@article{Sánchez-Reyes2022, 
author = {Javier Sánchez-Reyes and Leonardo Fernández-Jambrina},
title = {On the reach and the smoothness class of pipes and offsets: a survey},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {5},
pages = {7742-7758},
keywords = {cut locus, level set, offset, pipe, progenitor, reach, smoothness class, spine},
url = {https://www.sciopen.com/article/10.3934/math.2022435},
doi = {10.3934/math.2022435},
abstract = {Pipes and offsets are the sets obtained by displacing the points of their progenitor    S (i.e., spine curve or base surface, respectively) a constant distance    d along normal lines. We review existing results and elucidate the relationship between the smoothness of pipes/offsets and the reach    R of the progenitor, a fundamental concept in Federer's celebrated paper where he introduced the family of sets with positive reach. Most CAD literature on pipes/offsets overlooks this concept despite its relevance, so we remedy this deficiency with this survey. The reach admits a geometric interpretation, as the minimal distance between    S and its cut locus. For a closed    S, the condition    d  <  R means a singularity-free pipe/offset, coinciding with the level set at a distance    d from the progenitor. This condition also implies that pipes/offsets inherit the smoothness class        C    k  ,    k  ≥  1, of a closed progenitor. These results hold in spaces of arbitrary dimension, for pipe hypersurfaces from spines or offsets to base hypersurfaces.}
}