This is not a duplicate of my other question in regard to this.
I really am not understanding this rotation of axes stuff. If we want to graph a 45 degree shifted ellipse for example, we can think of it as in the XY plane as a normal ellipse but then when we convert the X's and Y's into equivalent forms in terms of x and y, the graph is graphed the way it was in the XY plane but now in the xy plane. It seems as if this would just give you the ellipse in the xy plane not shifted at all. Every X and Y is given a value in terms of x and y, so aren't these coordinates giving you the same exact thing?
Here is a picture of rotating an ellipse's axis:
Note that although it looks like $x = Y$ and $y = X$, this is not the case. Notice that the coordinates $(x, y)$ are the rotated values of $(X, Y)$ through $\phi$ (which is $30^\circ$ for my picture):
$$ x = X\cos\left(\phi\right) - Y\sin\left(\phi\right) \\ y = X\sin\left(\phi\right) + Y\cos\left(\phi\right) $$
...which would be $x = \frac{\sqrt{3}X - Y}{2}$ and $y = \frac{X + \sqrt{3}Y}{2}$ for my picture ($\phi = 30^\circ$).