Catenary Curve Calculator

In Geometry, a catenary is an idealized curve that a hanging chain or cable assumes under its weight when it is merely supported at its ends. The curve has a U shape and appears to be a parabola on the surface, but it is a (scaled, rotated) graph of the hyperbolic cosine. You may graph a catenary curve using the provided sag, length, and increment values with the help of this online catenary curve calculator.

\( y = a \space cosh \dfrac{x}{a} \)

Catenary Definition

A cable takes on a catenary shape when it is supported at both ends and is only pulled in one direction by its weight. As a result, the cable’s center dips due to the uniform gravitational force, creating a symmetrical curve on both sides of the minimum point. The catenary curve is intriguing because it frequently appears in the environment.

A natural catenary has two functional characteristics:

  1. The horizontal force (Fx) in the cable is constant along its entire length; and
  2. The vertical force (Fy) in the cable is equal to the weight of the cable being carried at any given position (i.e. Fy is zero at the bottom of the loop).

Resolving these forces yields the angle of the wire at any point, for example,

\( tan^{-1}(\dfrac{Fy}{Fx}) \)

Equation of the catenary curve:

\( cosh \space x = \dfrac{e^x + e^{-x}}{2} \) OR \( y = a \space cosh(\dfrac{x}{a}) + b \)

x is the abscissa and y is the ordinate of a point on the curve
e is Euler’s number,
a is a constant: horizontal tension divided by the weight per unit length
And b is the integration constant impacting the y-intercept of a graphed catenary curve.

Applications of the catenary curve

Catenaries are used in engineering and architecture, for instance, to create hanging bridges or arches when they are turned upside down. The St. Louis Gateway Arch is one of the most striking examples. Catenaries are also present in the natural world, such as the curvature of a spider web.


What are catenary curves used for?

Kilns frequently incorporate catenary arches in their design. The intended measurements of a hanging chain are transferred to a form to generate the desired curvature. The form is then used to guide the placement of bricks or other building materials.

Is the catenary curve a parabola?

Catenary curves have uniform forces over the length of the rope, but parabolas have uniform forces when the horizontal length is considered. The catenary is more prevalent in self-suspending constructions (think of the Golden Gate Bridge).

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