Equivalent Descriptions of Generalized Cylinders

The "equivalence" problem for shape descriptions is that a single three-dimensional shape may have several different descriptions. The Slant Theorem (Shafer) for equivalent generalized cylinder descriptions was proven under the restrictions that the same radius function and the same axis be used for all the descriptions. A proof is given that the theorem still holds when the "same radius function" condition is removed. It does not hold when the "same axis" condition is removed. The ellipsoid is a. counter-example