Imagine there are two unrotated ellipses in 2d with different major and minor axes (that is to say different ellipses, but also consider case where ellipses have proportional major and minor axes, so same ellipses just that one is bigger than the other one) centered at origin. Should the shortest distance from one ellipse to the other be in the direction of either major or minor axes, or is it something completely else?
Overlapping ellipses centered at origin:
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conic-sections