I am trying to find a description of a rotation in a three-dimensional space with a matrix that uses only 2 angles. It is easy to find one which uses three angles, since I can always consider the rotation on one singular axis at a time and then multiply them together. Still I am sure it can be done with one angle less.
What in my opinion is a starting point is that the group of orthonormal matrices $$\not O(n,\mathbb{R})=R(n,\mathbb{R}),$$ where R is the group of rotations. Now this equality yields a series of equations on which I have been working a little (In this same way you show that a rotation on the plane uses only one angle), but I couldn't find a solution. It would be appreciated to use only basic geometry notions since I am only in the second semester of my math studies!
[Math] Rotation matrix in 3-dimensional space with two angles.
geometrylinear algebramatricesrotations
Best Answer
OK, so you're convinced you can do it with three angles. Since you need at least three parameters to describe all rotations, you're going to have to figure out what to replace that missing information with.
In fact, you can use a single angle, as long as you're willing to encode the rest of the information in a line that acts as an axis for the rotation. To accomplish this, you can just use a point away from the origin. Drawing a line through the origin and that point determines the axis of rotation. You only need one angle in a plane normal to the axis to determine the entire rotation.
So for example, $((1,1,1),\pi/4)$ could be interpreted as parameters to rotate about the axis through $(0,0,0)$ and $(1,1,1)$ with a $\pi/4$ angle. The most sensible way to rotate would be so that the vector $(1,1,1)$ is the upward normal to the plane, and so a positive angle of rotation would be counterclockwise looking "down" onto the plane.
This is already a pretty crisp way to look at rotations, but if you really want to use two angles, you can probably contrive something similar. You could use a point in the $x-y$ plane to determine a line through the origin, which you could then incline/decline by an angle to get an axis of rotation, and you could add the angle of rotation in the plane determined by the axis. However, this is really not much different from the three-angle representation, since the line in the $x-y$ plane could be described as the line $y=0$ rotated in the $x-y$ plane by a certain angle.