Unraveling the Catenary: How a Kite’s String Shapes the Curve

Catenary

Yesterday on Zhihu I saw an intriguing question about the shape of a kite’s line when wind varies with height, and many answers suggested using the catenary curve.

Definition

The catenary (Catenary) is a curve commonly used to describe a uniformly dense, inextensible chain hanging between two supports under its own weight, forming a downward‑bending shape.

Although its outline looks similar to a parabola, Galileo (1638) proved that the tension varies along the rope, making the true shape a hyperbolic cosine. Hooke derived its mathematical properties in 1670, and later Leibniz, Huygens, and Johann Bernoulli refined the model.

Derivation

Consider the lowest point of the rope where the horizontal tension is T₀. For any point on the curve, resolve forces vertically and horizontally, relate the tangent slope to the tension, and use the linear mass density of the rope. This yields the differential equation whose solution is

y = a \cosh\left(\frac{x}{a}\right) + b

where a is a constant determined by the rope’s weight and tension, and b sets the vertical offset. The half‑length of the rope and the half‑distance between supports appear in the boundary conditions.

History

In 1489 Leonardo da Vinci, while painting a portrait, wondered about the natural curve of a necklace, a problem that remained unsolved for over a century. Galileo guessed a parabola, but Hooke proved it was not. After the invention of calculus by Newton and Leibniz, the problem attracted many mathematicians. In the 1690s the Bernoulli brothers, Euler’s teacher, and others finally provided a complete solution.

Back to the original kite problem

The catenary assumes the rope is suspended solely by its own weight, which is unrealistic for a kite line that also experiences aerodynamic forces. When wind is strong, the weight of the rope becomes negligible compared with wind drag.

To address this, one can extend the derivation to include wind pressure (see Huxley’s answer) or employ numerical simulation tools such as Matlab Simulink (see the “Flying Fish” answer).

Further derivation reference: Huxley’s answer on Zhihu.

Simulation reference: Matlab Simulink model of a kite line.

Spring has arrived—perfect time to fly a kite!