Graphics Reference
In-Depth Information
k
1
U
-
[
]
Cp p
=
,
+
1
i
i
i
=
1
is called a
polygonal curve
in
X
defined by the vertex sequence. Each
p
i
is called a
vertex
of the curve. If
p
1
=
p
k
, then
C
is called a
closed
polygonal curve.
Let
C
be a closed polygonal curve in a plane
X
in
R
n
that is homeomorphic to a
circle. By the Jordan curve theorem the curve
C
divides the plane into two connected
closed sets
A
and
B
, so that
XA BA BC
=»
,
«=
,
and one of the parts, say
A
, is bounded and the other is not.
Definition.
The bounded part
A
is called the
polygon
with
vertices
p
i
defined by the
polygonal curve.
A polygon is actually a polyhedron that is homeomorphic to a disk. This fact, that
an imbedding of a circle in the plane extends to an imbedding of the disk, is the so-
called Schoenflies theorem that was partially proved by A. Schoenflies in 1906.
Definition.
If
P
is a polygon and if
P
1
,
P
2
,..., and
P
k
, k > 0, are disjoint polygons
contained in the interior of
P
, then removing the interior of a
P
i
creates a
hole
in
P
and
(
)
cls
PP P
-- --
K
P
1
2
k
is called a
polygon with k holes
.
From the comments above, polygons and polygons with holes are two-
dimensional polyhedra.
Definition.
If s is a simplex, then the simplical complex ·sÒ={t Ô t ≺ s} is called
the
simplicial complex determined by
s
.
For example, ·
v
0
v
1
v
2
Ò={
v
0
,
v
1
,
v
2
,
v
0
v
1
,
v
1
v
2
,
v
0
v
2
,
v
0
v
1
v
2
}.
Next, let us isolate the maps that are naturally associated to complexes.
Definition.
Let K and L be simplicial complexes. A
simplicial map
f : K Æ L is a
map f from the vertices of K to the vertices of L with the property that if
v
0
,
v
1
,...,
and
v
k
are the vertices of a simplex of K, then f(
v
0
), f(
v
1
),..., and f(
v
k
) are the ver-
tices of a simplex in L. If f is a bijection between the vertices of K and those of L, then
f is called an
isomorphism
between K and L and the complexes are said to be
iso-
morphic
. We shall use the notation K
L to denote that complexes K and L are
isomorphic.
6.3.2. Proposition.
Composites of simplicial maps are again simplicial maps.
Proof.
Easy.
We show how simplicial maps induce continuous maps of underlying spaces. Let
