Graphics Reference
In-Depth Information
(a)
(b)
(c)
Figure 14.31.
Problems for nondegenerate cases.
(a)
(b)
Figure 14.32.
Biasing a function toward zero.
f vanishes, which basically corresponds to the case where entire simplices are mapped
to 0 and the contour is not one-dimensional. The second is where the contour passes
through a vertex of the triangulation. We refer the reader to the paper for ways to
handle these cases. However, there are some potential problems even in the nonde-
generate case if one does not choose the triangulation well. Some of the ways that the
algorithm can produce results with incorrect topologies are shown in Figure 14.31.
Figure 14.31(a) shows how disconnected contour components can lead to a connected
answer. Figure 14.31(b) shows the opposite. Figure 14.31(c) shows how a component
can be lost entirely if its interior does not contain any vertices. It is therefore impor-
tant to choose the “resolution” of the triangulation carefully. It must be noted,
however, that although one would hope to have an algorithm of this type produce the
correct results for f's that appear in
real
applications, one should
not
expect this for
arbitrary f because one can find differentiable f that have
arbitrary
compact subsets
of
R
2
as their contour. Furthermore, consider Figure 14.32 which shows the contour
of the function f(x,y) = xy. That example shows that increasing the resolution is not
good enough by itself, because, as shown in Figure 14.32(a), the algorithm would rep-
resent the actual connected contour by two disjoint pieces. That means that, given a
starting point on the contour, the algorithm would only trace out one of the pieces
and leave out the other one. One way to deal with this is to add a bias to 0, meaning
that we replace the function values at vertices that are less than some e by 0. Figure
14.32(b) shows how this can correct the problem in Figure 14.32(a), but with the
potential effect of blacking out some squares. Furthermore, it is not easy to choose
the correct e. Making it too big would black out too many squares. [DLTW90] sug-
gests a trial and error method as the best approach here.
