Computation of Singularities for Engineering Design

The computation of singularities or critical points of polynomial and other more complex vector fields in a finite subdomain of the n-dimensional Euclidean space is the underlying fundamental process behind several important engineering and scientific problems. These include, for example, design, analysis, scientific visualization, and manufacture of complex objects in a computer environment. This paper starts with a review of extant solution techniques and focuses on recent research by the Design Laboratory in this general area. Specifically, we summarize the algorithmic techniques we have developed on computation of solutions of systems of non-linear polynomial equations and other more complex equations involving irrational functions. Such equations arise in shape interrogation problems including intersections of sculptured objects, symmetry transforms, distance function computations, visualization of rational and offset or parallel surfaces, stationary point computations of maps of physical properties, and in detailed analysis of differential geometry properties of complex free-form surfaces. Examples illustrate our techniques and their applications.