What Is a Catenary Curve?
A catenary is the curve formed by a perfectly flexible, uniform chain or cable hanging freely under its own weight between two fixed supports. The word "catenary" comes from the Latin word catena, meaning "chain." Although it superficially resembles a parabola, the catenary is a fundamentally different curve described by the hyperbolic cosine function.
The equation of a catenary with its vertex at (0, a) is y = a * cosh(x/a), where a is the catenary parameter that determines the shape of the curve. A larger value of a produces a flatter, more gently curving catenary, while a smaller value produces a tighter, more steeply curving shape.
Key Catenary Formulas
Catenary Equation
The fundamental equation describing the shape of a hanging chain, where a is the catenary parameter.
Sag
The vertical distance between the support points and the lowest point (vertex) of the catenary.
Arc Length
The total length of the curve between two points, calculated using the sinh function.
Catenary Parameter
The parameter a = H/w, where H is the horizontal tension and w is weight per unit length.
Catenary vs. Parabola
For centuries, mathematicians assumed that a hanging chain formed a parabola. It was not until 1691 that Leibniz, Huygens, and Johann Bernoulli independently proved that the true shape is a catenary. While the two curves look similar for small sag-to-span ratios, they diverge significantly as the sag increases.
The key mathematical difference is that a parabola is described by y = x2/(4a), while a catenary uses y = a * cosh(x/a). For engineering purposes, the parabolic approximation is often used when the sag is less than about 10% of the span, as the error is negligible in that range.
Practical Applications
Overhead Power Lines
Electrical engineers use catenary calculations to determine the sag, tension, and clearance of overhead transmission lines. The sag must be carefully calculated to ensure the cables do not touch the ground or nearby structures, while also keeping the tension within safe limits for the conductor material.
Architecture
The inverted catenary (a catenary flipped upside down) is the ideal shape for an arch that supports only its own weight. This principle was famously used by Antoni Gaudi in designing the arches of the Sagrada Familia and by Eero Saarinen in the Gateway Arch in St. Louis, which is a weighted catenary.
Bridge Cables
Suspension bridge cables naturally form a catenary shape when supporting only their own weight. When the bridge deck is hung from the cables, the shape changes to approximately a parabola because the load is distributed uniformly along the horizontal span rather than along the cable length.
Catenary in Nature
Spider webs, clotheslines, necklaces, and any flexible chain or rope hanging under gravity all form catenary curves. Even soap films between parallel rings form a surface called a catenoid, which is the surface of revolution of a catenary.
Properties of the Catenary
- The catenary has the unique property that the tension at any point is proportional to the height y at that point: T = w * y.
- The catenary minimizes potential energy for a flexible chain of given length between two fixed points.
- The derivative of cosh is sinh, and the derivative of sinh is cosh, making catenary calculus particularly elegant.
- The area under a catenary arc equals a * sinh(x/a), which also equals the arc length from the vertex to that point.
- The radius of curvature at the vertex is equal to the parameter a.
Tips for Catenary Calculations
- The parameter a must be positive for a physically meaningful catenary.
- For small x/a ratios (shallow curves), cosh(x/a) is approximately 1 + x^2/(2a^2), recovering the parabolic approximation.
- When given span and sag, finding a requires solving a transcendental equation, typically using numerical methods.
- The vertex of the catenary (lowest point) occurs at x = 0, where y = a.