How to rotate at all axes simultaneously with a 3x3 rotation matrix?

Question

I've been working on an OpenGL project and currently I'm attempting to rotate the camera by using the input from the cursor movement. To rotate the orientation of the camera I need to edit a 3 length vector by multiplying it with a 3x3 rotation matrix but instead of making a separate rotation matrix for each axis I have to calculate all the rotations simultaneously with one 3x3 matrix. I am not using GLM to do the rotation but instead I'm using my own function. Currently, I have made a function which will rotate the camera on the X axis which won't work due to the fact it only rotates on a singular axis. The original glm function redirects to a quaternion rotate function that takes a 4x4 matrix, an angle and a vector as prerequisites. Does anyone have a solution?

What I have so far: $$ \begin{bmatrix} cos(angle) & -sin(angle) & 0 \\ sin(angle) & cos(angle) & 0 \\ 0 & 0 & 1 \ \end{bmatrix}* \begin{bmatrix} v.a & v.b & v.c \end{bmatrix} $$

What the rotation is supposed to look like: https://drive.google.com/file/d/1MJKgO_3tQPFxVLlO007csKyyyM8ih_J8/view?usp=sharing

Currently it is only able to rotate on a single axis.

Answer 1

Yep, this is quite doable! To be precise, you wouldn't be rotating about "all axes", you'd be representing your rotation as a rotation about a single axis that would be a generic unit vector not necessarily pointing along a Cartesian axis. The details are a bit messy, but easy enough to plug into and you can find them on Wikipedia's "Rotation matrix", subsection "Conversion from rotation matrix to axis–angle":

https://en.wikipedia.org/wiki/Rotation_matrix#Conversion_from_rotation_matrix_to_axis%E2%80%93angle

I won't write out the details, but the idea is that your rotation matrix should have the following properties:

• it's a $3\times3$ real matrix (so it turns a real three-vector to a real three-vector).
• it has determinant one (so it rotates without stretching).
• it maps the axis of rotation to itself.
• its trace is $1 + 2 \cos \theta$, where $\theta$ is the angle of rotation (this is clearly true for rotations about Cartesian axes, and the trace is independent of the axis).
• it rotates in the proper direction: that is, clockwise or counterclockwise.