Graphics Reference
In-Depth Information
Figure 7.18.
Using curve normals
for subdivision.
7.18. The computation for finding the points was similar to the way that silhouettes
are found, except that now one allows other angles. More precisely, one solves
n
n
Ê
Ë
ˆ
¯
z
x
-
1
q
i
=
tan
or
n
sin
q
-
n
cos
q
= 0
.
x
i
z
i
We can precompute the values sin q
i
and cos q
i
. This gives a whole new set of edge
trackers. All the edge trackers above must be linked together appropriately to get a
polygonal approximation to the intersection curve. Links are created as we pass crit-
ical points. When we pass saddle points we get
breaks
that must be repaired by “side-
ways” iteration. One moves along level curves of constant y but changing q. A new
edge tracker is created.
Using this approximation, level curves at a scan line are a chain of edge trackers
that can be specified by an index into a table of sines and cosines. The angle rotates
and we would expect that the index changes by 1 as we move from one edge tracker
to the next (except obviously when we pass from the end of the table to the begin-
ning). There is one
other
case however. The angle q can have local maxima or minima
as we move around. See Figure 7.19. This occurs at inflection points of cross-section
curves. Mathematically, if
v
is the tangent to the level curve, the directional derivative
of q in the direction
v
vanishes: D
V
q=0. We need to create trackers to follow this
function.
The x-sampling described above sometimes made the polygonal nature of the
approximation obvious in the output. Nevertheless it seemed to be reasonably good
and did not use an excessive amount of time to compute. For smooth shading, most
of the time was spent in the y-iteration. Blinn's approach made surfaces look better
than if one had used a polygonal approximation to them.
A Summary of the Blinn Algorithm.
It is a scan line algorithm that generalizes
polygon visible surface algorithms and involves solving equations determined by
various geometric features, such as
