Graphics Reference
In-Depth Information
parameterization in u and v to define a Bézier parameterization for such a patch. See
[Mart82], [MaPS86], [Prat90], [Boeh90], [DuMP93], [Prat95], and [KraM00]. This is
very useful for representing cyclides in a CAGD program. One factor restricts the prin-
cipal cyclide patch, namely, its four corners lie on a circle. This means that once one
has picked two adjacent sides of a patch one has only one degree of freedom to pick
the fourth corner since it must lie on the circle determined by the other three. One
can also define triangular Bézier patches ([AlbD97]). Conditions for obtaining com-
posite cyclide patches that join with G
1
continuity are discussed in [KraM00].
For cyclide intersections see [MaPS86] and [John93].
We already mentioned at the beginning of this section that cyclides are useful in
blending. They are also useful in controlling the fairness of a surface. The advantage
of cyclides is that they provide a manageable representation of a larger piece of a
surface. Tensor product Bézier or B-spline patches deal with smaller pieces. This has
led to a search for other general fourth-degree algebraic surfaces that might be useful
in CAGD. One such class of surfaces are the
supercyclides
that are projective images
of Dupin cyclides. These are special cases of so-called
double-Blutel
surfaces. For more
on these generalizations see [Dege94], [Prat96], and [Prat97]. A unified theory of
supercyclides is described in [Dege98].
12.14
Subdivision of Surfaces
Like in the case of curves, being able to subdivide surfaces is important in a variety of
applications. Subdivision problems come in two flavors. In one case we have a para-
metric surface and in the other we have no parameterization but simply a polygonal
surface defined by an arbitrary (not necessarily rectangular) grid of points. This section
makes a few comments about the first case. The second is dealt with in Section 12.17.
At one level, subdividing a parametric surface simply amounts to subdividing its
domain. On the other hand, if we are dealing with a surface defined by control points
or control points and knots, then the more interesting question is how one can add new
control points or knots. The tensor product surface case is quite straightforward and
reduces to subdividing curves in the u- and v-direction and hence is basically a one-
dimensional problem. The triangular surface case is more involved. Blossoms come in
very handy here. One way to subdivide a triangle is shown in Figure 12.24. We add new
vertices at the midpoints of the edges. The four new triangles give rise to four new tri-
angular nets. The main issue is to do computations with respect to the smaller trian-
gles as efficiently as possible by judicious use of the de Casteljau algorithm. We refer
the interested reader to [Gall00]. There are other ways to subdivide triangles and
[Fili86] suggests that the choice of subdivision should be made adaptively.
Figure 12.24.
Subdividing triangular domains.
