OFFSET
0,1
COMMENTS
The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.
The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.
LINKS
Eric Weisstein's World of Mathematics, Isoperimetric Quotient
Wikipedia, Elliptic integral
FORMULA
Equals ((Pi^2)/2) * exp(-1-2*Integral_{x=0..1} log(E(x)) dx).
EXAMPLE
0.916816923382168248...
MATHEMATICA
First[RealDigits[Pi^2/2*Exp[-1 - 2*NIntegrate[Log[EllipticE[x^2]], {x, 0, 1}, WorkingPrecision -> 100]]]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Tian Vlasic, Jun 25 2023
STATUS
approved