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Laying Out and Cutting the Joint

Laying out both wings Arranged like this, the pieces start to look like a boomerang. I always put the lead arm on top of the dingle arm at the elbow. Although people have doubted it, I think that this way, if the wood should shrink, it will do so in a way that increases the angle of attack on the arms rather than decreasing it, which I think is preferable.
Shooting the edges straight and square

I shoot the edges of the two boards flat and square, using the planing aid like a shooting board. Only the inside edge of each wing is important, as that is the part involved in the joint.

The Mujingfang plane is not really well suited for this (the sides are not perfectly flat and square), but with a little effort, it can be done.
Checking the edges for square I check for square against a light source. This is good enough for me.
Scribing the first shoulder of the joint

Eyeballing the angle - I usually aim for something like 75° - I put the boards together at the elbow and scribe a line along the edge of one board. This will be the shoulder of the lap joint.

Technically, I should scribe the other way around, with the bevel of the knife to the waste side. The way I'm doing it, the bevel of the knife puts a tiny chamfer on the shoulder of the lap joint. But those are considerations for people who can work way more accurately than me ;-)
Layout lines for the first half of the lap joint

I square down the shoulder line and add a scribe line showing half the thickness of the wood, marking out what will become the cheek of the joint. To make that line, I set the marking gauge to the exact half width by trying from both sides of the board and making sure I hit the same spot.

After marking the first half of the joint, I leave the marking gauge at the same setting and use that to mark the second wing, too. I mark both wings from the back of the boomerang. That way, I try to get a joint that is as flat as possible on the back of the boomerang, while on the front, there is more room for corrections in the shaping process.  
Using a guide block for the shoulder cut First, I want to cross-cut the shoulder of the lap joint. To guide the saw cut, I clamp on a piece of scrap that is straight and square and align it with my scribed line.
Cutting the shoulder Using the guide, I cut the shoulder, taking care not to go too deep at either end of the cut.
Finished shoulder cut This is what the shoulder cut looks like.

Cleaning Up and Fitting the Joint

There are several ways to remove the waste in the lap joint. I prefer to make a rip cut as close to the layout line as I dare, just like the rip cut used to make the two boards. This is what it looks like afterwards.

Another possibility would be to make multiple crosscuts and chisel out all of the waste. I have tried that, but I have found it to be too tedious and more inaccurate than my current technique.

A variation would be to chisel out most of the waste, but leave the final trimming of the cheek to a small router plane (such as the Stanley 271). I haven't tried that so far.

Here's why I use a wide chisel. Using light, paring cuts, I clean up the cheek of the lap joint. Know where your fingers are, and make sure they are not in front of your chisel!
Occasionally, I check for flatness from different angles. Decide for yourself how accurate you can be. I usually give up at some point with a "not quite perfect, but good enough" joint.
When checking the cheek for flatness, I mark the high spots that need some more work with the chisel; being patient here pays off in the final result.
There will usually be some extra material right in the corner between the two faces I have just cut. Extra care must be taken to clean that up, otherwise the joint will not close properly. 
Before working the end grain along the shoulder, I chamfer the far end, or otherwise the wood will splinter if I plane, file or pare the shoulder in that direction.
This is where the chisel needs to be scary sharp. I check the shoulder with a ruler or against the mating piece, and pare the high spots very carefully, trying to make paper-thin shavings.

Although the spirit of this article is to show people how to make hardwood boomerangs with a moderate investment in tools, this is the one tool which I would put first on the list of "really nice tools I want to buy someday" every time. It is a Veritas Medium Shoulder Plane, or to put it another way, the perfect tool to trim the cheek and the shoulder of the lap joint. 

While this plane is not cheap, it is really a great value for the money, and there aren't any reasonable inexpensive alternatives. I tried fettling some wooden rabbet planes which you get for some small change from the flea market around here. These always look like they have been used as door stops for a couple of centuries and I never got them into a good enough shape that I could use them confidently for fine work.

Depending on where you live, you might get a nice Stanley 92 or something comparable from Ebay.

This is the shoulder plane cleaning up the, well, shoulder. To do this, you need to chop off the far corner of the joint, otherwise there will be blow-out.

Note that I didn't cheat on you at this point :-) - I am just showing how to use the plane, I did not actually use it on this particular boomerang.
At some point in the proceedings, the joint really looked this bad. Don't worry too much - try to pare the two faces flat as far as possible, and check for flatness frequently.  
If the cheek of the joint is flat, it is time to start test fitting. No need to worry about the tear-out on the cheek, that part will be invisible anyway. (Or so I thought; see the picture of the joint below.)

This is the joint starting to look good. Note that there is still a gap, which can have multiple reasons: perhaps there is some material left in the corner between the shoulder and the cheek, or the shoulder and/or the mating piece are not perpendicular to the cheek.

This part is where I spend most of the time, usually. It is just a matter of test-fitting, paring, correcting, more test fitting, etc. 

If the shoulder and the mating edges of the other wing are not parallel both on the front and back, I leave the shoulder of the joint alone and rather try to fit the mating edge. This can be done much more easily with the plane than fiddling around with the shoulder that we just worked hard to get straight.

Usually I don't use a file for fitting the shoulder or cheek, because one tends to round over parts of the joint. Try if it works for you; if you do, work slowly and take care to keep the file straight and square to the cheek.

By the way, I forgot to include this file in my tool overview. It is intended for soft metals such as copper or brass. I got it for some change at the flea market. It works quite well on wood and leaves a very smooth surface.
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