Catenary Curve Calculator
Compute y on a catenary curve y = a·cosh(x/a) from parameter a and x-coordinate. Free online catenary calculator with interactive curve chart, formula, and physics explanations.
About This Calculator
About the Catenary Curve
A catenary is the natural curve formed by a uniform flexible chain, cable, or rope hanging freely under its own weight between two fixed points. The word "catenary" comes from the Latin catena meaning "chain." Unlike a parabola, which is often mistakenly assumed to be the shape of a hanging cable, the true shape is described by the hyperbolic cosine function.
The Catenary Equation
The standard catenary equation is y = a · cosh(x / a), where a is the catenary parameter equal to T0/w (horizontal tension at the vertex divided by the weight per unit length), x is the horizontal distance from the vertex, and y is the vertical height. The vertex of the curve is at the point (0, a).
Physical Meaning of the Parameter a
The parameter a determines the curve's shape and steepness. A small a (e.g., a = 1) produces a tightly curved, deeply sagging line -- like a heavy chain hanging with low tension. A large a (e.g., a = 10) produces a flat, almost straight line -- like a tightly stretched cable under high tension. In engineering, a is the ratio of horizontal tension to the cable's weight per unit length.
Real-World Applications
Catenaries appear in suspension bridges (the main cables between towers), overhead power transmission lines, telephone and fiber-optic cables, mooring lines for ships, and even in architectural design -- the Gateway Arch in St. Louis is an inverted catenary. Understanding the catenary is essential in civil, structural, and mechanical engineering for designing cable-supported structures.
Catenary vs. Parabola
While a catenary and a parabola look similar near the vertex, they are mathematically distinct. The catenary uses the hyperbolic cosine function (cosh), while a parabola is a quadratic. For shallow curves (small sag relative to span), a parabola is a reasonable approximation, but the true hanging-cable shape is always a catenary. The difference becomes noticeable for deeper sags.
How to Use This Calculator
Enter the catenary parameter a (a positive number) and the x-coordinate at which you want to evaluate the curve. Click Calculate to see the y value and view the interactive line chart showing the full catenary shape. The chart plots the curve across a range of x values centered around the input. All inputs are saved to the URL for easy sharing.
Related Formulas
The arc length from the vertex to a point at x = X is s = a · sinh(X / a). The derivative (slope) at any point is dy/dx = sinh(x / a). The radius of curvature at the vertex is equal to a, meaning the curve's sharpness at its lowest point is directly determined by the parameter a.
Frequently Asked Questions
What is the formula for a catenary curve?
The standard catenary equation is y = a·cosh(x/a), where a is the catenary parameter and cosh is the hyperbolic cosine. The curve describes the shape of a uniform flexible chain or cable hanging under its own weight between two supports.
How is a catenary different from a parabola?
A catenary (y = a·cosh(x/a)) and a parabola (y = x^2) look similar near the vertex but differ mathematically. The catenary flattens more gradually and uses hyperbolic cosine. For shallow curves a parabola is a reasonable approximation, but the true hanging cable shape is always a catenary.
What does the catenary parameter a represent?
The catenary parameter a equals T0/w, where T0 is the horizontal tension at the vertex and w is the weight per unit length. A larger a means higher tension and a flatter curve; a smaller a means lower tension and a tighter curve. The vertex is at (0, a).
Where are catenary curves used in real life?
Catenary curves appear in suspension bridge cables, overhead power lines, telephone wires, hanging chains, and even the Gateway Arch (inverted catenary). In architecture, inverted catenaries provide optimal compression in arches and domes.
How do you calculate arc length of a catenary?
The arc length from the vertex to x = X is s = a·sinh(X/a). The total arc length between x = -X and x = X is 2a·sinh(X/a). This comes from integrating the arc length element with dy/dx = sinh(x/a).
Can a hanging cable ever form a perfect parabola?
No. A uniform hanging cable under gravity always forms a catenary, not a parabola. A parabola only appears when weight distribution is uniform along the horizontal span, which happens in suspension bridge decks but not in free-hanging cables.
What is the Gateway Arch and its relation to catenary?
The Gateway Arch in St. Louis is designed as an inverted weighted catenary -- specifically a flattened catenary (y = A·cosh(Bx) - C). This shape optimally distributes compression loads, making the structure both strong and material-efficient.
How does the catenary curve change as parameter a varies?
Small a (e.g., a = 1) produces a steep, tight U-shape. Large a (e.g., a = 10) produces a flat, gentle curve. The vertex y-value equals a, so the entire curve scales vertically with a. The chart in the calculator lets you visualize this directly.