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} (E(x)^2)/sqrt(1 - x^2) dx).
EXAMPLE
0.87892065082960412...
MATHEMATICA
First[RealDigits[Pi^2/(4 * NIntegrate[EllipticE[x^2]^2/Sqrt[1 - x^2], {x, 0, 1}, WorkingPrecision -> 100])]]
PROG
(PARI) Pi^2/(4*intnum(x=0, 1, (ellE(x)^2)/sqrt(1 - x^2))) \\ Hugo Pfoertner, Jun 25 2023
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Tian Vlasic, Jun 25 2023
STATUS
approved