A Small Dovetail Problem, Graphical vs Mathematical Solution

This was an interesting problem I came up against the other week. I needed to determine the angle of some tiny dovetails used to secure the front sight of an air rifle. I used two methods: one, a graphical solution that leveraged the power of my CAD program, the other a purely mathematical one.

I measured the dovetail over some PeeDee thread wires. Then I repeated the measurement with a second set of different diameter wires.

Here’s the result. By drawing the diameters and distances of the wires in my CAD program I was able to draw a line tangent to the two circles (the thread wires) and measure the angle. It’s very quick. The alternative?

The Machinery’s Handbook formula for measuring dovetail slides is x=D(1+cot 1/2θ)+a

I used the pair of measurments (x1,D1, x2, D2) to solve the unknown “a” and then used that value to solve for θ. 

 ((x-a)/D)-1 = cot 1/2θ
((x1-a)/D1)=((x2-a)/D2)
Solve for a, then solve for θ I won’t spoil the fun for you. Plus I might have made a humiliating mistake somewhere.

This process nets the same result of the graphical solution after much scribbling, erasing and calculator use — so before reaching for the calculator it’s always a good idea to see if a graphical solution will be easier to achieve. 

In any case, the dovetails are likely 60 degrees, but the measurements were thrown off by the small magnitude of the measurements, burrs, etc. which highlights another limitation of such calculations.

Machinery’s Handbook [Industrial Press]
Street Pricing [Google Products]
Via Amazon [What’s This?]