How to convert quaternion to spherical angles of rotation?

Question

Mathematics

Good afternoon!
Please tell me where you can find a mathematically painted transformation of quaternions into spherical rotation angles and vice versa. I found such a conversion program:
but I could not figure it out

Answer 1

In any book on 3d graphics, for example for game developers. In this example, the following happens. A quaternion is a rotation around a vector, it is normalized, i.e. its length is 1, and it also has a fourth coordinate, unlike a 3d vector, so it's a 4d vector. If we say that the vector around which the rotation occurs V and the angle alpha, then q == { sin(alpha / 2) * Vx, sin(alpha / 2) * Vy, sin(alpha / 2) * Vz), cos(alpha / 2) }. The first line actually highlights the fourth component, although it would be more accurate:
half_cos_alpha = q->w;
The second line gets the sine of the half angle, according to the Pythagorean theorem, and given that V is a unit vector, thus:
half_sin_alpha = sqrt(1.0 - half_cos_alpha * half_cos_alpha)
And so on :) (considering that the text is clearly not complete and sa is not initialized at all). The idea is probably to rotate the point (unit vector) {0, 0, 1} using this quaternion, and then determine the latitude and longitude of this point (as a unit vector).