Enter the catenary parameter (a), x-range, and step size to compute points along a catenary curve — the shape formed by a hanging rope or chain under gravity. You get the y-values at each x, the arc length, the sag, and a plotted curve — all based on the classic formula y = a · cosh(x/a).
Arc Length (L)
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Sag (Max Deflection)
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y at X Start
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y at X End
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y at Vertex (x=0)
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Max Tension (at Endpoints)
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Number of Points Computed
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A catenary curve is the shape a perfectly flexible, uniform rope or chain forms when it hangs freely under its own weight between two points. The word 'catenary' comes from the Latin 'catēna', meaning chain. Despite resembling a parabola, it is a distinct mathematical curve described by the hyperbolic cosine function.
The standard catenary equation is y = a · cosh(x/a), where 'a' is the catenary parameter equal to the ratio of horizontal tension to weight per unit length (a = Tx/w). The vertex of the curve sits at (0, a), the lowest point of the hanging rope.
The arc length L of a catenary from x₁ to x₂ is given by L = a · [sinh(x₂/a) − sinh(x₁/a)]. For a symmetric span from −x to +x, this simplifies to L = 2a · sinh(x/a). This calculator computes the arc length automatically from your entered parameters.
Both look similar but have different equations. A parabola (y = x²) describes the path of a projectile under uniform horizontal loading, while a catenary (y = a·cosh(x/a)) describes a rope hanging under its own distributed weight along its length. The catenary is 'flatter' at the bottom and rises more steeply toward the endpoints.
The parameter 'a' equals horizontal tension (Tx) divided by weight per unit length (w): a = Tx/w. A larger 'a' produces a flatter, wider curve (less sag), while a smaller 'a' results in a tighter, more pronounced sag. The minimum y-value of the curve equals 'a', occurring at x = 0.
Catenary curves appear in architecture (the Gateway Arch in St. Louis is an inverted catenary), suspension bridge cable profiles, power line sag calculations, ship anchor chains, and even in nature like spider webs. Engineers use catenary equations to compute tension, sag, and required cable length in structural designs.
The horizontal tension Tx is constant throughout the rope. The total tension T at any point x is T = a·w·cosh(x/a) = Tx·cosh(x/a), which is maximum at the endpoints where the curve is steepest. The vertical component Ty at each end equals w times half the arc length supported by that end.
Sag is the vertical difference between the attachment points and the lowest point of the rope. For a symmetric catenary with endpoints at x = ±x_max, sag = y(x_max) − y(0) = a·cosh(x_max/a) − a. A greater weight-to-tension ratio increases sag.