Numerical Methods for Implicitisation and Their Applications

Modern methods in computer aided design and manufacturing (CAD/CAM) allow geometries of almost arbitrary shape and topology to be created. However, CAD/CAM software has long been plagued by problems of robustness when it comes to modelling the results of operations such as intersections. Such geometries can lack watertightness and correct topology, requiring costly repair procedures to be applied prior to analysis or manufacturing. One technique that can support the robustness and consistency of geometric computations is implicitisation. The mathematical foundations for implicitisation algorithms were developed in the theory of algebraic geometry during the nineteenth century. These methods were reintroduced into computer aided geometric design (CAGD) in the 1980s, with a particular view to computing accurate intersections. Since then, many other approaches to implicitisation have been introduced. In this chapter we present an accessible discussion of several approaches to computing implicitisations numerically. This includes both exact methods for low degree curves, and approximate methods for higher degree surfaces and envelopes. We will also discuss applications of implicitisation in fields such as robotics, computer graphics and CAD/CAM.