TABLE OF CONTENTS Many techniques for modeling solids are limited because they do not represent the interior of a solid. They cannot represent internal properties or offer ways of representing internal behavior. These techniques assume internal homogeneity of the model. For many purposes, they are adequate. However, in this chapter, we investigate the mathematics and geometry supporting a more complete solid representation scheme. There are several well-known methods for creating simple solid shapes, simple in the sense that the solids are not Boolean constructions or otherwise complex assemblies of even simpler solids. The discussion here includes parametric solids, instances and parameterized solids, sweeps, and solids generated by some form of nonlinear transformation. Mortenson (1995) provides valuable supporting discussions for solid shape transformations.
Perhaps the simplest and most direct mathematical approach we can use to model a solid is by continuous, three-parameter, single-valued functions of the form
where the parametric variables u, v, and w are constrained to the interval u, v, w ?[0, 1]. These functions define the coordinates of the set of points comprising the solid, exterior as well as interior. The terms hyperpatch and parametric solid are often used interchangeably, although historically the term hyperpatch has a narrower meaning.
We will develop the mathematics of these solids as we did for curves and surfaces. In doing this, we find that fixing the value of one of the parametric variables results in a surface within or on the boundary of the solid in...
