Approaches to Geometric Modeling - Computer Graphics and Geometric Modeling - page 177

5.3.6 Parametric Representations

Many of the representations of solids rest on a representation of their boundaries.
That was true even in the case of the csg-rep. Although the primitives were solids,
in practice one only had equations or parameterizations for their surfaces, and
the interior of the solid was not referenced explicitly. As far as parameterizations are
concerned, there is no reason why we have to limit ourselves to parameterizations
of two-dimensional objects. If we want access to interior points, we can define
three-dimensional parameterizations just as easily. For example,

( ) = ( ) Π[

]

Π[

]

Π[

]

pr , q , z r cos q , r sin q , z , r 01 , , q 02 , p , z 02 , ,

is a parameterization of a solid cylinder of radius 1 and height 2 with axis the z-axis.
If we allowed such parameterizations, then we could also generate interior points of
the object at will. Chapter 12 describes a number of basic surfaces and their para-
meterizations. Similarly, one could describe a corresponding basic collection of solids
and their parameterizations. In other words, three-dimensional parameterizations are
a representation scheme for solids. See [Mort85] for a discussion of what he calls a
tricubic parametric solid . This is a space parameterized by a function p(u,v,w) of the
form

3 3 3

  Â

j

Π[] ( ) =

i k 3

puvw ,, a uvw , uvw ,, 01 , and a ΠR .

ijk ijk

i = 0 j = 0 k = 0

This is the most general cubic parameterization, but one can look at special cases
such as Bezier or spline forms, just like in the surface case. See [HosL93].

5.3.7 Decomposition Schemes

Decomposition representation schemes represent objects as a union of quasi-disjoint
pieces. These representations come in two flavors: object-based or space-based . The
object-based versions present a subdivision of the object itself. The space-based ver-
sions, on the other hand, subdivide the whole space and then mark those pieces that
belong to the object. The hatched cells in Figure 5.21(b) define a space-based decom-
position representation of the object in Figure 5.21(a). Figure 5.21(c) shows an object-
based decomposition of the same object.
Another distinction between decomposition schemes is whether they use a
uniform or adaptive subdivision . The choice is driven by the geometry of the object.
For example, at places where an object is very curved it would be advantageous to
subdivide it more to get a more accurate representation. Object-based decomposition
schemes tend to be adaptive.

Cell Decompositions. This is a very general object-based decomposition represen-
tation. Here the primitive pieces that an object is broken into can be arbitrary (curved)
cells, typically triangles in the two-dimensional case or tetrahedra in the three-
dimensional one. The idea is to find triangular or tetrahedral pieces each of which