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\ge1 $, of a closed progenitor. These results hold in spaces of arbitrary dimension, for pipe hypersurfaces from spines or offsets to base hypersurfaces.