Local and global analysis of parametric solid sweeps

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 nondecomposable. 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.

A Computational Framework for Boundary Representation of Solid Sweeps

“…In such cases, the correct construction of the envelope requires appropriate trimming of the contact-set. We refer the reader to [4] for a comprehensive mathematical analysis of the trimming and the related subtle issues arising due to local/global intersections of the family . In this paper, we focus on the case of simple sweeps.…”

“…We make the following key assumptions about : (i) the solid is smooth, and (ii) the tuple is in a general position (see [4]). The action of (at time in ) on is given by .…”

“…Much of the mathematics of self-intersection, of passing body-check and of overall geometry have been described in the earlier work [4]. This paper uncovers the topological aspects of the solid sweep and its construction as a solid model.…”

“…This, coupled with [4], which constructs the geometry/parametrizations of the surfaces, was used to build a pilot implementation of the solid sweep using the popular ACIS solid modeling kernel [3]. We give several illustrative 1…”

“…The brep of course is the topological data of vertices, edges and co-edges, loops bounding the faces and orientation of these, and the underlying geometric data of the surfaces and curves. As we show, the brep of , while intimately connected to that of , has several intricate issues of orientation and parametrization which need resolution.Much of the mathematics of self-intersection, of passing body-check and of overall geometry have been described in the earlier work [4]. This paper uncovers the topological aspects of the solid sweep and its construction as a solid model.…”