Calculations at a catenary dome or a body of revolution of a catenary. This form is created from a catenary arc with the underlying catenary of the form y=a*cosh(x/a) in the interval x ∈ [ -b ; b ], which is rotated about its height and the closing top surface.
Enter the shape parameter a (a>0) and the maximum input value b (corresponding to the radius of the circle of the top surface). Choose the number of decimal places, then click Calculate.
Formulas:
h = a * [ cosh(b/a) - 1 ]
L = 2 * π * a * [ b * sinh(b/a) - a * cosh(b/a) + a ]
A = L + π * b²
V = π * a * [ ( b² + 2 * a² ) * cosh(b/a) - 2 * a * b * sinh(b/a) - 2 * a² ]
pi:
π = 3.141592653589793...
Shape parameter, radius a and height have the same unit (e.g. meter), lateral and surface area have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit -1.
A catenary dome without the circular top can also be called a bowl. In architecture, this form is used for vaults and, as the name suggests, domes. In fact, the term catenary dome is primarily an architectural term. Body of revolution of a catenary curve, on the other hand, is the correct mathematical term, but it is more cumbersome and less self-explanatory.
The catenary dome is described by the hyperbolic functions hyperbolic sine (sinh) and hyperbolic cosine (cosh). It is a solid of revolution, meaning it is rotationally symmetric about its axis of height and mirror-symmetric about every plane in which this axis lies.
Like a paraboloid, unlike a hemisphere, a catenary dome does not have a constant curvature, but rather a curvature that increases steadily from the vertex to the edge. As a result, its volume, relative to its height and base radius, is somewhat smaller than that of a corresponding spherical cap, while its surface area is larger. This property can be advantageous in some technical applications, such as when a large surface area is required within a limited volume, as in reflectors or certain architectural structures.