A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.
Description
The mathematics
While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, , a sum of two exponential functions. One parabola is , and hyperbolic cosine is . The curves are unrelated.
The line of thrust
Unlike a catenary arch, the parabolic arch employs the principle that when weight is uniformly applied above, the internal compression (see line of thrust) resulting from that weight will follow a parabolic curve. Of all arch types, the parabolic arch produces the most thrust at the base. Also, it can span the widest area. It is commonly used in bridge design, where long spans are needed.
Compared to catenary arches
When an arch carries a uniformly distributed vertical load, the correct shape is a parabola. When an arch carries only its own weight, the best shape is a catenary.
Uses
In nature
A hen's egg can be fairly well described as two different paraboloids connected by part of an ellipse.