There is a simple relationship
Hi. I'm a brand new member here, and this is an old thread. I don't really see an answer to the original question here. So it seems I have something to offer.
Many decades have passed since my F in shop class, I have 4 college degrees, and I am doing much better now. I am a "re-beginner" at woodworking.
Here is the math. If the explanation below is hard to visualize, please tell me. I can put an illustrated version on my web site.
Draw a sketch of the interface between the two dovetail-joined parts. It should look like a jagged line of opposing trapezoids. I'll try to do that with ascii art here (if you used a fixed-width font, it might work):
----- ----- -----
\ / \ / \ /
----- ----- -----
It doesn't matter whether this is a blind or through dovetail joint. The analysis below focuses only on the geometry of the joint interface. (How that lines up with your workpiece determines what kind of dovetail joint it is.)
Define some variables:
h = depth of cut (distance from top to bottom of my diagram above)
d = diameter of router bit at widest point
p = period of the dovetail pattern (distance from corner of one tail (or pin) to the next tail (or pin)
e = angle of dovetail bit. This is the angle between any of the slanted lines in my diagram and the vertical, and corresponds to the way bit angles are defined in typical bit specifications. (The total angle subtended by the bit, a pin, or a tail is 2e.)
The universal relationship between these variables is
h sin(e) = d - p/2
I just purchased an Incra Incremental Positioning Jig Model 25971. It comes with a couple of templates, one for box joint and another for dovetails. These templates are only good for two particular dovetail bit geometries and a couple of stock thicknesses. Rather than buy more templates, I can use this formula to make my own, for any bit geometry and stock thickness.
The distance between any line and the next in an Incra template is p/2, and the distance between any A line and the next A line or any B line and the next B line is p.
All I have to do is check and maybe calibrate the scaling accuracy of my printer and I can crank out any template for any geometry with a simple computer program. The printer scaling can even be slightly off, and the Incra incremental positioning jig will get it perfect as long as p is an integral multiple of 32nds of an inch.
Thomas D. Shepard, Sc. D.
"Where are your hands?"