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.
         Ellipsoids Crossing http://cs.smith.edu/%7Eorourke/MathOverflow/EllipsoidsX.jpg
But I may well be misinterpreting the intent...

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.
         
But I may well be misinterpreting the intent...