Would a bounding box be a possibility?

It certainly cuts down on the runtime tests for those points outside the bounding box, but if you imagine that there is nothing of interest outside the bounding box; it becomes the edges of the image.

Ie. The image is scaled so that the bounding box of the clipped region becomes coincident with the screen or viewport. Then you back to square (or rectangle :) one.

I'm just wondering if there is any mathematical way to "combine" the formulae of two or more partially overlapping ellipses? Trouble is, I haven't a clue what to search for.