Graphics Reference
In-Depth Information
resented as a labeled planar polygon and that we can normalize that polygon to be a
regular k-gon.
6.5.2. Lemma.
Given a surface
S
we can always find a labeled complex (L,m) =
(L
S
,m
S
) satisfying:
(1) ÔLÔ=
Q
k
for some k.
(2) The vertices of L are precisely the points
w
0
(k),
w
1
(k),..., and
w
k-1
(k).
(3) There is a homeomorphism h :
X
(L,m)
Æ
S
.
(4) If y
S
:
Q
k
Æ
S
is given by y
S
= h
°
p
(L,m)
, then y
S
Ôint(
Q
k
) and y
S
Ô
e
i
(k) are
one-to-one.
Proof.
See [AgoM76].
Think of the labeled complex (L
S
,m
S
) in Lemma 6.5.2 as being derived from
S
by
cutting along y
S
(∂
Q
k
). Conversely,
S
can be reconstructed from
Q
k
by pasting together
those edges of
Q
k
that are mapped onto the same set in
S
by y
S
.
The main consequence of Lemma 6.5.2 is that the study of surfaces has been
reduced to the study of certain labeled complexes because each surface
S
has
an associated labeled complex (L
S
,m
S
), which in turn determines the surface since
S
X
(L
S
,m
S
)
.
Before describing an even simpler and more compact representation for
S
we need
some notation. Let
S
+
=
{
}
A
12
,
,
K
be the infinite set of distinct symbols A
i
. Define
}
»
{
}
S=
{
-
1
-
1
AA
,
,
K
A A
,
,
K
,
12
1
2
where each expression A
i
-1
is considered as a purely formal symbol and no algebraic
significance is attached to the superscript “-1.” We shall identify the symbol (A
i
-1
)
-1
with A
i
. With this identification a
-1
will belong to S whenever the symbol a does. Let
W
=
the set of all nonempty finite strings a a
K
a
,
where a
Œ
S
.
12
q
i
For example, the strings
(
)
-
1
-
1
--
1
1
-
1
AA
=
A A
,
AA A A
,
and
AAAA AA
11
1
1
12
1
111
1
13
2
belong to W.
Returning to our surface
S
, choose a labeled complex (L
S
,m
S
) and map y
S
:
Q
k
Æ
S
as described in Lemma 6.5.2. Given an “admissible” labeling of the edges of
Q
k
,
define a string
waa
k
S
=
K
Œ
W
12
for
S
by letting the element a
i
be the label of the ith edge
e
1
(k) of
Q
k
. Here is an infor-
mal description of how such admissible labelings are obtained. Label the first edge
