How do you calculate the axis-aligned bounding box of an ellipse?

Question

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? 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. 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. **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.**

Answer 1

First of all: get a graph of the ellipse for which you need the bounding box so that you can get a general idea of what you need.

The general equation for an ellipse is Ax²+Bxy+Cy²+Dx+Ey+F=0 and B²--4AC=4(e²-1)<1.

To answer your question then you will need to find the maximum & minimum points which will give you the sides of the bounding box parallel to the x-axis. The other sides will be a bit trickier to do, but what you need to do is find the minimum distance from the ellipse to a line of the form y=constant; that will give you the points you need for the other sides of the bounding box.

This will get you what you need. There may be an easier way, especially if you can find the relationship between the equation of the ellipse and foci. Good luck!