Graphics Reference
In-Depth Information
The
boundary
of a 1D mesh defined as a
formal sum
of the vertices of the mesh
in which the coefficient of each vertex is determined as follows: Each edge from
vertex
i
to vertex
j
adds
+
1 to the coefficient for
j
, and
−
1 to the coefficient for
i
;
we sometimes write that the boundary of the edge
ij
is
j
i
. Applying this to the
mesh above, we find that the boundary formal sum is (reading edge by edge, and
writing
v
i
for the
i
th vertex)
−
v
2
v
3
(
v
2
−
v
1
)+(
v
3
−
v
2
)+(
v
4
−
v
3
)+(
v
1
−
v
4
)+(
v
6
−
v
5
)
,
(8.1)
v
1
v
0
which simplifies to
v
6
−
v
5
. Informally, we say that the boundary consists of ver-
tices 5 and 6.
The reason for the formalism arises when we consider more interesting
meshes, like the one shown in Figure 8.4. The boundary in this case consists of
v
1
+
v
2
+
v
3
+
v
4
+
v
5
−
v
4
v
5
5
v
0
.
Figure 8.4: A wagon-wheel-
shaped mesh. An arrow from
vertex i to vertex j indicates that
(
i
,
j
)
is an edge of the mesh, and
not
(
j
,
i
)
.
A 1D mesh whose boundary is zero (i.e., the formal sum in which all coeffi-
cients are zero) has the property that it's easy to define “inside” and “outside” by a
rule like the winding number rule for polygons in the plane. Such a mesh is called
closed.
A 1D mesh where each vertex has degree 2 (i.e., where each vertex has an
arriving edge and a leaving edge) is said to be a
manifold mesh:
In the abstract
graph, every point has a neighborhood (a set of all points sufficiently near it) that
resembles a small part of the real number line. A point in the interior of an edge,
for example, has the edge interior as such a neighborhood. A vertex has the union
of the interiors of the two adjacent edges, together with the vertex itself, as such a
neighborhood.
We use the term “manifold mesh” to suggest that such meshes are like
man-
ifolds,
which we will not formally define; there are many topics that introduce
the idea of manifolds with the appropriate supporting mathematics [dC76, GP10].
Informally, however, an
n
-dimensional manifold is an object
M
with the property
that for any point
p
∈
M
, there's a neighborhood of
p
(i.e., a set of all points in
M
close to
p
, defined appropriately) that looks like the set
R
n
:
}
(the “open ball”) in
R
n
. “Looks like” means that there's a continuous map from the
ball to the neighborhood and back. (These continuous maps are also required to be
“consistent” with one another wherever their domains overlap; the precise details
are beyond the scope of this topic.) For example, the unit circle in the plane is a
1-manifold because one neighborhood of the point with angle coordinate
{
∈
x
<
1
x
θ
con-
sists of all points with coordinates
θ −
0. 1 to
θ
+
0. 1; the correspondence to the
unit ball in
R
(i.e., the open interval
)
. Similarly,
familiar smooth surfaces in 3-space like the sphere, or the surface of a donut, are
2-manifolds. An atlas (i.e., a book showing maps of the whole world) is a kind of
demonstration that the sphere is a manifold: Each page of the atlas gives a corre-
spondence between some region of the globe (e.g., Western Europe) and a portion
of the plane (i.e., the page of the atlas that shows Western Europe).
Shapes with corners (like a cube) can also be manifolds, but they are
not
smooth
manifolds, which is what's usually meant when the term is used
informally—a continuous map that takes a small region around the corner of
the cube and sends it to the plane ends up distorting things too much for all the
required conditions for “smoothness” to hold.
−
1
<
x
<
1) is
u
→
10
(
u
− θ
