It's funny how the seemingly easy can be ridiculously hard. A relative wanted a replica of a serving tray that belonged to a family member, and I was asked to work out the mitre angles because I'm the math guy. "Sure, no problem" I naively said. All that was required was to tilt the tops of all the sides outward so that the timber was 25 degrees from vertical.

Angled Frame 
Before going too far it seemed prudent to cut a test using some scrap wood. The first step was to cut a 25 degree taper on the edge of the moulding that's in contact with the base. That was easy. I then came unstuck trying to calculate the angles required to make the corners fit together at 90 degrees. I ended up using an awesome
online calculator to calculate the angles for convenience. It may seem easy, but in this case the mitre angle needed to be cut at about 22.9 degrees, while at the same time the head of the saw needs to be tilted at 39.9 degrees. It took me a while to figure out where those numbers come from, but suffice to say it's not the simplest of maths.

Undercut Mitre 
The moulding wasn't to easy to work with as it doesn't have many flat surfaces.

Undercut Mitre 
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Mitre 
The corners don't fit together that well as I didn't take extremely accurate measurements. It also doesn't help that the parallel edges aren't exactly the same length.

Frame Corner 
One problem I currently have is that I don't have a good way to measure angles. That should be solved soon. Combining the results so far with a little more accuracy should make the job a lot easier.

Mitre Corner 
In the image below you can see the complex shape of the wood I'm using for the test. The actual serving tray will be made out of plain rectangular material making things a lot easier.

Profile Shape 
I do have a method to calculate all the angles needed to make compound mitre cuts. It makes use of vectors and exploits some of their abilities, but I'll leave that for another time.
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