Irregular pyramids puzzle

Question

I'm working on a 3d framework, especially on a conversion from an exotic coordinate system to the Cartesian one.

So I tried to simplify the problem with a pyramid. Simply, on the figure bellow:

we know EF, the alpha and beta angles only. the angle ABE and EBC are right angles.

And we need to calculate BA, BC and EB. It's obviously fully constrained, but I don't find the trick. Some help? :D

Thanks in advance.

Answer 1

Assuming that AFCB is a rectangle, we have by the 3D Pythagorean theorem $$ EF^2 = AB^2 + BC^2 + BE^2 $$ and by trigonometry $$ AB = BE \tan \alpha \qquad BC = BE \tan \beta \tag{1} $$ so $$ EF^2 = BE^2 (\tan^2 \alpha + \tan^2 \beta + 1) \quad \Rightarrow \quad BE = \frac{EF}{\sqrt{\tan^2 \alpha + \tan^2 \beta + 1}}. $$ Once we know $BE$, $AB$ and $BC$ can be obtained from the trigonometric relationships (1).