Catenary

You can add the calculus for the stress at the minimun curve value and some safety factor to the horizontal tensions. Further, add a real manufacturer table with usual values for cables and show how to select the appropiate, at least, for the mechanical properties of the cable.

Can relate also the safety factor with some increments on the weight, as the ice.

Also can write the problem as the differential equation of the equilibrium of the tensions for a small cable element, but it is assumed that you want a geometry view point (and post here, in "Algebra & Geometry")

I don't remember if is in the Elgostz's variational calculus book (which have not at hand) or in some other russian author that there are a very nice example that minimizing a functional the result is the catenary, but can't remember which functional it is or how to recostruct the example, but even use differential calculus it is a very 'geometric' example.

For geometrics also, can add some wind and see the curve into the 3d space, or as a vibrational question, checking that resonance frequencies are not closed to the natural freq of the cable.

Regards. Alvaro.