I was reminded that I had intended to show how the compound joints are generated geometrically.
To start with, most of the finger is just a rectangle with a cylindrical top, and the base where the fingers meet the main body is also a cylinder. The challenging part is the compound curve where the tops of the fingers meet the base, which is not entirely obvious.
This compound curve can be generated by starting with a circle corresponding to the curvature of the base, taking the point where it touches the main body of the finger, and sweeping upward in a circular arc over the top of the finger (which is already a cylinder).
When sweeping the circle in this way, it must stay horizontal. If it were to tilt, for example if you revolve around the cylinder on top of the finger, it would produce a torus and that is not the shape we’re after. It must stay flat, i.e. it looks like this when viewed from the end.
The disc by itself would leave some extra material behind (circled below) so the circles are extended to a hull between two circles, so it doesn’t leave any garbage behind after performing the subtraction.
It is tricky to explain why this compound surface works, partly because I understand almost, but not quite 100% why this works. To test that it works I took a solid block and subtracted two copies of the finger, offset and rotated corresponding to how a box joint is constructed. By taking this negative shape, it would be the ideal opposite mating finger shape. And I confirmed that the mating shape and the finger shape are the same shape, so it mates perfectly with itself.
Now for simplicity it’s much easier not to worry about all the compound curve, and simply cut a channel so you are guaranteed clearance. This channel is deep enough to remove the entire top cylindrical part, which is half the finger width. The width of the channel must be half the finger width too, to remove all the ‘green’ portion in these diagrams, although it can be wider.
I have only attempted these cylindrical box joints but it would also be possible to make rectangles with rounded corners. Conceptually, you can split the finger in half down its length and insert a rectangle, where then the finger width is more than twice the roundover radius. You would widen the space between the fingers by the same amount.