A255899 -id:A255899 - cp's OEIS frontend

A348564 Decimal expansion of the distance between the centers of two unit-radius circles such that the position of centroid of each of the two lunes created by their intersection is on its boundary.

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

Offset: 0

Amiram Eldar, Oct 22 2021

A lune is the crescent-shaped region bounded by two circular arcs. Since the shape is concave, its centroid may lie either inside the shape, outside it, or on its boundary, depending on the radii of the arcs and the distance between them.

Let alpha be the angle between two segments, the first is connecting the center of one of the circles to one of the lune's vertices, and the second is connecting the centers of the two circles (see the illustration in the links section). Then, this constant is equal to 2*cos(alpha), where alpha = 1.360434... radians or 77.947146... degrees.

			0.41762783583513893254998577691877778554776832295757...

Amiram Eldar, Illustration.
Wikipedia, Centroid.
Wikipedia, Lune (geometry).

Cf. A255899.

RealDigits[2*Cos[x] /. FindRoot[Pi*(2 + 1/(Cos[x] - 1)) + Sin[2*x] - 2*x == 0, {x, 1}, WorkingPrecision -> 110], 10, 100][[1]]

Equals 2*cos(x) where x is the smaller of the two positive roots of the equation Pi * (2 + 1/(cos(x)-1)) + sin(2*x) - 2*x = 0 (the larger root is Pi/2).

A336199 Decimal expansion of the distance between the centers of two unit-radius spheres such that the volume of intersection is equal to the sum of volumes of the two nonoverlapping parts.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jul 11 2020

Solution to the three-dimensional version of Mrs. Miniver's problem.

The intersection volume is equal to 2/3 of the volume of each sphere, i.e., 8*Pi/9.

			0.452147427578415981828610831183181263247509153259677...

Eric Weisstein's World of Mathematics, Sphere-Sphere Intersection.
Wikipedia, Mrs. Miniver's problem.

Cf. A019673, A019699, A133749, A156546, A255899.

RealDigits[4 * Sin[ArcCos[-1/3]/3 - Pi/6], 10, 100][[1]]

Equals 4 * sin(arccos(-1/3)/3 - Pi/6).

The smaller of the two positive roots of the equation x^3 - 12*x + 16/3 = 0.

cp's OEIS Frontend

A348564 Decimal expansion of the distance between the centers of two unit-radius circles such that the position of centroid of each of the two lunes created by their intersection is on its boundary.

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