Catenary Curve Calculator

What Is a Catenary Curve?

A catenary is the mathematical curve assumed by a flexible, uniform cord, chain, or cable hanging freely under the influence of its own weight between two fixed supports. First investigated by Galileo Galilei, Christiaan Huygens, Gottfried Leibniz, and Johann Bernoulli in the 17th century, the catenary shape minimizes gravitational potential energy across its entire length.

Mathematical Equation of the Catenary

The canonical Cartesian equation of a catenary with parameter $a$ and vertex at $(0, a)$ is given by the hyperbolic cosine function:

$$y(x) = a \cosh\left(\frac{x}{a}\right) = a \cdot \left(\frac{e^{x/a} + e^{-x/a}}{2}\right)$$

When calibrated relative to the lowest point (the vertex where $y(0) = 0$), the equation becomes:

$$y(x) = a \left[\cosh\left(\frac{x}{a}\right) - 1\right]$$

Key Geometric and Physical Parameters

• Catenary Parameter ($a$): Defined physically as $a = T_0 / w$, representing the ratio of the horizontal tension $T_0$ to the cable's weight per unit length $w$.
• Horizontal Span ($L$): The total distance between the two supporting towers ($x = \pm d = \pm L/2$).
• Sag / Dip ($s$): The vertical distance from the support elevation to the lowest point of the curve: $$s = a \left[\cosh\left(\frac{L}{2a}\right) - 1\right]$$
• Cable Arc Length ($S$): The total physical length of the hanging conductor: $$S = 2a \sinh\left(\frac{L}{2a}\right)$$
• Support Tangent Angle ($\theta$): The angle of the cable relative to the horizontal at the support: $$\tan\theta = \sinh\left(\frac{L}{2a}\right), \quad \cos\theta = \frac{a}{a + s}$$

Catenary vs Parabola: What Is the Difference?

While visually similar, a catenary and a parabola arise from different physical loading conditions:

• Catenary: Occurs when load is uniformly distributed along the arc length of the cable itself (such as overhead power transmission lines or freestanding suspension chains).
• Parabola: Occurs when load is uniformly distributed along the horizontal span (such as suspension bridge cables carrying a flat, heavy roadway via vertical suspender cables).

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