TABLE OF CONTENTS Global properties of complex solid models refer to volume, surface area, moment of inertia, center of gravity, and so on. In earlier sections, we investigated global properties of curves and surfaces, such as arc length and area, by direct integration. This method is not effective, even if possible, for determining the global properties of complex solids, but other methods are available. Computing global properties of a solid requires us to evaluate the triple integral
where ? is the property we require, f (p) is a vector function describing ?, and integration is over the entire volume of the solid.
Two theorems are useful for relating integrals of different types (line, surface, and volume): Gauss s theorem (also called the divergence theorem), which relates an integral over a closed surface to the integral over the corresponding enclosed volume, and Green s theorem, which relates a closed-path integral to the surface integral over the enclosed region. We will use these theorems particularly in the TimmerStern method, but we will not explore their development, since it is treated in many calculus texts.
A natural association exists between certain characteristic properties of a representation scheme and the algorithms for computing the global properties of models produced by that scheme. Let us look at some examples that support this important concept.
Instances and parameterized shapes or primitive-instancing modeling schemes define families of objects using a finite number of parameters. Assigning a value to each of...
