Computation of the Geometric Properties of Cams

Abstract

In this chapter we discuss the accurate and fast computation of the first three moments—volume, vector first moment, and inertia matrix—of regions bounded by 2-D contours and 3-D surfaces. The approaches that have been reported in this regard vary in nature, although most of them refer to a discrete integration, namely, a summation of integrals over a set of subregions (Lee and Requicha, 1982a and b; Woodwark and Quinlan, 1982). Some methods resorting to the Gauss Divergence Theorem (GDT) have also been reported (Messner and Taylor, 1980; Miles and Tough, 1983; Angeles, 1983; Angeles et al. 1990; Lien and Kajiya, 1984; Ota et al., 1985). The latter are especially attractive because they allow the reduction of area or volume integrals to either line or surface integrals. Whereas formulas for the transformation of integrals defining the volume and the first moment of solid regions bounded by closed surfaces into surface integrals have been available for some time (Brand, 1965; Hurley, 1981), computer-oriented algorithms making use of such formulas are somewhat recent. In fact, Messner and Taylor (1980) are recognized as the first to make full use of the GDT in this context. Lien and Kajiya (1984) proposed simple formulas based on the GDT that allow an exact evaluation of the moments of arbitrary polyhedra. Although the formulas they propose are quite general, for they allow the computation of moments of any order—see Section 8.3 for a definition of these—, they are limited to polyhedra; on the other hand, they do not include explicit formulas for the functions required to evaluate the surface integrals involved.