Automatic Fitting of Conical Envelopes to Free-Form Surfaces for Flank CNC Machining

Michael Bartoň, Pengbo Bo, Helmut Pottmann


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.


  • Pengbo Bo, Michael Bartoň, and Helmut Pottmann.
    Automatic fitting of conical envelopes to free-form surfaces for flank CNC machining.
    Computer Aided Design, 91:84–94, 2017.

Prof. Dr. Helmut Pottmann   +

Projects: C01
University: TU Wien
E-Mail: pottmann[at]
University: King Abdullah University of Science and Technology
E-Mail: helmut.pottmann[at]