Rotation & angle of universal joint?

Rotation & angle of universal joint?
Since I got such quick and helpful advice on my piston rod question, here is my next challenge: universal joints!

Specifically, how do I work out the compound angles of a double u-joint connection? See the image below, where I need to connect axle A to axle B by using the short connecting axle that's floating nearby.

Here are some parameters:
  • Axle A is oriented parallel the model's Z axis; axle B is oriented at an angle to it. The connector will have to be rotated in both the X and Y directions, but you can't just set one and then the other, because changing either affects both. I'm guessing the 't' word, trigonometry, will come in again here, but because of the compound angle, I'm not sure where to place the triangles.
  • Axles A and B are rotated 22.5 degrees with respect to their longitudinal axes; my assumption is that the U-joint will transmit the same angle to the connector axle, and that while this rotation needs to be accounted for, it won't change the method needed to determine the connector angle.
  • I guess I need to determine the rotation of the U-joint hubs at A and B, around both of their local X and Y axes. Again, my assumption is that the hubs themselves, however they may be rotated, will always remain at the same coordinates in absolute space; that is, they always form the point around which the connector axle is angled at each end.
  • The end of the axle at B may need to be slid downward along its own longitudinal axis in order to correctly intersect the connector axle; this unknown position makes the trigonometry harder to figure out for me.
Of course, I can (and probably will) just download this model from the OMR and steal the answers from there, but again, I'm keen to know the right way to solve these challenges on my own. And I could also probably work it out by iteratively tweaking each angle until everything appears to line up, but I like to know that I've actually calculated it precisely instead of just doing it by eye.

Looking forward to everyone's suggestions!
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Rotation & angle of universal joint? - by N. W. Perry - 2019-10-02, 3:36

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