Characterizing envelopes of moving rotational cones and applications in CNC machining

Abstract
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.

Citation
Skopenkov, M., Bo, P., Bartoň, M., & Pottmann, H. (2020). Characterizing envelopes of moving rotational cones and applications in CNC machining. Computer Aided Geometric Design, 83, 101944. doi:10.1016/j.cagd.2020.101944

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.

Publisher
Elsevier BV

DOI
10.1016/j.cagd.2020.101944

arXiv
2001.01444

Additional Links
https://www.sciencedirect.com/science/article/pii/S016783962030131X

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