Hi eric - try to imagine that the network lines must be in the same kind of a grid pattern as longitude and latitude lines on a map.

Your center horizontal line and vertical line are like that, but the outer closed curve is unlike any kind of latitude or longitude line - it touches all the other curves. So because of that it isn't possible for that arrangement to be constant u and constant v curves in a 2D grid.

It could work to have 4 small curves coming to the center point from the closed curve - that would make a pole like at the north pole or something, but once you have the 2 other vertical curves thrown in that won't work anymore.

Split the long closed curve into halves by the horizontal line and then you'll have a networkable layout - the ends will be like the pole areas of a globe.

I'll try to give you some more explanation later.

- Michael