# Parabolic arch

A parabolic arch is an arch in the shape of a parabola. Such arches are used in bridges, cathedrals, and elsewhere in architecture and engineering.

## 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, cosh(x), a sum of two exponential functions. One parabola is $f(x)=x^{2}+3x-1$ , and hyperbolic cosine is $\cosh x={\frac {e^{x}+e^{-x}}{2}}$ . 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

A hen's egg can be fairly well described as two different paraboloids connected by part of an ellipse .

### Cathedrals and churches

A few examples of parabolic arches.:

### Architects

Parabolic arches were used by architects Oscar Niemeyer, and Antoni Gaudí, who also used catenary arches.