OFFSET
0,1
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.
LINKS
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).
EXAMPLE
0.41762783583513893254998577691877778554776832295757...
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]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 22 2021
STATUS
approved