# Catenary curves

A catenary is the curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force. The curve has a U-like shape, superficially similar in appearance to a parabola, but it is not a parabola. The word catenary is derived from the Latin word for ‘chain.’ The curve is also called the alysoid and chainette.

One of the most well-known architectural examples of catenaries is Antonio Gaudi’s hanging-chain funicular models. Gaudi’s method exploits the property of a catenary curve, which describes a chain in pure tension. When the catenary is inverted it becomes a curve in pure compression.

Within Rhino, catenaries can be generate using either Grasshopper’s in built catenary component or via the Kangaroo plug-in. Grasshopper’s catenary component is more accurate as it is calculated mathematically. Kangaroo, on the other hand, is a live physics engine and will therefore give an approximation of the catenary curve. Note that the Kangaroo examples shown below are using version 0.099. If you are using the latest version, 2.02, you’ll need to modify the definitions substantially.

For simple definitions it is easier to use the Grasshopper catenary component. For this to work, ensure that the length of catenary curve is greater than the distance |AB|. To get a hanging chain affect, set the gravity vector to 0,0,-1. To invert the curve, set the gravity vector to 0,0,1.

To create a simple catenary curve using Kangaroo, firstly reference two Rhino points. These will be the anchor points for the simulation. Ensure these points are flattened when connecting them to the ‘AnchorPoints’ input. Next we need to apply a force to this line generated from these two points. Since the force we require is gravity, we can apply a ‘Unary’ force to it. However, to make the element flexible so that we get the hanging chain affect, we must first subdivide the line into smaller pieces and model each segment as a separate spring force at each of these ‘kinks’. We can then extract the start and end points of each of these kinks. Since some of the points will be coincident we can use the ‘removeDuplicatePts’ component to simplify the geometry.

Although we have a Unary force already applied, we also need to create a spring force between each kink. Note there is no Kangaroo input for the start length of a spring – it simply uses the length of the curve input to springs. The rest length of the spring, also called natural or slack length, is the length it ‘wants’ to be. Ensure that the force objects input is flattened. Finally, connect a timer and a Boolean toggle for the simulation to run.

The benefit of using Kangaroo over Grasshopper’s components is that we can create complex catenary structures. For example, it is possible to create a catenary membrane. In this instance, we need to use a mesh instead of a surface. This is because Kangaroo does not process NURBS geometry. If NURBS geometry is required, use a mesh and rebuild it into a NURBS surface after the simulation.

It is also possible to create nested catenary structures. This entails hosting the anchor points of one chain to another chain. By default, Kangaroo joins overlapping control points. Therefore, in order to prevent the curve network from separating during the animation, ensure that that the start and end points of the nested catenary curve rests on the division points of the primary catenary.