Characterizing envelopes of moving rotational cones and applications in CNC machining

Mikhail Skopenkov, Pengbo Bo, Michael Bartoň, Helmut Pottmann

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address developable surfaces and ruled surfaces. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.
Original languageEnglish (US)
Pages (from-to)101944
JournalComputer Aided Geometric Design
StatePublished - Oct 20 2020

Bibliographical note

KAUST Repository Item: Exported on 2020-10-21
Acknowledgements: Three of the four coauthors are grateful to King Abdullah University of Science and Technology, where they met altogether and started this project. The authors are also grateful to R. Bryant, S. Ivanov, and A. Skopenkov for useful discussions. The first author has been supported within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018-2019 (grant N18-01-0023) and by the Russian Academic Excellence Project “5-100”. The second author has been partially supported by the National Natural Science Foundation of China (61672187) and the Shandong Provincial Key R&D Program (2018GGX103038). The third
author has been partially supported by Spanish Ministry of Science, Innovation and Universities: Ram´on y Cajal with reference RYC-2017-22649 and the European Unions Horizon 2020 research and innovation programme under agreement No. 862025.


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