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

Author

Amiram Eldar, Oct 22 2021

Keywords

• nonn
• cons

Comments

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.

Examples

			0.41762783583513893254998577691877778554776832295757...
		

Links

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

Crossrefs

Cf. A255899.

Programs

Mathematica

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

Formula

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).