OFFSET
0,4
COMMENTS
This difference far exceeds the difference between the parabola and the quarter arc of a circle.
Catenary - Shape of a cable in its natural state between two points.
Funicular - The shape of a cable under a load or loads between two points.
Parabola - The shape of a cable between two points under uniform load.
Arc - The shape of a cable between two points being at a constant distant from a third point.
The area under the Parabola minus the area under the Catenary.
LINKS
David Griffin, Catenaries, Parabolas and Suspension Bridges
Math Forum, Ask Dr. Math, Catenary and Parabola Comparison
Wikipedia, Catenary
Wikipedia, Parabola
FORMULA
Equals 2/(e-1)^2 - 2/3. - Jean-François Alcover, Feb 18 2014
EXAMPLE
0.010727108010265122454156368548309422002905984483643365507680033...
MAPLE
evalf(2/(exp(1)-1)^2-2/3, 150); # Alois P. Heinz, Jul 16 2021
MATHEMATICA
(* focus at y=0 *) NIntegrate[x^2 - (Cosh[x] - 1)/(Cosh[1] - 1), {x, 0, 1}, WorkingPrecision -> 2^7, MinRecursion -> 2^10, MaxRecursion -> 2^12]
(* to view the three curves, parabola, catenary and a ¼ arc of a circle *) Plot[{(Cosh[x] - 1)/(Cosh[1] - 1), x^2, -Sqrt[-x^2 + 1] + 1}, {x, 0, 1}]
Join[{0}, RealDigits[2/(E-1)^2-2/3, 10, 128][[1]]] (* Jean-François Alcover, Feb 18 2014 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Robert G. Wilson v, Aug 14 2013
EXTENSIONS
a(128) corrected by Georg Fischer, Jul 16 2021
STATUS
approved