Local and global analysis of parametric solid sweeps |
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Affiliation: | 1. Institute of Applied Geometry, Johannes Kepler University, Linz, Austria;2. MTU Aero Engines GmbH, Munich, Germany;1. Escola d''Art i Superior de Disseny d''Alcoi, Barranc de Na Lloba, E-03801 Alcoi, Spain;2. Institute of New Imaging Technologies, Departamento de Lenguajes y Sistemas Informáticos, Universitat Jaume I, Campus de Riu Sec, E-12071 Castellón de la Plana, Spain;3. Institut d''Informàtica i Aplicacions, Universitat de Girona, Campus Montilivi, E-17071 Girona, Spain;1. Department of Electrical and Computer Engineering, Seoul National University, Seoul 151-744, South Korea;2. Computer Science Department, Technion, Haifa 32000, Israel;3. Department of Computer Science and Engineering, Seoul National University, Seoul 151-744, South Korea |
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Abstract: | In this work, we propose a structured computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a “local analysis”, which covers parametrizations and singularities, as well as a “global theory” which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the “funnel” which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface–surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as decomposable and non-decomposable. Further, we construct an invariant function θ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between θ, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach of testing points, θ is a computationally robust global function. It is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included. |
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Keywords: | Solid sweep Boundary representation Parametric curves and surfaces Solid modelling |
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