5, 4, 5, 7, 4, 1, 8, 9, 8, 2
Offset: 0
Examples
0.8079455065990344186379234801326308858044719291481968445...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..10000
- Max Chicky Fang, Closed Form for Half-Area Overlap Offset of 2 Unit Disks, arXiv:2403.10529 [math.GM], 2024.
- Eric Weisstein's World of Mathematics, Circle-Circle Intersection
Programs
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Mathematica
d0 = d /. FindRoot[ 2*ArcCos[d/2] - d/2*Sqrt[4 - d^2] == Pi/2, {d, 1}, WorkingPrecision -> 110]; RealDigits[d0][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 26 2012, after Eric W. Weisstein *) -
PARI
default(realprecision, 100); solve(x=0,1, 2*acos(x/2) - (x/2)*sqrt(4-x^2) - Pi/2) \\ G. C. Greubel, Nov 16 2018
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PARI
d=solve(x=0,1,cos(x)-x);sqrt(2-2*sqrt(1-d^2)) \\ Gleb Koloskov, Feb 27 2021
Formula
Equals sqrt(1+A003957) - sqrt(1-A003957) = sqrt(2-2*sqrt(1-A003957^2)) = 2*A086751. - Gleb Koloskov, Feb 26 2021