Question

I hope this is the proper location to ask this question which is the same as this one, but expressed as pure math instead of graphically (at least I hope I translated the problem to math correctly).

Considering:

• two vectors that are orthogonal: Up (ux, uy, uz) and Look (lx, ly, lz)
• a plane P which is perpendicular to Look (hence including Up)
• Y1 which is the projection of Y (vertical axis) along Look onto P

Question: what is the value of the angle between Y1 and Up?

As mathematicians will agree, this is a very basic question, but I’ve been scratching my head for at least two weeks without being able to visualize how to project Y onto P… maybe now too old for finding solutions to school exercises.

I’m looking for the trigonometric solution, not a solution using a matrix. Thanks.

Edit: I found that I needed to determine the sign of the angle, relative to a rotation axis which had to be Look. I posted the final code on my linked question (see link above). Thanks to those who helped. I appreciate your time.

Answer 1

I’m just doing this on paper. I hope it’s right.

Let’s assume Up and Look are normalized, that is, length 1. Let’s say that plane P contains the origin, and L is its normal. Y is (0, 1, 0)

To project Y onto P, find its distance to P…

d = Y dot L = ly

…and then scale the normal by -d to get the Y1 (that is, the projection of Y on P)

Y1 = (lx * ly, ly * ly, lz * ly)

Now normalize Y1, that is, scale it by (1 / length). If its length was 0 then you’re out of luck.

The dot product of Y1 and Up = the cosine of the angle. So

angle = acos(Y1 dot Up)