Graphics Reference
In-Depth Information
(2) Next, do a curve interpolation on the rows of the array of points
r
sj
to get the
desired control points
q
st
.
Of course, like in the curve case, one is not given the knots in practice. Unfortu-
nately, things get more complicated here because we have to produce
one
set of knots
u
i
for
all
of the curves p(u,v
j
) with fixed v
j
, j = 0, . . . n, and similarly for the v
j
. One
typically uses some sort of averaging process, but that may not work very well if our
data is not well spaced. See [PieT95] or [Fari97] for a much more thorough discus-
sion. [PieT95] also discusses of interpolation of curve networks.
Data sets are not always rectangular. For example, one might have gotten rows of
unequal number of data points from a sampling of slices of an object. One approach
for this case is to use spline curves that interpolate the rows and then use a skinning
surface (see Section 14.7) for these curves for our interpolating surface. One poten-
tial problem is that the number of column control points might get very large. An
approach that alleviates this problem can be found in [PieT00].
12.13
Cyclide Surfaces
This section discusses one final class of surfaces that can be defined by equations,
namely, the cyclides. See Figure 12.21. Interest in cyclides has waxed and waned over
time. In 1982 R.R. Martin ([Mart82]) showed that they were useful in CAGD and since
then interest in these surfaces has revived. They have proved especially useful for
certain blending operations. We shall describe a few such applications in Section 15.6.
In this section we shall discuss a few of their properties relevant to CAGD. We can
only present a brief overview. More details can be found in [ChDH89], [Prat90], and
[Boeh90]. Another good reference is [KraM00]. Throughout this section, the term
“cyclide” will mean “Dupin cyclide.” Only at the end will we make a few comments
about a more general related class of surfaces also called cyclides.
Cyclides are defined in Section 9.13 in [AgoM04] by means of geometric con-
structions that make it easier to deduce some of their geometric properties, but for
computation purposes it is useful to have both a parametric and an implicit defini-
tion. We give such definitions for central cyclides whose spine curves are an ellipse
and hyperbola and which are in standard position as shown in Figure 12.22. More
precisely, we assume that the ellipse
E
and hyperbola
H
with branches
H
1
and
H
2
are
defined by equations
Figure 12.21.
A central ring cyclide.
