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the surface characterization driven by the critical points of the height function has found sev-
eral applications in the analysis of terrain modeling [
9
]. Figure
8.3
represents the critical points
over a mathematical surface and a triangle mesh representing a terrain-like surface; red points
correspond to maxima, blue points to minima and green points are saddles.
e computation of critical points on discretized surfaces received considerable attention
in the literature. Banchoff [
11
] introduced critical points for height functions defined over poly-
hedral surfaces, by using a geometric characterization of critical points. A simplicial model in
which linear interpolation is used on the triangles of the underlying mesh is the most common
example of a polyhedral surface. Starting from the observation that a small neighborhood around
a local maximum or minimum never intersects the tangent plane, as shown in Figure
8.2
(a), while
a similar small neighborhood is split into four pieces at non-degenerate saddles, as shown in Fig-
ure
8.2
(b), the number of intersections is used to associate an
index
with each discrete critical
point.
Consider the two-dimensional simplicial complex
in
R
3
with a manifold domain, and
the height function
WR
3
!R
with respect to the direction
in
R
3
;
is called
general for
if
.v/¤.w/
whenever
v
and
w
are distinct vertices of
. Under these assumptions, critical
points may occur only at the vertices of the simplices and the number of times that the plane
through vertex
p
and perpendicular to
cuts the link of
p
is equal to the number of
1
-simplices
in the link of
p
with one vertex above the plane and one below (see Figure
8.2
). Point
p
is called
middle
for
for these
1
-simplices. en, an indexing scheme is defined for each vertex of
as
follows [
10
]:
i.v;/D1
1
2
.number of 1simplices with v middle for /:
(8.1)
(a)
(b)
Figure 8.2:
Configuration of vertices around a maximum point (a) and a non-degenerate saddle (b)
[
24
].
Discrete critical points
are at the vertices of the simplicial model and are defined as points
with index different from
0
. In particular, the index is equal to
1
for maxima and minima, while
it can assume an arbitrary negative integer value for saddles.
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