Mario: I'm aware of that, but the jig I'm thinking of building requires much 'tighter' understanding of the geometry than your swing-style jig, which is elegantly simple, but I just don't see how I can fit it here. I'll double-check all the measurements, as it's the easiest option, but I'll delve into the scribbling and high-school geometric thinking anyway, as an intellectual exercize at the very least.
OK, I'm fairly certain I'm overthinking this entire thing, but I'll do it anyway. I hope my Geometry's not totally off, but if it is, please correct me!
Basically, when you meaure your compound radius fingerboard with a guage set at 90 degrees to the face of the 'board, you're measuring a conic section not perpendicular to the centreline, which my geometry tells me is actually an ellipse, not a circle. Not actually something that has a radius, per se, rather has two axes and two focal pointy things. A very, very round ellipse, but an ellipse still.
That's a view of where the radius (section of circle) 'actually' is, mislabled 'theoretical R' in the top diagram. The bottom picture has, on the left, an exagerrated 'dead on' view of where the circular section in a compound radius board lies, ie demonstrating the end-to-end curvature. The 'straight on' measurements, looking like straight lines, are elliptical sections, not circular.
Take a look at the cone at the top right; that's how your jig (I think) and John Hows is set up; the arms are perpendicular to the surface of the 'board, not to the imaginary 'centreline' of the cone, which is defined by the line drawn between the two pivot points. This means that to get the actual proper 'radius' for that cone, you'd have to use a guage perpendicular to that imaginary 'centreline'. Otherwise you're measuring the elliptic. If the arms were set up as they are on the left hand side (making for a far bulkier, unhandy jig), the length of the swing would actually equate to the 'actual' radius, ie portion of a circular conic section.
To get the 'proper' radius using the John How/Mario method, the swing arms would have to be longer than the desired radius by a factor determined by the angle between the centreline and a line along the 'sides' of the cone (call it one of the imaginary strings). The example I used was Warmoth's 10"-16" compound radius; that's the mess at the bottom, showing a theoretical arm length of 10.6" for the short, and 17" for the long for the desired 10"-16" radius over an 18" stretch.
What it boils down to is all this:
A radius jig that essentially is 'guided' in routed slots, as opposed to acutally pivoting on an arm, would have to have curvature at each end, determined by the angle of the perpendicular, circular section to the 'string' line. That's the top picture. One way to achieve this is make the disks that carry the routed 'slots' at the correct locations (which should be themselves routed on a jig featuring both an angled pivot and the angled cut relative to the 'face') angle so they represent the perpendicular to the centreline.
This will automatically generate the curvature through the vertical travel the carrier board would have to go through when following these 'angled' surfaces.
Now, ultimately, I suspect these elliptical sections differ to a really, really, really tiny degree from the 'true' circular sections, so that we're talking small fractions of milimeters difference at crown heights and curvatures, but for a jig like the one sketched (very much in terms of idea, rather than execution) in the last picture, the angles in particular need to be 'right' if the whole thing is to be stable, rigid, and not subject to the workpiece moving around in incorrect ways.
Thoughts? Corrections? Anything?
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