Dan Asimov wrote:

If we replace R^3 by its "projective compactification" by adding the P^2 of the "lines at infinity", the result is P^3, projective 3-space. ... Then the conic sections become the intersection of one of these conoids with an "affine projective plane" in P^3. All the nondegenerate intersections are equivalent to ellipses, but the segmentlike degenerate ellipse at issue becomes a circle. (At least it's compact.)

This is a nice point of view. Thanks, Dan! Jim Propp