Calculations for a straight catenary arc or catenary segment. This is a figure formed by a catenary of the form y = a * cosh(x/a) in the interval x ∈ [ -b ; b ] and the closing straight line. cosh is the hyperbolic cosine, meaning the curved line is hyperbolic.
Enter the shape parameter a (a>0) and the maximum input value b (corresponding to half the span) and choose the number of decimal places. Then click Calculate. The parameter a determines the curvature of the catenary. The larger a, the flatter the curve.
Formulas:
l = 2a * sinh(b/a)
s = 2b
h = a * [ cosh(b/a) - 1 ]
u = l + s
A = 2a * [ b * cosh(b/a) - a * sinh(b/a) ]
Shape parameter, input value a, height, catenary arc length and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The catenary curve describes the shape of an ideally flexible, homogeneous chain suspended at two points under its own weight. When this shape is reversed, it creates a catenary arch, which is used in architecture because it optimally dissipates compressive forces and is very stable. Prominent examples of its architectural application include the hanging vaults of Antoni Gaudí's Sagrada Família in Barcelona, the Gateway Arch in St. Louis as an inverted chain arch, and the cable curves of modern suspension bridges such as the Golden Gate Bridge in San Francisco.
Although not a hyperbola, the catenary curve is described by the hyperbolic cosine, whose mathematical definition is based on the same exponential functions from which the equation of the hyperbola is derived. One aspect of the catenary curve is the ratio between its area and its length. The area A is directly proportional to the product of the shape parameter a and the difference between the projected arc length and the actual chord. Furthermore, the catenary curve can be defined as the curve whose center of gravity, for a given arc length, has the lowest possible potential energy.