OFFSET
0,1
COMMENTS
Consider two circular frames each of diameter D and with a separation of d.
If d/D < r = 0.527697..., then a catenoid gives the absolute minimum area.
If r < d/D < L = 0.66274... (Laplace limit), there are 3 minimal surfaces of revolution passing through the frames: 2 catenoids and the so-called Goldschmidt discontinuous solution consisting of the 2 disks.
If d/D > L, there remains only the Goldschmidt solution.
LINKS
Robert Ferréol's MathCurve, Catenoid
Eric Weisstein's MathWorld, Laplace Limit
Eric Weisstein's MathWorld, Minimal Surface of Revolution
FORMULA
arccosh(u)/u, where u = 1.21136... is solution to u*sqrt(u^2-1) + arccosh(u) - u^2 = 0.
Solution of 2*cosh((x^2+1)/2) = x+1/x. - Robert FERREOL, Feb 07 2019
EXAMPLE
0.5276973969625715285724233433631805779688537906314195417222751595...
MATHEMATICA
digits = 105; u0 = u /. FindRoot[u*Sqrt[u^2-1] + ArcCosh[u] - u^2 == 0, {u, 6/5}, WorkingPrecision -> digits+5]; r = ArcCosh[u0]/u0; RealDigits[r, 10, digits] // First
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Apr 21 2015
STATUS
approved