The tops and backs of so called flattop guitars and similar instruments are generally not flat at all, but are spherically domed. The radius of the doming is long, typically between 12' and 30'. A plate is domed by arching the surfaces of the braces that will be glued to the plate, forcing the plate into a dished form, and then gluing the arched braces onto the plate. When the glue dries the plate can be removed from the form and it will still retain the doming.
Traditionally the braces were arched by planing, and checked against a template. This is time consuming, so some folks (I don't know who) thought up a scheme to do this precisely and quickly. The rectangular, uncarved brace is back bent to assume the needed arch and held like that in a jig. Then the concave face of the brace is sawed, sanded and/or planed flat. When the brace is released from the jig it springs back straight, but the surface that was lopped off while the brace was bent into the jig now assumes a nice uniform arch.
A note to folks with math backgrounds. This process uses splines to approximate circular arcs and the radius of the resulting arc will indeed be smaller than that of the curve the jig bends it in to, and will depend on the height of the brace. But the radii we are dealing with here are large and the practical context is woodworking, so the accuracy of this process for this purpose is more than adequate.
Instructions are included here for building the same style of brace arching jig as described by Jon Sevy here. See John's page for nice pictorial instructions for how such a jig is used to arch braces. It is a very useful jig. Since it has a flat bottom and a flat back, the jig can be used against a bandsaw fence to quickly flatten the brace top; it can be used up against a jointer fence to finish off the brace top; or it can be run through a planer to the same effect.
Last updated: September 11, 2018
The first step is to cut two 1.5" x 0.75" x 24' long strips of stiff, dimensionally stable wood. I'm using solid mahogany marine plywood here, which is amazingly stable (and amazingly expensive, too). Cut two 1.5" x 1.5" x 0.75" blocks of the same material (or of really hard wood such as ebony) as well.
Next, measure 12" from one end and mark the center line on the broad surface of one of the long strips. On the same face, mark a line parallel to and 1" from each end.
In the next fews steps we'll mark a line on this surface with the long radius curve needed to arch the brace. We'll do this by bending a spline supported by two end points 22" apart – the two end lines marked in the preceding step. Before doing this we'll need to figure out and mark on the center line how far we need to bend the spline. Here are the displacement values needed for all the commonly used plate doming radii:
12' (144") radius: displacement is 0.42" for 22"
15' (180") radius: displacement is 0.34" for 22" length;
20' (240") radius: displacement is 0.25" for 22" length;
25' (300") radius: displacement is 0.20" for 22" length;
28' (336") radius: displacement is 0.18" for 22" length;
30' (360") radius: displacement is 0.17" for 22" length;
If you want to build a jig for some other radius then use this formula and calculator to calculate the displacement value needed for that radius at 22" length.
If the radius of the arch will be 20' or less then mark the appropriate displacement from one of the long edges on the already marked center line of the strip:
We need at least 0.25" displacement here, so if the desired radius is longer than 20' just mark 0.25" from the edge here at the centerline, then subtract the displacement value from 0.25" and mark the resulting value from the edge at both of the lines that are 1" from the ends of the strip. So for example if we are building a jig for a 30' radius arch the center line would be marked at 0.25" from an edge and each of the end lines would be marked 0.08" (0.25" - 0.17") from that edge too.
If the radius we are using is 20' or less then temporarily clamp one of the two little blocks to the outside of each of the two end lines so they are flush with the top surface of the strip. Then bend a long flexible spline (I'm using a flexible yardstick here) from those blocks down to the displacement line at the center line:
Mark the curve of the spline on the strip from one block to the other. If the radius we are using is longer than 20' then either clamp the blocks so that the top inside corners are located at the end displacement marks, or tack in a brad just under both of these end displacement marks to use as end supports for the spline:
Then bend the spline from the pins or blocks down to the center line displacement point and trace the spline as described above.
After the curve of the spline is traced saw out the waste between the reference edge and the curved line on the bandsaw. Sand down to the line using a spindle sander or drill press mounted drum sander. Check to be sure the curve is smooth by bending your spline into it and holding it at the center. The spline should be in contact with the curved surface of the strip for its entire length.
Now glue the strip just fashioned to the other long strip so all the straight edges line up:
I made sure to line up the ends and keep the bottoms of both strips flat on the table saw extension before clamping here.
Now a mortise is made in the front face of the curved strip to accept one on the small blocks. The block is positioned and the face is marked:
The lines are carried up to the top curved surface. The depth of the mortise should be about 0.625", or so that approximately 0.125" of the curved strip will remain.
The mortise is cut with the band saw and chisel and cleaned up so the block will fit square and snug but is easy to slip in and out.
Take the block out of the mortise. The center of the back strip is marked and a 0.75" hole is drilled 0.375" deep there to provide clearance for a 1/4-20 x 1.75” long hex bolt and washer. The hole is drilled with a Forstner bit. Using the center point of this blind hole, drill all the way through the jig using a 1/4" bit. Put the block back in the mortise and mark through the hole in the jig for drilling the block. Drill the block with whatever diameter bit you need to insert a 1/4-20 threaded insert into the block, but drill this hole at a slight angle so that, when the block is bolted to the jig it will tip slightly toward the back of the jig. This will give the block face a better grip on the brace when the jig is used. Since the block is only supported by the one screw it will tip away from the back when it is tightened against a brace. To keep it from sticking out the bottom of the jig when it does this, slice off the bottom of the block at an angle.
Insert the threaded insert into the block.
Assemble the jig to make sure everything fits right and adjust as necessary. Put a couple of small strips of stick on sandpaper on the part of the block that protrudes above the curve and on the back strip opposite. This will help give the “jaws” part of the jig better grip.
Here's a closeup of the jig holding a brace blank ready for flattening.
For really thick and tall braces it is a good idea to save a piece of the off cut when you trimmed a brace blank to length. Tucking this piece under the clamping block below the screw will keep the block from canting as it is tightened and will keep the clamping face of the block parallel to the side of the brace. There are some dimensions of brace blank that are simply too stiff to be shaped in this jig though and these will have to be marked and either planed by hand or shaped using the belt sander.