Confusion about Rotation matrices from Euler Angles

Question

I am trying to learn more about Euler Angles so as to help myself in understanding how I can control my camera better in the game.

I came across the following formula that converts Euler Angles to rotation matrices:

In the equation, I could see that the first matrix from the left is the rotation matrix about x-axis, the second is about y-axis and the third is about z-axis.

From my understanding about ordinary matrix transformations, the later transformation is always applied to the right hand side. And if I'm right about this, then the above equation should have a rotation order starting from rotating about z-axis, y-axis, then finally x-axis.

But, from the symbols it seems that the rotation order start rotating about x-axis, then y-axis, then finally z-axis. What should the actual order of the rotation be?

Also, I am confuse about if the input vector, in this case, would be a row vector on the left, or a column vector on the right?

Answer 1

It really depends on how you write your vector. DirectX uses row vectors, OpenGL uses column vectors. The matrices need to be transposed accordingly.

If you consider your vector a column vector, it goes to the right: R_x R_y R_z v. But if it's a row vector, it needs to go on the left: v R_x R_y R_z.

The order also changes depending on this definition, because the matrix "closest" to the vector is applied first. So for a column vector, the order would be z, y, x, but for a row vector it's x, y, z.

And then it also depends on how your Euler angles are defined. Wikipedia gives a bunch of different options with corresponding matrices: http://en.wikipedia.org/wiki/Euler_angles#Matrix_orientation

Answer 2

With matrix transformation, it always works right to left. The right most matrix is 'applied' first.

Also, input vector would be column on the right.

Rule of thumb: the matrix closest to the vector goes first.