Graphics Reference
In-Depth Information
Inline Exercise 23.1:
Convince yourself that after one level of subdivision any
mesh becomes a quad mesh, and that each newly introduced edge vertex
e
i
has
degree four. Then show that in further subdivision, each newly introduced face
vertex also has degree four.
The special case
n
=
4 (which is the most common, as shown by the preceding
exercise) is worth examining.
In this case, Equation 23.7 becomes
i
e
i
4
f
i
4
v
=
1
2
v
+
1
+
1
4
,
(23.8)
f
'
1
4
v
e
'
1
which says that
v
is a weighted average of
v
, the average of the adjacent edge
points, and the average of the adjacent face points, just as for curve subdivision
the new vertex location was an average of the old vertex location and the average
of the adjacent edge points.
Figure 23.5: For a quad mesh,
the face vertices from the previ-
ous level of subdivision are oppo-
site
v
in each quad.
This situation (after the first level of subdivision) is shown in Figure 23.5.
In this case, we can rewrite the subdivision formula for
v
in terms of
v
,
{
e
i
}
,
f
i
}
and
{
f
i
}
instead of using
{
; all we need to do is substitute
f
i
=
v
+
e
i
+
e
i
+
1
+
f
i
(23.9)
4
to get
i
e
i
4
f
i
4
v
=
1
2
v
+
1
+
1
4
(23.10)
4
i
e
i
4
(
v
+
e
i
+
e
i
+
1
+
f
i
)
=
1
2
v
+
1
+
1
4
/
4
(23.11)
4
4
i
e
i
4
v
4
+
i
e
i
+
i
f
i
16
=
1
2
v
+
1
+
1
4
(23.12)
4
8
32
i
64
f
i
.
16
v
+
3
9
e
i
+
1
=
(23.13)
Corresponding formulas for
e
i
and
f
i
in terms of the pre-subdivision vertices are
1
16
[
6
v
+
6
e
i
+
e
i
−
1
+
e
i
+
1
+
f
i
−
1
+
f
i
]
and
e
i
=
(23.14)
f
i
=
1
4
[
v
+
e
i
+
e
i
+
1
+
f
i
]
.
(23.15)
If we list the coordinates of the central vertex
v
, then the edge vertices
e
1
,
e
2
,
e
3
,
e
4
, and then the face vertices
f
1
,
...
,
f
4
ina9
×
3 matrix,
V
, we can
summarize by saying that
