Editing Arch (section)

=== Parabolic ===
The popularity of the arches using segments of a circle is due to simplicity of layout and construction,{{sfn|Mark|1996|p=387}} not their structural properties. Consequently, the architects historically used a variety of other curves in their designs: [[elliptical curve]]s, [[hyperbolic cosine]] curves (including [[catenary]]), and [[parabolic curve]]s. There are two reasons behind the selection of these curves:{{sfn | González | Samper | Herrera | 2018 | p=185}}
# they are still relatively easy to trace with common tools prior to construction;
# depending on a situation, they can have superior structural properties and/or appearance.
The [[hyperbolic curve]] is not easy to trace, but there are known cases of its use.{{sfn | González | Samper | Herrera | 2018 | p=185}} The non-[[circumference|circumferential]] curves look similar, and match at shallow profiles, so a catenary is often misclassified as a parabola{{sfn | Bradley | Gohnert | 2022 }} (per [[Galileo]], "the [hanging] chain fits its parabola almost perfectly"{{sfn|Osserman|2010|p=220}}). González et al. provide an example of [[Palau Güell]], where researchers do not agree on classification of the arches or claim the prominence of parabolic arches, while the measurements show that just two of the 23 arches designed by [[Gaudi]] are actually parabolic.{{sfn | González | Samper | Herrera | 2018 | pp=174, 184}}

<gallery>
File:Palau Güell, Barcelona 07.jpg|Palau Güell: Parabolic{{sfn| González | Samper | Herrera | 2018 | p=183 | loc = #14}}
File:Palau Güell, Barcelona 114.jpg|Palau Güell: Hyperbolic{{sfn| González | Samper | Herrera | 2018 | p=182 | loc = #9}}
File:043 Palau Güell, c. Nou de la Rambla 3-5 (Barcelona), baixos.jpg|Palau Güell: [[Rankine curve]]{{sfn | González | Samper | Herrera | 2018 | p=183 | loc = #18}} (a.k.a. [[weighted catenary]])
File:Barcelona Palau Güell Orgel (1).jpg|Palau Güell: Elliptical{{sfn| González | Samper | Herrera | 2018 | p=183 | loc = #19}}
</gallery>

Three parabolic-looking curves in particular are of significance to the arch design: [[parabola]] itself, [[catenary]], and [[weighted catenary]]. The arches naturally use the inverted (upside-down) versions of these curves.

A parabola represents an ideal (all-compression) shape when the load is equally distributed along the span, while the weight of the arch itself is negligible. A catenary is the best solution for the case where an arch with uniform thickness carries just its own weight with no external load. The practical designs for bridges are somewhere in between, and thus use the curves that represent a compromise that combines both the catenary and the [[funicular curve]] for particular non-uniform distribution of load.{{sfn | Benaim | 2019 | p=501}} The practical free-standing arches are stronger and thus heavier at the bottom, so a weighted catenary curve is utilized for them. The same curve also fits well an application where a bridge consists of an arch with a roadway of packed dirt above it, as the [[dead load]] increases with a distance from the center.{{sfn|Osserman|2010|p=224}}

<gallery>
File:Tyne Bridge - Newcastle Upon Tyne - England - 2004-08-14.jpg|A [[through arch bridge]] ([[Tyne Bridge]] in [[Newcastle upon Tyne]], [[England]]): parabolic-looking arches with multiple deck supports distributing the load
File:Arch Reflection 1 (5263608871).jpg|[[Gateway Arch]] is stronger at the bottom: weighted catenary curve
</gallery>