While creating a perch for our parrot, I stumbled upon something non trivial : if you want to assemble a box with non vertical sides, what cut angle are you supposed to use to have a nice assembly ? It’s obvious that it should be a trapeze, but what would be the α angle ?
Easy peasy, let’s dig into the maths!
On the schematics, once assembly is done, we have B=B’ and A,B, B’, A’ in same plane.
Supposing that we have an angle β with vertical (meaning a standard box has β=0), this means
as OA = OA’, we have trivially
If you need to be convinced, the animation above is done with β=30°. It’s been realized with pov and the code can be found on my POV repository on bitbucket.
And by the way, why did I need that ? Just to make it real :
Bruce's hideout 
And for once I’ll even comment, even if it’s to complain. Your plane figure isn’t the clearest – in particular, I’m not sure if $alpha$ is AOB or BOB’ (and I’m too lazy to figure that out for myself), and it would be nice to have $beta$ as well in there.
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alpha is AOB, beta is not in this plan so can’t be displayed “trivially”
I wrapped my mind several times on how to represent it and I haven’t found a decent solution. A wireframe 3D is hard to draw and impossible to read, 3 faces view only speaks to mechanical engineers… I should actually transform the post into a video, I’ll probably work on this.
Thanks for the feedback,
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I made a video to explain alpha and beta and never posted it: shame on me.
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