Let's cover an example showing the different rotations in different spaces:

1. Open ch04_02_model-view-rotation.html in your browser:

2. As we did in the previous example, we will see the following:
   • A cone at the origin of the world
   • The camera is located at [0, 2, 50] in world coordinates
   • Three sliders that allow us to rotate either the world or the camera
   • A matrix where we can see the result of different rotations
3. Let's see what happens to the axis after we apply a rotation. With the World coordinates selected, rotate the world 90 degrees around the x-axis. What does the Model-View matrix look like?
4. Let's see where the axes end up after a 90 degree rotation around the x-axis:
   • By looking at the first column, we can see that the x-axis has not changed. It's still [1, 0, 0]. This makes sense since we are rotating around this axis.
   • The second column of the matrix indicates where the y-axis is after the rotation. In this case, we went from [0, 1, 0], which is the original configuration, to [0, 0, 1], which is the axis that is coming out of the screen. This is the z-axis in the initial configuration. This makes sense since we are now looking from above, down at the cone.
   • The third column of the matrix indicates the new location of the z-axis. It changed from [0, 0, 1], which as we know, is the z-axis in the standard spatial configuration (without transforms), to [0, -1, 0], which is the negative portion of the y-axis in the original configuration. This makes sense since we rotated around the x-axis:

5. As we've just seen, understanding the rotation matrix (the 3x3 upper-left corner of the Model-View matrix) is simple: the first 3 columns always tell us where the axis is.
6. Where are the axes in the following transformation? Take a look at the following diagram:

7. Check your answer by using the sliders to achieve the rotation that you believe produces this matrix.
8. Let's see how rotations work in Camera space by changing the coordinates, selection.
9. Increase the angle of rotation in the x-axis by incrementing the slider position. What do you notice?
10. Using the sliders, try different rotations in camera space.
11. Are the rotations commutative? That is, do you get the same result if you rotate, for example, 5 degrees on the x-axis and 90 degrees on the z-axis, compared to the case where you rotate 90 degrees on the z-axis and then 5 degrees on the x-axis?
12. Return to World space. Please remember that when you're in World space, you need to reverse the rotations to obtain the same pose, for example, if you were applying 5 degrees on the x-axis and 90 degrees on the z-axis, verify that when you apply -5 degrees on the x-axis and -90 degrees on the z-axis, you obtain the same result.

What just happened?

We've just learned that the Camera matrix rotation is the inverse of the Model-View matrix rotation. We've also learned how to identify the orientation of our world or camera after analyzing the rotation matrix (3x3 upper-left corner of the correspondent transformation matrix).