Graphics Reference
In-Depth Information
(a)
(b)
(c)
Figure 7.3:
Examples of cell complexes. (a) Circle built starting with a point and attaching a
1
-cell.
(b) Circle built starting with two points and attaching two
1
-cells. (c) Sphere built starting with a
point and attaching a
2
-cell [
24
].
Cell complexes
As simplicial complexes,
cell complexes
are constructed from basic building
blocks, called cells, that generalize the concept of simplices. A
-cell
e
corresponds to the
closed unit ball
B
Dfx2R
jkxk1g
of dimension
. Cells are glued together via
at-
taching maps
. Attaching the cell
e
to a space
Y
by the continuous map
'WS
1
!Y
, with
S
1
Dfx2R
jkxkD1g
the boundary of
B
, requires taking
Y
S
B
, where each point
x2S
1
is identified with the point
'.x/2Y
. e space so obtained is denoted by
Y
S
'
e
. It
is important to note that different attaching maps
'
can lead to different spaces.
e space
X
obtained by subsequently attaching finitely many cells is a finite
cell complex
.
is means that there exists a finite nested sequence
;X
0
X
1
:::X
k
DX
such that,
for each
iD1;2;:::;k
,
X
i
is the result of attaching a cell to
X
i1
. Further details can be found
in [
90
].
Examples of cell complexes are given in Figure
7.3
where the same circle is constructed
through cell adjunction in two different ways. In (a), we start with a
0
-cell, i.e., a point, and we
connect a
1
-cell via the attaching map that identifies the boundary of
B
1
with the starting point.
In (b), two
1
-cells are attached to two
0
-cells. In Figure
7.3
(c) the sphere is obtained by attaching
a
2
-cell directly to a
0
-cell, that is, identifying the boundary of
B
2
with a point.
7.1.1 CONCEPTS IN ACTION
Data representation
From a historical perspective, the first type of model used in computer
graphics and CAD/CAM was the
wireframe
model, which consists of the representation of edge
curves and points on the object boundary. is incomplete model evolved further into
surface
or
boundary
models, that provide an unambiguous representation of the geometry of the 3D
shape boundary, and into
solid
models, that encode a shape as a composition of volumes. e
corresponding computational models generally imply the discretization of a shape into a simplicial
or a cell complex. A model is called
simplicial
if its domain is discretized by a simplicial complex,
while it is called
regular
if the domain is discretized through a
regular grid
, i.e., a cell complex
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