Hybrid design - 3D skin projection

Hybrid design: shaped sail
Skin projection concerns the sail's true shape compared to the aerodynamic shape of the foil in flight, when it is arched and when the planform tapers with different profile thicknesses towards the tips. The purpose of the arch/anhedral would be to provide tension along the width of the foil as lift produced by the downbent tips pull outward on the foil providing structural integrity also adding a certain amount of directional flight stability. It can however also deform the panels of the sail between each rib or spar. If the panels were shaped to fit the anhedral curve excess skin billow can be reduced. This would imply that unwanted billow causing parasitic drag is reduced also.
Have a look at a rain gutter pipe for example(the metal ones that have been joined at angles where it kinks or change shape), you could argue that a straight pipe could have been bent to kink that way except it would "wrinkle" and fold on the inside of the bend where there is excess material and loose structural integrity and add flow resistance. It would also "stretch" the material a little on the outside of the bend, again weakening the structure. If it was a heavy pipe carrying lots of rainwater it could tear apart!
Now you could imagine the parafoil skin to form such a tube with tapering edges and smaller inside diameters, almost flattened towards the tip if you will. If that was bent, say due to the curvature of its inflight shape and the bridle you'd get ugly wrinkles and kinks where the excess fabric is compressed or where the shape of the fabric is incorrect to allow such natural deformation. The topskin would stretch in places as well as causing loose fabric in others which would bulge and mishape the intended airodynamic profile.
The problems arrise because of those kinks, bulges and wrinkles. Not only do they produce non-aerodynamic qualities but the also cause stresses within the parafoil that threaten to tear it apart on impact or in strong winds. Single skin hybrids aren't that much different from parafoils except that they only have topskins. But that is the nice part of it! You only have to do half the skin projection calculations.
To achieve this I tried the following methods:
First I tried something a little different. I calculated an imaginary rib between the two ribs you want to calculate the skin of. With an average chord and average thickness of the two ribs and place it in the middle running parallel between the two ribs at the average angle offset. Then I use lines to connect like profile points on the two ribs to like profile points on this imaginary centre rib. When the skin is projected flat, the line where this imaginary profile would've been sewn on is a straight line. Then I calculate the length of these profile lines, firstly in the front view of the kite(this ignores any wingsweep or size difference except thickness) and then lastly in the top view using the length from the first calculation to include wingsweep or chord size differences. Lines connecting like profile points on the two ribs are projected as straight lines using like profile points on the imaginary rib as a reference point only. It is possible to then lessen cell bulge by making the imaginary rib a few percent smaller, placing a little extra tension on the skin(like decreasing the vertical dia. on the circular single cell example on Gene Matocha's site) Very difficult to project onto final templates but possibly as accurate as you'll ever get.

The blue line is the intended distance being calculated. The yellow and green lines were calculated from the given distance in red. This image shows no arch but the calculations are not that much different.
Visualise a crescent planform hybrid(with no anhedral in this case) where the lines connecting like points on the profile describe nearly concentric crescent lines(contour lines) running together or across the wingtips so that the camber runs nearly parallel to the LE. This is the preferable design method. Giving up trying to manipulate the maths to project a skin for this I happened on a near enough accurate solution. Draw the topview of the kite the way it would look if it was inflated/framed. Draw parallel lines horizontally across this at 20(10 is usualy enough) or so equal intervals. See picture below. Draw the aerofoils flat(at the batten locations or in between, your preference) as the would appear if viewed from the side, this way the profile height can be measured at the intervals. The angular/projected width of the skin can now be calculated as you have the difference in height between ribs and the width in the horizontal plane making for an uncomplicated Pythagoras equation.

I would have prefered the camber to run nearly parallel to the LE tapering off into the tips. The following methods were tried for attempting to calculate the best shap for the wingtip panels:
a.This implementation was the prefered method but it is a bit hard calculate and project but can be done with the help of CAD.
b. I drew lines from the 20 or so equal intervals on the wingtip-side battens/profile to the wingtip and using the same method as before(difference in height, horizontal distance) to calculate the projected skin.
Notice how a section of the LE cannot be calculated though!
c. This is the most practical and does not involve major calculating. I mearely chose the effective to average depth of the wingtip panel to divide in the 20 intervals required to produce an accurate contour and calculated the true distance form profile points to points on the LE as divided by this mean line.
Another solution comparable to this is to take the angle offset from the previous panel's contour lines and draw lines from the wingtip-side battens/profile towards the LE adding that offset angle again. It gives the foil a slightly different contour but not alltogether wrong or inefficient either.
Again I used the same method as before(difference in height, horizontal distance) to calculate the projected skin.
d. For purposes of a prototype I merely used the last method which does not use contour lines but merely parallel lines acrros the kite and calculating the lengths thereof. This involved the simplest calculations and only took an hour to plot through Excel. It helps to have the profile section templates measured and cut full size as the profide the measurements for this calculation. My kite is 1100mm high, same as the biggest profile chord, the second being 1000mm. I used roughly 100mm segments to calculate the projected distances using the profile height at that position. A thin carbon fibre spar was used to plot a smooth curve between the points for an accurate approximation of the seam line.
Later I will create an Excel spreadsheet to create the skins for the desired sizes.