Carving shapes with ruled surfaces for rough machining

Abstract

We propose a novel method for constructing a small set of ruled surfaces, each strictly outside the input shape, for rough machining. To generate such ruled surfaces, our algorithm consists of two phases: (1) generate a small set of covers for the input shape and (2) fit each cover patch using ruled surfaces without colliding with the input shape. Since the normal changes fastest along one of the principal curvature directions and the ruled surfaces cannot be deformed along their rulings, making the rulings and the principal curvature direction where the normal changes fastest orthogonal is more likely to increase the cover areas when using ruled surfaces to fit the surfaces. Meanwhile, we can trace another principal curvature direction to form the base curve. Accordingly, our cover generation step is driven by a smooth cross field approximating the principal curvature direction field. Central to the second stage is an optimization-based fitting to adaptively reduce the approximation error while keeping the ruled surfaces collision-free with the input shape. Our method’s feasibility and practicability are demonstrated through various examples, including three physical manufacturing models using hot wire cutting.