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 $C^k$, $k\ge 1$, of a closed progenitor. These results hold in spaces of arbitrary dimension, for pipe hypersurfaces from spines or offsets to base hypersurfaces.