Editing Catenary (section)

==Mathematical description==

===Equation===
[[Image:catenary-pm.svg|thumb|350px|right|Catenaries for different values of {{mvar|a}}]]

The equation of a catenary in [[Cartesian coordinate system|Cartesian coordinates]] has the form<ref name="Lockwood122"/>

<math display=block>y = a \cosh \left(\frac{x}{a}\right) = \frac{a}{2}\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right),</math>
where {{math|cosh}} is the [[hyperbolic function|hyperbolic cosine function]], and where {{mvar|a}} is the distance of the lowest point above the x axis.<ref>{{cite web |url=http://mathworld.wolfram.com/Catenary.html |title=Catenary |last=Weisstein |first=Eric W. |author-link=Eric W. Weisstein |work=MathWorld--A Wolfram Web Resource |access-date=2019-09-21 |quote=The parametric equations for the catenary are given by x(t) = t, y(t) = [...] a cosh(t/a), where t=0 corresponds to the vertex [...] }}</ref> All catenary curves are [[Similarity (geometry)|similar]] to each other, since changing the parameter {{mvar|a}} is equivalent to a [[uniform scaling]] of the curve.

The [[Whewell equation]] for the catenary is<ref name="Lockwood122"/>
<math display=block>\tan \varphi = \frac{s}{a},</math>
where <math>\varphi</math> is the [[tangential angle]] and {{mvar|s}} the [[arc length]].

Differentiating gives
<math display=block>\frac{d\varphi}{ds} = \frac{\cos^2\varphi}{a},</math>
and eliminating <math>\varphi</math> gives the [[Cesàro equation]]<ref>[[#MathWorld|MathWorld]], eq. 7</ref>
<math display=block>\kappa=\frac{a}{s^2+a^2},</math>
where <math>\kappa</math> is the [[curvature]].

The [[radius of curvature]] is then
<math display=block>\rho = a \sec^2 \varphi,</math>
which is the length of the [[normal line|normal]] between the curve and the {{mvar|x}}-axis.<ref>[[#Routh|Routh]] Art. 444</ref>

===Relation to other curves===
When a [[parabola]] is rolled along a straight line, the [[roulette (curve)|roulette]] curve traced by its [[Conic section#Eccentricity, focus and directrix|focus]] is a catenary.<ref name="Yates 13"/> The [[Envelope (mathematics)|envelope]] of the [[Conic section#Eccentricity, focus and directrix|directrix]] of the parabola is also a catenary.<ref>Yates p. 80</ref> The [[involute]] from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the [[tractrix]].<ref name="Yates 13"/>

Another roulette, formed by rolling a line on a catenary, is another line. This implies that [[square wheel]]s can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any [[regular polygon]] except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.<ref>{{cite journal |last1=Hall |first1=Leon |last2=Wagon |first2=Stan |author2-link=Stan Wagon|year=1992 |title=Roads and Wheels |journal=Mathematics Magazine |volume=65 |issue=5 |pages=283–301 |jstor=2691240 |doi=10.2307/2691240}}
</ref>

===Geometrical properties===
Over any horizontal interval, the ratio of the area under the catenary to its length equals {{mvar|a}}, independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the {{mvar|x}}-axis.<ref>{{cite journal |last=Parker |first=Edward |year=2010 |title=A Property Characterizing the Catenary |journal=Mathematics Magazine |volume=83 |pages=63–64 |doi=10.4169/002557010X485120 |s2cid=122116662 }}</ref>

===Science===
A moving [[electric charge|charge]] in a uniform [[electric field]] travels along a catenary (which tends to a [[parabola]] if the charge velocity is much less than the [[speed of light]] {{mvar|c}}).<ref>{{cite book| last=Landau| first=Lev Davidovich| url=https://books.google.com/books?id=X18PF4oKyrUC&pg=PA56| title=The Classical Theory of Fields| year=1975| publisher=Butterworth-Heinemann| isbn=978-0-7506-2768-9| page=56}}</ref>

The [[surface of revolution]] with fixed radii at either end that has minimum surface area is a catenary

<math display="block">y = a \cosh^{-1}\left(\frac{x}{a}\right) + b</math>

revolved about the <math>y</math>-axis.<ref name="Yates 13">{{cite book |title=Curves and their Properties |first=Robert C. |last=Yates |publisher=NCTM |year=1952 |page=13}}</ref>