Catenary

Source inspiration: (Mathew 2000-2019).

Animations

The animation below shows the catenary \(y = c\cosh(x/c)\) alongside its parabolic approximation \(y = x^2/(2c) + c\) as the sag parameter \(c\) increases from 0.5 to 4. The catenary is the curve formed by a flexible cable hanging under gravity; the parabola is the first two terms of its Maclaurin series.

Julia source scripts that generated these animations are linked under each case.

Case 1 — Catenary vs parabola approximation as \(c\) varies

Behavior: For large \(c\) (nearly taut cable), the two curves are nearly indistinguishable near \(x=0\). For small \(c\) (slack cable), the catenary hangs deeper and the parabola diverges noticeably at the endpoints. The lowest point is always at \((0, c)\).

Julia source

Catenary y=c cosh(x/c) compared with its parabolic approximation y=x^2/(2c)+c as c sweeps from 0.5 to 4; the two curves nearly coincide near the apex for large c

Derivation Notes (Planned)

Short derivations will be added to explain the core equations and assumptions.

References

Mathew, John H. 2000-2019. Numerical Analysis - Numerical Methods Modules. https://web.archive.org/web/20190808102217/http://mathfaculty.fullerton.edu/mathews/n2003/NumericalUndergradMod.html.