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

Javier Sánchez-Reyes; Leonardo Fernández-Jambrina; Javier Sánchez-Reyes; Leonardo Fernández-Jambrina

doi:10.3934/math.2022435

AIMS Mathematics

2022, Volume 7, Issue 5: 7742-7758. doi: 10.3934/math.2022435

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Survey

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

Javier Sánchez-Reyes ^{1
,
,},
Leonardo Fernández-Jambrina ²

1.
IMACI, ETS Ingeniería Industrial, Campus Universitario, Edificio Politécnico, Universidad de Castilla-La Mancha, 13071-Ciudad Real, Spain
2.
Dep. de Matemática e Informática Aplicadas a las Ingenierías Civil y Naval, Universidad Politécnica de Madrid, Av. de la Memoria 4, 28040-Madrid, Spain

Received: 13 December 2021 Revised: 07 February 2022 Accepted: 10 February 2022 Published: 17 February 2022
MSC : 53-A04, 53-A05, 65-D17

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.
- cut locus,
- level set,
- offset,
- pipe,
- progenitor,
- reach,
- smoothness class,
- spine
Citation: Javier Sánchez-Reyes, Leonardo Fernández-Jambrina. On the reach and the smoothness class of pipes and offsets: a survey[J]. AIMS Mathematics, 2022, 7(5): 7742-7758. doi: 10.3934/math.2022435

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

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