I have yet to incorporate catenary curves to any shelters or tarps, but I may just give it a shot on my next shelter... whatever that is.
Most know (and some don't) that catenary curve allows for more equal pressures along a line. On my pyramid shelter I wasn't surprised to see that most of the pressure was on any stakes between the corners of the shelter, and less on the corners themselves. A catentary curve on each corner seam would help even out the pressures. One can figure out a 'cat-curve' mathematically, but the way I would do it is to start with a board that is the same length of the seam I would be using it on. Hang the board on the side of the garage, then hang a small chain with each end of the chain attached to the corners of the board. The amount of curve is whatever the distance is from the top of the board to the center of the chain. Trace a line on the board along the chain. Cut the board along the line. I now have a pattern to use on all eight pieces that make up the four corners.
Some put way too much curve because "It looks cool", but I only see needing about two inches on an eight foot line.

How do you know how long of chain? 2" or 12" or no inches is simply a result of how long the chain is and how much tension there is - but I agree - the curve or profile will be correct. I've used for rigging - push pins and lamp chain on a cork board. Works great.

quote billconner: "How do you know how long of chain? 2" or 12" or no inches is simply a result of how long the chain is and how much tension there is - but I agree - the curve or profile will be correct. I've used for rigging - push pins and lamp chain on a cork board. Works great."

I get you. I should have explained that the chain needs to be longer than the board (or seam) and have X" of sag in it.

The trouble is defining how much "extra" length do you add? 2", 4" 20"- they all produce a catenary curve.

Essentially the same technique used by Gaudi

Gaudi "chain" model

Hanging chain model by Antonio Gaudi, for Sagrada Familia in Barcelona. This naturally creates a curve called a "catenary", which perfectly distributes gravity loads in tension.

Sagrada Familia

(Actually, I have my own pictures of these, but grabbing them online was quicker than finding them).

dd

quote dentondoc: "Essentially the same technique used by Gaudi"

An excellent architectural and physical reference, but let's hope Portagekeeper's work proceeds more swiftly!

quote DanCooke: "The trouble is defining how much "extra" length do you add? 2", 4" 20"- they all produce a catenary curve."

Enough to create the amount of sag needed for the seam to tighten up. Of course for me it's guesswork until I get a feel for how much is enough for a certain length seam. In this case I figure 2" of sag in the chain is enough.

I think a good rule of thumb is that the depth of the cat cut should be 4-6% of the length of the seam. And bending a fair curve into a batten works just as well as a true catenary and you don't need to go thru the extra steps of making a different template for each length. Just find the center of the seam measure up "x" bend your batten so it intersects both corners/ends and the x dimension in the middle, then trace. This isn't a true catenary but I don't think it makes a difference from a performance point of view.