There's a reason why upright basses are the size they are. The wave lengths for the lowest notes are so long that the top doesn't even start to become efficient at making those sounds until it's four feet long or more. If you're thinking of an electro-acoustic that's another matter, since you can rely on the amp the fill in the low tones.
Designing arches using the 'curtate cycloid' method is really easy. In fact, if you download Mike Mahar's free 'Luthier Lab' Android app it's got a utility for that. You can work up an outline (also in LL), specify the lengthwise arch, and it will print out a full sized contour map for you. I've always done it the old fashioned way, but that's not too hard either. First; about 'curtate cycloids'.
If you mark a spot on the rim of a wheel, and trace out the path of that spot as the wheel rolls along a straight line, you get a 'cycloid' curve. Cycloids were sexy math a generation or so before Strad worked, since they were the basis of the Ptolemaic system of astronomy, and the math was all worked out then. By Strads' day it was being used in crafts. The full cycloid curve is a series of humps up from the points along the plane where the rim touches the surface. If you mark a spot partway in from the rim toward the axis of the wheel you get a 'curtate cycloid'. Several years ago somebody got the idea that this may have been the system the old boys used to draw arches, and checked it out. Given the distortions of age and normal uncertainties, it's about as close as any other system, and it's much simpler than many.
Basically, you start by specifying the lengthwise arch along the outline of the instrument. The high point will be at the bridge, and I usually make the low points around the edge at the inside edge of the liners. You thus specify the height of the arch at any line across the body, and the length between the low points.
What you need now is a disc that will roll once along that line from one end to the other. If you remember that pi~22/7 (as any good craftsman would) it's not too hard to make one from some card stock. If you punch a hole in the disc that's 1/2 of the arch height out from the pivot point you can stick a pencil in the hole and use it to draw out the cross arch by rolling the disc along a straightedge. I find it helps a lot to rosin the straight edge, so the disc doesn't slip so much. draw that on some card stock and cut it out to use as a template. Remember that you're drawing the outside of the arch, from the lowest point around the edge to the top, and subtract the thickness of the plate at the edge from the arch height off the surface. That is, if the arch as measured over the top is 18 mm, and the edge will be, say, 3.5 mm, you punch the hole in the disc (18-3.5)/2 mm from the pivot, or 7.25 mm.
You can lay out as many cross arches as you like on an outline of the top. Find all the places along the cross arches that are 2 mm above the low point at the edges, and connect those with a nice smooth line. Repeat at 2 mm intervals until you've got all of the contour lines. Label these clearly with the height above the bench, adding in the thickness of of the edge at the low point. If that's 3.5 mm than the lines are at 3.5, 5.5, and so one. Here's a shot of a couple of different contour plots for the same size of guitar but different long arches.
These patterns can be placed over the wood to trace the outlines of the contours. I remove most of the wood with a drill press planer, starting with establishing the high point at the top and working down by positioning the half pattern on the center line using the high point, and flipping it. This produces the stepped blank.
I note that I don't cut down to the lowest (3.5 mm) contour this way, to allow for extra thickness at the edge for the binding. I've used the Dremel setup for cutting violin purfling slots, and a 1/8" bit, to make a channel in from the edge almost down to that level for guidance here. It's a (relatively) easy matter to smooth out the arch from there.
Hope this helps.
