Catenary and parabola

This static comparison uses two ideally flexible cables between supports at equal height. The horizontal coordinate is x ∈ [−s,s], so the full span is 2s. Both curves have the same sag f. The inputs are a = 0.4–4 m, s = 1–6 m and catenary weight w = 1–40 N/m of cable. There is no stretching, bending stiffness, strength assessment or combined self-weight and deck-load model.

With S = s/a, the sag is f = a(cosh S−1) = 2a sinh²(S/2). Depth is positive downwards from the supports: y꜀(x) = a[cosh S−cosh(x/a)] and yₚ(x) = f[1−(x/s)²]. Both are exactly zero at x = ±s and equal f at x = 0. The diagram uses x/s and depth/f so small and large sags remain readable. Its horizontal and vertical scales are not equal in metres. The physical span, sag and lengths remain available as numbers.

The catenary carries a constant weight w per metre of cable; its total weight is G = wL꜀. For an explicit comparison at the same total weight, the parabola carries the constant load q = G/(2s) per horizontal metre. This does not mean that both cables have the same weight per cable metre. Their horizontal tensions are H꜀ = wa and Hₚ = qs²/(2f). Each support carries the vertical force G/2. Maximum tensions are T꜀ = √(H꜀²+(G/2)²) and Tₚ = √(Hₚ²+(G/2)²). Along the cables, the tensions are √(H꜀²+[wa sinh(x/a)]²) and √(Hₚ²+(qx)²).

The optional downward arrows show load per horizontal metre: q꜀(x) = w cosh(x/a) and qₚ(x) = q. Their shaft lengths share one linear scale, with a reference arrow and its value in N/m. The curves are sampled at alternating horizontal positions when both loads are shown, to keep arrows distinct. Arrow length is a load symbol, not a physical displacement. Changing the displayed curve or arrows does not change the calculated comparison.

Lengths are analytic: L꜀ = 2a sinh(s/a). For k = 2f/s, Lₚ = s[√(1+k²) + asinh(k)/k]. The nonnegative gap Δ(x) = y꜀(x)−yₚ(x) is zero at the supports and centre. It has equal maxima at ±x*, where z = x*/a is the unique positive solution of sinh(z)/z = 2(cosh S−1)/S² within (0,S). The reported maximum comes from this interior solution, not from the plotted sample grid. The normalized relative gap is Δmax/f; the separate gap plot remains in metres. Marked segments in the combined shape view identify these same locations.

Calculate builds a static result. Editing a, s or w clears old results; Reset keeps the inputs. Returning within the app preserves the result and display choices. Full reloads restore valid numeric inputs only and require another calculation.

Sources: University of Cambridge, From parabolas to catenaries: solution derives the catenary, force balance and arc length; the page is labelled as a draft. NPTEL / IIT Kharagpur, Cables, Lesson 31 §31.3 derives the parabola and tensions for a uniform horizontal load. Equal total weight, the parameter ranges and normalized display are choices of this tool.