A straight ridge line in a tarp.
Consider the ridge line of the simple tarp shown here. There are crinkles all along the length. When it rains, these crinkles fill up with water (yes, a hose test confirmed that). Also, when the wind blows, the tarp is going to flap around the ridge line. Can one do anything about this?
Curved Ridge Lines
The solution is of course well-known: you cut a curve into the ridge line, as shown in these two pictures, solving the problem - provided the curve is done correctly for that tarp or tent.
Example of a non-catenary curve on a ridge line.
However, it would be a serious mistake to think that the curve has to be a catenary curve. Other solutions to this problem exist, as shown here. This is a Macpac Olympus mountain tent, and I can testify that it can handle a storm without moving. In this case the ridge line is far more complex and is designed to keep the tent poles in exactly the right place.
However, the catenary curve has become traditional, and many people would like to be able to create one for their MYOG designs.
Mathematics - the Curve
The origin of the catenary curve is appropriate for this: it is the curve taken by a freely-hanging chain suspended from the two ends. The simple version of the mathematics is for the case where the two ends are at the same height. Now while this seems appropriate, there are several cautions to be mentioned.
First, many tarps do not have their two ends at the same height. We may ignore this for our purposes, as the next point will dominate.
Second, the curve is for a freely-hanging chain, not for a chain with all sorts of tension being applied along its length. A tarp has guy ropes at various points, which will always interfere with the pure curve. Once again, we may ignore this for our purposes provided the curve is not too severe.
The basic catenary curve.
The curve itself is described by a COSH(theta) function. Fortunately, this function is available in most spreadsheets. We have plotted out a version here just as an example. You can calculate your own curve using this spreadsheet. You must change the 'a' and 'b' coefficients to suit your case. In this case, 'a' is the HALF-length of the curve, while 'b' is the amount of sag in the curve. Note that you MUST use the same units for the two parameters. The units can be feet, metres, furlongs or hand-widths - the math doesn't care.
Once you have fiddled the curve to describe what you want, print out the table of XY values and plot them out on your fabric. Note that the curve generated may be used to describe the seam on your fabric OR the profile of the ridge line. If you put a three-inch sag in the curve on your fabric, the actual sag in the ridge-line will be less than this. However, this doesn't matter one iota.
I suggest you experiment a bit before cutting out the curve on the ridge line of your fabric. Cut the fabric out with a straight ridge line - no curvature. Then mark out the line for a small sag, pin it together and string it up. Now you can see what that looks like in reality. If there isn't enough sag, take it down, increase the sag parameter (b) a little, and try again. If you approach this from the 'not-enough-sag' side, rather than overestimating and having to reduce the sag, the pin holes will always be on the bit you cut off.
Once you have what you want, add the hem allowances, cut, and sew.