<p>If the major axis of the ellipse is vertical or horizontal, it's easy to calculate the bounding box, but what about when the ellipse is rotated?</p>

<p>The only way I can think of so far is to calculate all the points around the perimeter and find the max/min x and y values. It seems like there should be a simpler way.</p>

<p>If there's a function (in the mathematical sense) that describes an ellipse at an arbitrary angle, then I could use its derivative to find points where the slope is zero or undefined, but I can't seem to find one.</p>

<p><strong>Edit: to clarify, I need the axis-aligned bounding box, i.e. it should not be rotated with the ellipse, but stay aligned with the x axis so transforming the bounding box won't work.</strong></p>

I think the most useful formula is this one. An ellipsis rotated from an angle phi from the origin has as equation:
where (h,k) is the center, a and b the size of the major and minor axis and t varies from -pi to pi.
From that, you should be able to derive for which t dx/dt or dy/dt goes to 0.