On the reach and the smoothness class of pipes and offsets: a survey

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.

Automated summary

This paper investigates the relationship between the smoothness of pipes and offsets and the reach of their progenitor. The reach is a fundamental concept that determines the minimal distance between the progenitor and its cut locus. The study shows that under certain conditions, pipes and offsets can inherit the smoothness of their progenitor.