I know this is an old thread, but.... Question. The height of the curve seems to be somewhat arbitrary. And the rule of thumb is that it's 1" of height for each running foot (width) of the span, so a 12' wide tarp would need a 12" depth. The calculator allows height to be an independent variable.
Seems to me that this curve should be based on the stretch of the fabric and the width, so a very stretchy fabric would need more height of curve.... But I'm really just guessing at this.
Second thought: What if instead of cutting and hemming the catenary, instead you just ran a raised gather(?) using very sturdy series of stitches, following the catenary curve. Then you wouldn't have to cut the fabric. These 'gathers' would be wider near the corners and wouldn't have to meet in the middle. This would stretch the fabric out more along the baseline, to match the tension along the catenary.... I'll include my sketch. (My sketch has two catenary gathered seams because Xtrekker's cool calculator makes it easy to calculate.)
Keep in mind the 'gaps' (white areas) along the catenary lines are to be pulled together and stitched. The top drawing was my original idea, using different panels, but then I realized you could probably just gather together fabric and stitch it. The 'gaps' at the bottom would be bast on the specific stretch of the particular fabric, so polyester would have a smaller gather than nylon (because nylon stretches more).
It seems to me that the issues with a flapping tarp are 1. the bottom edge, and 2. an general area in the tarp where tension seems to drop because it's spread out. It's an interesting puzzle that once sorted should allow for tarps of any shape. (and likely this has already been figured out and I've been googling the wrong terms while trying to reinvent the wheel.....)
Will
Catenary Conjecture.pdf
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