Computational Topology
Homework 1 (due 9
/
29
/
09)
Fall 2009
1. Recall that a simple closed curve is
polygonal
if its image is the union of a ﬁnite number of
line segments. A
polygon
is the closure of the interior of a simple closed polygonal curve. The
boundary of a polygon
P
is denoted
∂
P
.
(a) Let
G
n
be a regular
n
gon centered at the origin, with one vertex at
(
1,0
)
. Describe a
homeomorphism
φ
n
: IR
2
→
IR
2
such that
φ
n
(
∂
G
n
) =
S
1
.
(b) Describe an algorithm for the following problem: Given a simple polygon
P
, construct a
homeomorphism
φ
:
P
→
G
n
such that
φ
(
P
) =
G
n
. The input polygon
P
is represented as an
array of
n
vertices in (say) counterclockwise order.
[Hint: Any polygon with holes can be
triangulated in
O
(
n
log
n
)
time. How do you want to represent the output homeomorphism?]
(c) Describe an algorithm to construct a homeomorphism
φ
: IR
2
→
IR
2
such that
φ
(
P
) =
G
n
.
Together with part (a), this proves the JordanSchönﬂies theorem for polygons.
2. A
cycle
in a topological space
X
is a continuous map
γ
:
S
1
→
X
. A
(free) homotopy
between
cycles
γ
and
δ
in
X
is a continuous function
h
:
[
0,1
]
×
S
1
→
X
, such that
h
(
0,
θ
) =
γ
(
θ
)
and
h
(
1,
θ
) =
δ
(
θ
)
for all
θ
∈
S
1
. Two cycles are
freely homotopic
if there is a free homotopy between
them.
(a) Loops and cycles are almost identical—any cycle can be turned into a loop by choosing a
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 Fall '08
 Staff
 Topology, Normal Curves, D. Eppstein, S. A. Mitchell

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