This is an attempt to illustrate VCF's two crossing congruent ellipsoids (from comments to Gerhard), one of which is shifted slightly in the $+x$ direction. It appears to me the shifting joins the top and bottom regions of the vertical ellipsoid, and so reducing the number of regions rather than increasing them. <br /> ![Ellipsoids Crossing][1]<br /> But I may well be misinterpreting the intent... [1]: http://cs.smith.edu/~orourke/MathOverflow/EllipsoidsX.jpg