Straight homogeneous generalized cylinders: Differential geometry and uniqueness results

Abstract

It is well known that certain objects, like ellipsoids or spheres, admit several, sometimes many, different parameterizations by generalized cylinders. Under what conditions does a given surface admit at most one description by a straight homogeneous generalized cylinder (SHGC)? Under what other conditions does a surface admit at most a few such descriptions? To answer these questions, it is first necessary to understand the geometry and intrinsic properties of SHGC's. In this paper, a necessary and sufficient condition for the regularity of the surface described by an SHGC is given, the Gaussian curvature of this surface is calculated, and it is proved that its parabolic lines are either meridians or parallels of the associated SHGC. These results are used in the second part of the paper to prove several new uniqueness results. It is first proved that if a surface is described by two SHGC's with the same cross-section plane and axis, then these SHGC's are necessarily deduced from each other through inverse scalings of their cross-sections and sweeping rule curve, and that the surface associated with an SHGC with at least two parabolic meridians and two parabolic parallels cannot be described by a different SHGC. Shafer's pivot and slant theorems are then extended to prove that the surface associated with a non-linear SHGC does not have any different SHGC parameterization with the same cross-section plane or the same axis. Finally, it is shown that a surface with at least two parabolic lines admits at most three different SHGC descriptions, and that a surface with at least four parabolic lines admits at most one SHGC description. In particular, a closed surface with at least one concave point admits at most three different SHGC descriptions.