I work in lighting. We rig bistro bulb strings a lot, and I'm trying to write a script that will calculate the arc length of cable we need to bring to different installs.
The givens I have to work with are:
- The heights of our rigging points from the ground (often different from each other)
- The span of the run (total horizontal distance)
- The clearance, which is the height of the lowest point of the line from the ground.
Formula for Catenary I'm using:
y(x) = a cosh(x/a) + c
Because we use the same materials for every install, physical factors such as linear weight and cross sectional area do not change. These constants determine the shape of the curve, and therefore its arc length, along with tension, which DOES change every installation, and scales linearly with a.
So far I have assumed the vertex lies at x = 0, which means the given clearance becomes c.
Because the vertex elevation also depends upon the physical properties of the chain, I believe defining an iterative value for tension should allow me to solve for an a that passes through all three defined points in space, but I can't figure out how to set this up. This is my question and what I need help with.
Once I have a process to determine a value for tension that gives me a, I should be able to solve the arc length formula for the newly defined catenary.
I've even found an online tool that calculates vertex position given only the difference between the two rigging points and the arc length, so I know that the reverse must be possible.
Ideally the solution will be generalized so I can compare proposed runs against the tension rating on our cables and know what is dimensionally possible within those physical constraints.