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)/4) * Integral_{x=0..1} sqrt(1 - x^2)/E(x)^2 dx.
EXAMPLE
0.933174653498462644...
MATHEMATICA
First[RealDigits[Pi^2/4 * NIntegrate[Sqrt[1-x^2]/EllipticE[x^2]^2, {x, 0, 1}, WorkingPrecision -> 100]]] (* Stefano Spezia, Jun 24 2023 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Tian Vlasic, Jun 24 2023
EXTENSIONS
More terms from Stefano Spezia, Jun 24 2023
STATUS
approved