A wire takes a form resembling an arc of a parabolaMathworldPlanetmathPlanetmath when suspended at its ends.  The arc is not from a parabola but from the graph of the hyperbolic cosineMathworldPlanetmath ( functionMathworldPlanetmath in a suitable coordinate systemMathworldPlanetmath.

Let’s derive the equation  y=y(x)  of this curve, called the catenaryMathworldPlanetmath, in its plane with x-axis horizontal and y-axis vertical.  We denote the of the wire by σ.

In any point  (x,y)  of the wire, the tangent lineMathworldPlanetmath of the curve forms an angle φ with the positive direction of x-axis.  Then,


In the point, a certain tension T of the wire acts in the direction of the value a.  Hence we may write


whence the vertical of T is


and its differentialMathworldPlanetmath (


But this differential is the amount of the supporting   σ1+(y(x))2dx (see the arc lengthMathworldPlanetmath).  Thus we obtain the differential equationMathworldPlanetmath

σ1+y2dx=ady, (1)

which allows the separation of variablesMathworldPlanetmath:


This may be solved by using the substitution (






This leads to the final solution


of the equation (1).  We have denoted the constants of integration by x0 and y0.  They determine the position of the catenary in regard to the coordinate axes.  By a suitable choice of the axes and the equation

y=acoshxa (2)

of the catenary.

Some of catenary

  • tanφ=sinhxa,sinφ=tanhxa  (cf. the GudermannianMathworldPlanetmath)

  • The arc length of the catenary (2) from the apex  (0,a)  to the point  (x,y)  is   asinhxa=y2-a2.

  • The radius of curvatureMathworldPlanetmath of the catenary (2) is  acosh2xa, which is the same as length of the normal lineMathworldPlanetmath of the catenary between the curve and the x-axis.

  • The catenary is the catacausticMathworldPlanetmath of the exponentialMathworldPlanetmathPlanetmath curve ( reflecting the vertical rays.

  • If a parabola rolls on a straight line, the focus draws a catenary.

  • The involute (a.k.a. the evolvent) of the catenary is the tractrixMathworldPlanetmath.

Title catenary
Canonical name Catenary
Date of creation 2014-10-26 21:25:30
Last modified on 2014-10-26 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