Local and Global Analysis of Parametric Solid Sweeps

TL;DR

This paper introduces a comprehensive computational framework for modeling the envelope of swept volumes in CAD systems, combining local and global analysis to improve robustness and efficiency in handling complex geometries.

Contribution

It presents a novel classification of sweeps, an invariant function for separation, and a complete structural understanding for stable and efficient trimming in solid sweep modeling.

Findings

Introduces a funnel-based parametrization for sweep analysis

Defines an invariant function to distinguish sweep types

Provides a robust method for computing singularities and trim curves

Abstract

In this work, we propose a detailed 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 parametrization 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 either decomposable or non-decomposable. Further, we construct an {\em invariant} function $\theta$ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between $\theta$, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach, $\theta$ is robust and 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.