Catenary

The catenary is the shape assumed by a hanging flexible inextensible chain or cable supported at its ends and subjected only to a uniform gravitational force. The name derives from the Latin word "catena," meaning chain. Its symmetric equation is given by the hyperbolic cosine function:
y = a * (cosh(x / a) - 1)

Interactive Experiment

Use the Catenary interactive module to explore how changes in tension, weight, and span affect the shape and length of the hanging chain.

1. Set Catenary scale a to 1, Half-span L to 1, and Probe fraction u to 0. Observe the selected point at the lowest point (0,0). The support sag is cosh(1) - 1 ≈ 0.543081 and the total length is 2 * sinh(1) ≈ 2.350402.
2. Change the Probe fraction u to -1. The selected point height now equals the right support sag, and the slope is -sinh(1), showing the reflection symmetry of the curve.
3. Set Catenary scale a to 0. The module rejects the nonpositive scale, demonstrating that tension must be strictly positive to maintain the shape; division by zero is avoided.
4. Restore Catenary scale a to 1, then set Half-span L to 0 with Probe fraction u at 1. The supports coincide at (0,0), and the span, sag, length, and slope all become 0. This is labeled the collapsed-span limit.
5. Set Catenary scale a to 0.5, Half-span L to 3 and Probe fraction u to 1. Read finite sag approximately 100.357818061 m and length 201.713157370 m; both supports and the low point fit the independent labelled axes.
6. Set Catenary scale a to 3, Half-span L to 0.00000001 and Probe fraction u to 1. The small positive sag is about 1.66666666667e-17 m, displayed in scientific notation rather than replaced with zero.
7. Compare the solid cyan catenary with the dashed parabola. While they look similar near the origin, they diverge significantly further out.