Dovetail math

I want to get the math down for cutting half blind dovetails down in one place, so here we go.

First, the variables from the diagram

• h --- the height of our cut
• bw --- the width of our dovetail bit, at the bottom
• p --- how far apart we cut dovetails
• a --- the angle of our dovetail bit
• top width --- how much wood is left at the top 

We need a few intermediate values, the first of which is the amount of wood the dovetail cuts.

d(h) = h * tan(a)
One quick sanity checks for the math is to see what happens with a straight bit: tan(a) = tan(0) = 0, which is what you'd expect. Another check is for a mythical 45 degree dovetail bit: tan(45) = 1, also as expected.
Now we need to figure out the top width, which is a function of h and p.

top width(h, p) = p - bw + (2 * d(h))
top width(h, p) = p - bw + (2 * (h * tan(a)))

Substituting for a straight bit we get the sensible value of p - bw.
Setting the top width to be equal to the bit width we get:

bw = p - bw + (2*h*tan(a))
2bw - (2*h*tan(a)) = p
p = 2(bw - (h*tan(a)))

Now we have to figure out how we want to round out the pins.

In this case it is just a matter of making two arcs of the same radius as the bit to get nicely rounded pins.