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Catenary Curve

A catenary is the curve formed by a hanging chain or cable under its own weight: y = a cosh(x/a). It differs from a parabola; inverted, it forms an optimal arch. Used in suspension bridges and Gaudí's architecture.

Concept Fundamentals
y = a cosh(x/a)
Equation
s = a sinh(x/a)
Arc length
dy/dx = sinh(x/a)
Slope
T = T₀ cosh(x/a)
Tension

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Catenary ≠ parabola: parabola has uniform horizontal load; catenary has load along the curve. Inverted catenary is the ideal arch—no bending moments. cosh(x) = (eˣ + e⁻ˣ)/2; sinh(x) = (eˣ − e⁻ˣ)/2.

Key quantities
y = a cosh(x/a)
Equation
Key relation
s = a sinh(x/a)
Arc length
Key relation
dy/dx = sinh(x/a)
Slope
Key relation
T = T₀ cosh(x/a)
Tension
Key relation

Ready to run the numbers?

Why: The catenary is the natural shape of a hanging chain. Inverted, it forms a stable arch. Leibniz, Huygens, and Bernoulli solved it in the 17th century. Used in bridges, power lines, and Gaudí's Sagrada Família.

How: y = a cosh(x/a) where cosh is hyperbolic cosine. Arc length from vertex: s = a sinh(x/a). Slope: dy/dx = sinh(x/a). Parameter a controls the 'sag' of the curve.

Catenary ≠ parabola: parabola has uniform horizontal load; catenary has load along the curve.Inverted catenary is the ideal arch—no bending moments.
Sources:Wolfram MathWorld — CatenaryWikipedia — Catenary

Run the calculator when you are ready.

Calculate CatenaryEnter parameter a and x to get height, arc length, slope

Catenary Curve Calculator

Calculate Catenary Curve Properties

Enter the parameters to visualize a catenary curve and calculate its properties at any point.

The parameter a defines the scale of the catenary. Larger values make the curve flatter.

The x-coordinate of the point where you want to calculate the catenary properties.

The lower bound of the x-axis for visualization.

The upper bound of the x-axis for visualization.

Results

Results at x = 2:

Height (y)
3.7622
Arc Length
3.6269
Slope
3.6269

Explanation:

  • Height (y): The vertical coordinate of the point on the catenary curve at x = 2.
  • Arc Length: The distance measured along the curve from the lowest point (x = 0) to this point.
  • Slope: The gradient of the curve at this point, representing the tangent's steepness.

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

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y = a cosh(x/a) — shape of hanging chain.

— Differential Geometry

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Inverted catenary = optimal arch (Gaudí).

— Architecture

How to Use This Calculator

  1. Enter a positive value for the parameter aa (default is 1).
  2. Set the x-coordinate where you want to calculate the catenary properties.
  3. Adjust the minimum and maximum x values for visualization if needed.
  4. Click "Calculate" to see the results and visualization.
  5. Use "Show Formulas" to view the mathematical equations.
  6. Click "Reset" to start over with default values.

Understanding the Catenary Curve

The catenary curve is often confused with a parabola, but they are different curves with distinct mathematical properties. The word "catenary" comes from the Latin word "catena," meaning "chain."

Mathematical Definition

A catenary curve is described by the hyperbolic cosine function. When a perfectly flexible, uniform cable or chain is suspended between two points and subjected only to its own weight, it naturally forms a catenary.

Historical Significance

The catenary curve was first mathematically analyzed in the late 17th century by mathematicians including Leibniz, Huygens, and Bernoulli. They were responding to a challenge posed by Jakob Bernoulli to find the curve formed by a hanging chain.

Applications in Architecture and Engineering

The catenary has important applications in architecture and engineering:

  • When inverted, the catenary forms an optimal arch shape that can support its own weight without generating bending forces.
  • Suspension bridges often incorporate catenary principles in their design.
  • Electrical transmission lines form catenary curves between support towers.
  • Antoni Gaudí used inverted catenary models to design the Sagrada Família in Barcelona.

Catenary vs. Parabola

While a catenary looks similar to a parabola, there are key differences:

  • A parabola is described by y=x2y = x^2 (with appropriate scaling and shifts).
  • A catenary is described by y=acosh(x/a)y = a \\cosh(x/a).
  • A parabola forms when a uniform load is distributed horizontally (like a suspension bridge deck).
  • A catenary forms when the load is distributed along the curve itself (like a hanging chain).

Catenary Curve Formulas

Catenary Equation:
y=acoshleft(fracxaright)y = a \\cosh\\left(\\frac{x}{a}\\right)

This is the fundamental equation of a catenary curve, where aa is a parameter that determines the shape of the curve and cosh\\cosh is the hyperbolic cosine function.

Arc Length:
s=asinhleft(fracxaright)s = a \\sinh\\left(\\frac{x}{a}\\right)

The arc length from the lowest point (where x = 0) to any point x on the curve.

Slope:
fracdydx=sinhleft(fracxaright)\\frac{dy}{dx} = \\sinh\\left(\\frac{x}{a}\\right)

The slope of the tangent to the curve at any point x.

Tension in Chain/Cable:
T=T0coshleft(fracxaright)T = T_0 \\cosh\\left(\\frac{x}{a}\\right)

The tension at any point, where T0T_0 is the tension at the lowest point.

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