Straight Homogeneous Generalized Cylinders: Differential Geometry and Uniqueness Results
Abstract
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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Cite
Text
Ponce. "Straight Homogeneous Generalized Cylinders: Differential Geometry and Uniqueness Results." IEEE/CVF Conference on Computer Vision and Pattern Recognition, 1988. doi:10.1109/CVPR.1988.196256Markdown
[Ponce. "Straight Homogeneous Generalized Cylinders: Differential Geometry and Uniqueness Results." IEEE/CVF Conference on Computer Vision and Pattern Recognition, 1988.](https://mlanthology.org/cvpr/1988/ponce1988cvpr-straight/) doi:10.1109/CVPR.1988.196256BibTeX
@inproceedings{ponce1988cvpr-straight,
title = {{Straight Homogeneous Generalized Cylinders: Differential Geometry and Uniqueness Results}},
author = {Ponce, Jean},
booktitle = {IEEE/CVF Conference on Computer Vision and Pattern Recognition},
year = {1988},
pages = {327-334},
doi = {10.1109/CVPR.1988.196256},
url = {https://mlanthology.org/cvpr/1988/ponce1988cvpr-straight/}
}