The author studies the differential geometry of straight homogeneous generalized cylinders (SHGCs). He derives a necessary and sufficient condition that an SHGC must verify to parameterize a regular surface, computes the Gaussian curvature of a regular SHGC, and proves that the parabolic lines of an SHGC are either meridians or parallels. Using these results, he addresses the following problem: under which conditions can a given surface have several descriptions by SHGCs? He proves several results. In particular, he proves that two SHGCs with the same cross-section plane and axis direction are necessarily deduced from each other through inverse scalings of their cross-sections and sweeping rule curve. He extends Shafer's pivot and slant theorems. Finally, he proves that a surface with at least two parabolic lines has at most three different SHGC descriptions, and that a surface with at least four parabolic lines has at most a unique SHGC description.
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