Automatic fitting of conical envelopes to free-form surfaces for flank CNC machining

Pengbo Bo, Michael Bartoň*, Helmut Pottmann

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


We propose a new algorithm to detect patches of free-form surfaces that can be well approximated by envelopes of a rotational cone under a rigid body motion. These conical envelopes are a preferable choice from the manufacturing point of view as they are, by-definition, manufacturable by computer numerically controlled (CNC) machining using the efficient flank (peripheral) method with standard conical tools. Our geometric approach exploits multi-valued vector fields that consist of vectors in which the point-surface distance changes linearly. Integrating such vector fields gives rise to a family of integral curves, and, among them, linear segments that further serve as conical axes are quickly extracted. The lines that additionally admit tangential motion of the associated cone along the reference geometry form a set of candidate lines that are sequentially clustered and ordered to initialize motions of a rigid truncated cone. We validate our method by applying it on synthetic examples with exact envelopes, recovering correctly the exact solutions, and by testing it on several benchmark industrial datasets, detecting manufacturable conical envelope patches within fine tolerances.

Original languageEnglish (US)
Pages (from-to)84-94
Number of pages11
JournalCAD Computer Aided Design
StatePublished - Oct 2017

Bibliographical note

Funding Information:
We thank Denys Plakhotnik for providing the datasets for Figs. 17 and 18. This research has been supported by the European Community's 7th Framework Programme under grant agreement 286426 (GEMS). The first author has been partially supported by The National Natural Science Foundation of China (Grant No. 61672187) and Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.201711). The second author has been partially supported by the Basque Government through the BERC 2014–2017 program and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

Publisher Copyright:
© 2017 Elsevier Ltd


  • 5-axis CNC machining
  • Flank milling
  • Free-form surface
  • Shape manufacturing
  • Tangential movability

ASJC Scopus subject areas

  • Computer Science Applications
  • Computer Graphics and Computer-Aided Design
  • Industrial and Manufacturing Engineering


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