Chapter 8
Solid Modeling
Solids represent a large variety of objects we see and handle. The chapters on curves and surfaces
treated earlier are intended to form the basis for solid or volumetric modeling. Solid modeling
techniques have been developed since early 1970’s using wireframe, surface models, boundary
representation (b-rep), constructive solid geometry (CSG), spatial occupancy and enumeration. A
solid model not only requires surface and boundary geometry definition, but it also requires topological
information such as, interior, connectivity, holes and pockets. Wire-frame and surface models cannot
describe these properties adequately. Further, in design, one needs to combine and connect solids to
create composite models for which spatial addressability of every point on and in the solid is required.
This needs to be done in a manner that it does not become computationally intractable.
Manufacturing and Rapid Prototyping (RP) both require computationally efficient and robust
solid modelers. Other usage of solid modelers is in Finite Element Analyses (as pre- and post-
processing), mass-property calculations, computer aided process planning (CAPP), interference analysis
for robotics and automation, tool path generation for NC machine tools, shading and rendering for
realism and many others.
8.1 Solids
The treatment in previous chapters on curve and surface design is purely geometric. However, we
would realize that it takes more than geometry alone to interpret solids. Solids are omnipresent, in all
possible forms, shapes and sizes. From the representation viewpoint, we may mathematically regard
a solid V to be a contiguous set of points in the three-dimensional Euclidean space E^3 satisfying the
following attributes:
(a) Boundedness: A set V of points must occupy a finite volume or space in E^3. The possibility of
solids with infinite volume thus gets eliminated.
(b) Boundary and interior: Letb (V) and I (V) be two subsets of V such that b(V)∪I(V) = V, wherein
b(V) comprises boundary points and I(V) is the set of interior points. A point p∈V is an interior
point (p∈I(V)) if there exists an open ball enclosingp that consists of points in V only.
Thus, if p 0 is the center of the ball B of arbitrary small radius r, and if pi,p∈B⊂V, then
|pi – p 0 | < r. A point p is a boundary point if p∈V and p∉I(V) (Figure 8.1). Note that for p as
a boundary point, if an open ball B 1 of radius r 1 is drawn around p,B 1 shall contain points
belonging to E^3 – V as well.