A procedural framework and mathematical analysis for solid sweeps

TL;DR

This paper introduces a procedural framework and mathematical analysis for accurately parametrizing solid sweeps and detecting self-intersections, enhancing CAD system capabilities with efficient and robust methods.

Contribution

It presents a novel procedural approach for envelope parametrization and a mathematical invariant for self-intersection detection in solid sweeps.

Findings

Procedural approach yields accurate envelope parametrization.

Mathematical invariant effectively detects self-intersections.

Method is computationally efficient and suitable for CAD integration.

Abstract

Sweeping is a powerful and versatile method of designing objects. Boundary of volumes (henceforth envelope) obtained by sweeping solids have been extensively investigated in the past, though, obtaining an accurate parametrization of the envelope remained computationally hard. The present work reports our approach to this problem as well as the important problem of identifying self-intersections within the envelope. Parametrization of the envelope is, of course, necessary for its use in most current CAD systems. We take the more interesting case when the solid is composed of several faces meeting smoothly. We show that the face structure of the envelope mimics locally that of the solid. We adopt the procedural approach at defining the geometry in this work which has the advantage of being accurate as well as computationally efficient. The problem of detecting local self-intersections is central to a robust implementation of the solid sweep. This has been addressed by computing a subtle mathematical invariant which detects self-intersections, and which is computationally benign and requires only point queries.