My proof above can be confusing because I don't make that adjustment until after the rotation with respect to the standard metric. The formula for doing a rotation of angle and dilation by factor k around the point (0, 0) is (x y) k( cos sin sin cos)(x y). This single correction factor does not allow for any rotation of the principal axes. I need the rotation calculation to somehow know how to rotate the camera down by 45 degrees and take direction into consideration too. in either the x, y, or 45 degrees from the x and y directions. Before rotating the camera I need to remember all its rotations, then only adjust the Rotate(45,currenty, currentz). As noted, that transformation isn't an isometry (in the Euclidean metric) it's a $45^\circ$ rotation composed with a dilation by $\sqrt)$ to preserve the length of the rotated vector. The problem is the camera moves and rotates.
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