I am beginning the process of making a tarp and have been working at making a template to help me layout the curves. In the process I have run across excel spreadsheets, the string method and other ways of making the curve. Since I have access to CAD I thought I would plot these curves in my cad program using the data generated from one of the spreadsheets and then print out a paper pattern that I could use to cut a Masonite template.
I went through a painstaking process of creating the length of the curve and making loft lines and then connecting those points into a curve through cad and I got a nice curve. Then I wondered how this curve differed from a simple radius. So I overlaid a radius on top of the cantenary cut I had plotted with lofting method. I found no significant difference between the curves. However there were very minor curve differences so I took one of the spreadsheets and took all the rounding out of the loft lines so the lines were rendered to .001 of an inch. I then did the same overlay and the lines were identical. I don't know if this makes creating the curves any easier as one of the curves plotted resulted in a radius of 339 inches (3 inch depth on 90 inch lengh) which amounts to a radius of 28 1/4 feet. It did get me thinking if I could take something like dyna glide or zingit and make and attach it to a far point enabling me to cut the radius I needed on my template with my router to give me a nice clean edge. I guess the perfectness of that curve would be ruined by my cutting ability anyway so I abandoned the idea.
The other thing that this showed me is that I could create a longer template and use it on any edge. Since it is essentially a simple radius it doesn't matter which part of the pattern I use to plot the curve. I can simply plop down the template making sure that both end points touch the template and then draw the line. I don't need to measure offsets from each end as I saw in one instructional video I watched.
I am not a mathematician or physics expert but it appears that as long as the depth of the cut is significantly less than the length of the cut a simple radius results. This also makes laying out designs much simpler in CAD. You simply do a single loft point on the middle of the edge and create a three point curve.
I am attaching a couple of pdf files that you can print on a regular printer and then tape together to form a template. There are corner marks on each page, you trim one page to those marks and overlay it on the next page lining up the hatch marks and curve as well and you should end up with a full length template that you can attach to your pattern material and cut out.
The radius for the 3 inch depth on 90 inch run is 339 inches
The radius for the 5.75 inch depth on 80 inch run is 141.75 inches
The radius for the 5 inch depth on 72 inch run is 132 inches