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Catenary

Catenary

The curve naturally formed by a slack rope or wire hanging between two fixed points. A catenary is NOT a parabola, even though it looks like one. Note: The graph of the hyperbolic cosine function is a catenary.

 

 

U-shaped curve of a catenary, symmetric about a vertical axis, with a smooth minimum at the bottom, wider at the top.

Key Formula

y=acosh⁡ ⁣(xa)=a2(ex/a+e−x/a)y = a \cosh\!\left(\frac{x}{a}\right) = \frac{a}{2}\left(e^{x/a} + e^{-x/a}\right)
Where:
  • aa = A positive constant that controls how "tight" or "loose" the curve is (related to the ratio of horizontal tension to the weight per unit length)
  • xx = The horizontal distance from the lowest point of the curve
  • yy = The height of the curve at position x
  • cosh⁡\cosh = The hyperbolic cosine function

Worked Example

Problem: A hanging cable forms a catenary described by y = 2 cosh(x/2). Find the height of the cable at x = 0 and at x = 2.
Step 1: Recall that cosh(t) = (e^t + e^{-t}) / 2, and that cosh(0) = 1.
cosh⁡(0)=e0+e02=1+12=1\cosh(0) = \frac{e^0 + e^0}{2} = \frac{1+1}{2} = 1
Step 2: Evaluate y at x = 0, the lowest point of the cable.
y(0)=2cosh⁡ ⁣(02)=2cosh⁡(0)=2×1=2y(0) = 2\cosh\!\left(\frac{0}{2}\right) = 2\cosh(0) = 2 \times 1 = 2
Step 3: Evaluate y at x = 2. First compute the argument: x/a = 2/2 = 1.
cosh⁡(1)=e1+e−12≈2.7183+0.36792≈1.5431\cosh(1) = \frac{e^1 + e^{-1}}{2} \approx \frac{2.7183 + 0.3679}{2} \approx 1.5431
Step 4: Multiply by a = 2 to get the height at x = 2.
y(2)=2×1.5431≈3.0862y(2) = 2 \times 1.5431 \approx 3.0862
Answer: At x = 0, the cable height is exactly 2. At x = 2, the cable height is approximately 3.09. The cable sags to a minimum height of 2 (the value of a) at its lowest point and rises on either side.

Frequently Asked Questions

Why is a catenary not a parabola?
A catenary is defined by the hyperbolic cosine function, y = a cosh(x/a), while a parabola is defined by a quadratic, y = ax² + bx + c. Near the bottom, the two curves look almost identical, but they diverge significantly as you move away from the center. A catenary grows exponentially at the ends, while a parabola grows only quadratically. They arise from different physical situations: a catenary comes from a chain hanging under its own uniform weight, whereas a parabola describes the shape of a cable supporting a uniformly distributed horizontal load (like the deck of a suspension bridge).
Where do catenaries appear in real life?
Any freely hanging chain, rope, or cable forms a catenary. Power lines between poles, spider silk between anchor points, and necklaces hanging from your neck all approximate catenaries. The Gateway Arch in St. Louis is an inverted catenary. Architects also use catenary arches because the shape distributes weight purely as compression, eliminating bending forces.

Catenary vs. Parabola

Both are U-shaped symmetric curves, but a catenary is described by y = a cosh(x/a) and a parabola by y = ax². A catenary forms when a uniform chain hangs under gravity; a parabola forms when a cable supports a load spread evenly along the horizontal (like a bridge deck). Near their vertices, the curves nearly overlap, but the catenary rises faster (exponentially) farther from the center. You can distinguish them: expand cosh in a Taylor series and the catenary matches the parabola only in the first two terms.

Why It Matters

The catenary is essential in engineering and architecture. Suspension cables, power lines, and anchor chains all follow this curve, so engineers must use the catenary equation—not a parabolic approximation—to calculate sag, tension, and cable length accurately. Inverted catenaries form ideal arches that carry loads in pure compression, a principle used from ancient Roman aqueducts to the Gateway Arch.

Common Mistakes

Mistake: Assuming a hanging cable forms a parabola.
Correction: A freely hanging uniform cable forms a catenary, not a parabola. A parabola only results when the load is distributed uniformly along the horizontal direction (as with a suspension bridge deck), not along the cable itself.
Mistake: Thinking cosh(x) and cos(x) are the same function.
Correction: The hyperbolic cosine cosh(x) = (e^x + e^{-x})/2 is defined using exponentials and always yields values ≥ 1. The ordinary cosine cos(x) oscillates between −1 and 1. They behave very differently despite similar names.

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