Catenary Curve Calculator

About the Catenary Curve Calculator

The catenary curve describes the shape formed by a flexible chain or cable hanging freely under its own weight, supported at its endpoints. Defined by the equation y = a·cosh(x/a), it is one of the most elegant curves in mathematics and engineering, appearing everywhere from power line design to architectural arches.

The parameter "a" in the catenary equation represents the ratio of horizontal tension to the weight per unit length of the cable. A larger value of "a" produces a flatter, more gently curving catenary, while smaller values create deeper, more pronounced sags. Understanding this relationship is critical for structural engineers designing suspension bridges and electrical engineers planning overhead transmission lines.

This calculator computes all essential catenary properties: the height at any point along the curve, total sag between supports, arc length of the cable, horizontal and maximum tensions, local slope, curvature, and radius of curvature. It also generates a detailed properties table showing how these values change along the curve's length.

Preset configurations for common real-world scenarios — power lines, suspension bridges, heavy chains, decorative cables, and clotheslines — let you quickly explore how different parameters affect the curve's shape. The interactive curve profile visualization shows the catenary's characteristic U-shape, and the reference table summarizes all key formulas.

Whether you're an engineer sizing cables, a physics student studying statics, or an architect designing hanging structures, it gives the precise catenary calculations you need.

When This Page Helps

Catenary equations involve the hyperbolic cosine function with parameters that interact nonlinearly — computing y-coordinates, arc lengths, sag depths, and horizontal tension from a cable's weight per unit length and support span requires careful algebra that is easy to mishandle. The page evaluates the full catenary profile, plotting the curve for any parameter a, computing sag, arc length, and tension, and letting you switch between metric and imperial units. Architects, structural engineers, and power-line designers use catenary math daily, and the workflow removes the manual algebra overhead so you can focus on design decisions.

How to Use the Inputs

1. Enter the catenary parameter "a" or choose a preset scenario.
2. Set the horizontal position x where you want to evaluate the curve.
3. Enter the total span between the two support points.
4. Provide the weight per unit length of the cable or chain.
5. Adjust the number of table points for more or less curve detail.
6. Read sag, arc length, tension, and slope results from the output cards.
7. Examine the curve profile and properties table for the complete view.

Example Calculation

Tips & Best Practices

• For gently hanging cables, parameter "a" is much larger than the span.
• Sag increases rapidly when span approaches or exceeds 2a.
• Horizontal tension is constant everywhere along the catenary.
• Maximum tension always occurs at the support points.
• The catenary approximates a parabola whenever sag is small relative to span.

The Catenary vs. the Parabola

A common misconception is that a hanging cable forms a parabola. Galileo originally believed this, but in 1691 Leibniz, Huygens, and Johann Bernoulli independently proved the true shape is a catenary: y = a cosh(x/a). A parabola is a good approximation only when the sag is small relative to the span. For deeper sags, the hyperbolic cosine diverges significantly from a parabolic fit. The parameter a = T₀/w represents the ratio of horizontal tension to weight per unit length and controls the curve's "flatness" — larger a means a flatter, tighter cable.

Structural and Architectural Applications

The inverted catenary is the ideal shape for a compression arch: the Gateway Arch in St. Louis, Gaudí's hanging-chain models for the Sagrada Família, and Roman aqueduct arches all exploit this principle. Power transmission lines, suspension bridge cables (under uniform horizontal load, these are parabolas, but under self-weight, catenaries), and overhead contact wires for electric railways all follow catenary geometry. Understanding sag is critical: excessive sag causes ground clearance violations, while too little sag (high tension) risks conductor fatigue or snap in cold weather.

Arc Length, Tension, and Design Trade-offs

The arc length of a catenary from x = −L/2 to L/2 is 2a sinh(L/(2a)), which is always longer than the span L. Engineers must order enough cable to cover this extra length. The tension varies along the curve: minimum at the lowest point (T₀ = wa) and maximum at the supports T = w√(a² + y²). These relationships drive conductor selection, tower height decisions, and span-to-sag ratio tables in power-line design. Wind and ice loading add uniform horizontal and vertical loads that shift the curve between a catenary and a parabola, requiring iterative sag-tension calculations.

Frequently Asked Questions

• What is a catenary curve?

  A catenary is the curve formed by a uniform flexible chain or cable hanging freely under gravity, described by y = a·cosh(x/a).

• How does a catenary differ from a parabola?

  A parabola results from uniform horizontal loading (like a suspension bridge deck), while a catenary results from uniform loading along the cable itself. They are nearly identical for small sag-to-span ratios.

• What does the parameter "a" represent?

  Parameter a equals horizontal tension divided by weight per unit length (T_h / w). Larger a means a flatter curve.

• How is catenary sag calculated?

  Sag = a · cosh(L / 2a) − a, where L is the span. This gives the vertical distance from the support level to the lowest point.

• Where is the maximum tension in a catenary?

  Maximum tension occurs at the support points, calculated as T_max = w · a · cosh(L / 2a).

• What are real-world examples of catenaries?

  Power lines between poles, anchor chains, spider webs, the Gateway Arch in St. Louis (an inverted catenary), and freely hanging necklaces all form catenary shapes.

• Can this handle inclined supports?

  This calculator assumes level supports. Inclined catenaries require asymmetric formulations with different support heights.