catenary
A wire takes a form resembling an arc of a parabola^{} when suspended at its ends. The arc is not from a parabola but from the graph of the hyperbolic cosine^{} (http://planetmath.org/HyperbolicFunctions) function^{} in a suitable coordinate system^{}.
Let’s derive the equation $y=y(x)$ of this curve, called the catenary^{}, in its plane with $x$axis horizontal and $y$axis vertical. We denote the of the wire by $\sigma $.
In any point $(x,y)$ of the wire, the tangent line^{} of the curve forms an angle $\phi $ with the positive direction of $x$axis. Then,
$$\mathrm{tan}\phi =\frac{dy}{dx}={y}^{\prime}.$$ 
In the point, a certain tension $T$ of the wire acts in the direction of the value $a$. Hence we may write
$$T=\frac{a}{\mathrm{cos}\phi},$$ 
whence the vertical of $T$ is
$$T\mathrm{sin}\phi =a\mathrm{tan}\phi $$ 
and its differential^{} (http://planetmath.org/Differential)
$$d(T\mathrm{sin}\phi )=ad\mathrm{tan}\phi =ad{y}^{\prime}.$$ 
But this differential is the amount of the supporting $\sigma \sqrt{1+{({y}^{\prime}(x))}^{2}}dx$ (see the arc length^{}). Thus we obtain the differential equation^{}
$\sigma \sqrt{1+{y}^{\prime 2}}dx=ad{y}^{\prime},$  (1) 
which allows the separation of variables^{}:
$$\int \mathit{d}x=\frac{a}{\sigma}\int \frac{d{y}^{\prime}}{\sqrt{1+{y}^{\prime 2}}}$$ 
This may be solved by using the substitution (http://planetmath.org/SubstitutionForIntegration)
$${y}^{\prime}:=\mathrm{sinh}t,d{y}^{\prime}=\mathrm{cosh}tdt,\sqrt{1+{y}^{\prime 2}}=\mathrm{cosh}t$$ 
giving
$$x=\frac{a}{\sigma}t+{x}_{0},$$ 
i.e.
$${y}^{\prime}=\frac{dy}{dx}=\mathrm{sinh}\frac{\sigma (x{x}_{0})}{a}.$$ 
This leads to the final solution
$$y=\frac{a}{\sigma}\mathrm{cosh}\frac{\sigma (x{x}_{0})}{a}+{y}_{0}$$ 
of the equation (1). We have denoted the constants of integration by ${x}_{0}$ and ${y}_{0}$. They determine the position of the catenary in regard to the coordinate axes. By a suitable choice of the axes and the equation
$y=a\mathrm{cosh}{\displaystyle \frac{x}{a}}$  (2) 
of the catenary.
Some of catenary

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$\mathrm{tan}\phi =\mathrm{sinh}\frac{x}{a},\mathrm{sin}\phi =\mathrm{tanh}\frac{x}{a}$ (cf. the Gudermannian^{})

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The arc length of the catenary (2) from the apex $(0,a)$ to the point $(x,y)$ is $a\mathrm{sinh}\frac{x}{a}=\sqrt{{y}^{2}{a}^{2}}$.

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The radius of curvature^{} of the catenary (2) is $a{\mathrm{cosh}}^{2}\frac{x}{a}$, which is the same as length of the normal line^{} of the catenary between the curve and the $x$axis.

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The catenary is the catacaustic^{} of the exponential^{} curve (http://planetmath.org/ExponentialFunction) reflecting the vertical rays.

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If a parabola rolls on a straight line, the focus draws a catenary.
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Title  catenary 
Canonical name  Catenary 
Date of creation  20141026 21:25:30 
Last modified on  20141026 21:25:30 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  29 
Author  pahio (2872) 
Entry type  Derivation 
Classification  msc 53B25 
Classification  msc 51N05 
Synonym  chain curve 
Related topic  EquationOfCatenaryViaCalculusOfVariations 
Related topic  LeastSurfaceOfRevolution 
Related topic  HyperbolicFunctions 
Related topic  Tractrix 
Related topic  EqualArcLengthAndArea 
Defines  catenary 