SIAM
J.
COMPUT.
Vol.
20,
No.
4,
pp.
603-621,
August
1991
1991
Society
for
Industrial
and
Applied
Mathematics
001
CONSTRUCTIVE
WHITNEY-GRAUSTEIN
THEOREM:
OR
HOW
TO
UNTANGLE
CLOSED
PLANAR
CURVES*
KURT
MEHLHORN
AND
CHEE-KENG
YAP
Abstract.
The
classification
of
polygons
is
considered
in
which
two
polygons
are
regularly
equivalent
if
one
can
be
continuously
transformed
into
the
other
such
that for
each
intermediate
polygon,
no
two
adjacent
edges
overlap.
A
discrete
analogue
of
the
classic
Whitney-Graustein
theorem
is
proven
by
showing
that
the
winding
number
of
polygons
is
a
complete
invariant
for
this
classification.
Moreover,
this
proof
is
constructive
in
that
for
any
pair
of
equivalent
polygons,
it
produces
some
sequence
of
regular
transformations
taking
one
polygon
to
the
other.
Although
this
sequence
has
a
quadratic
number
of
transformations,
it
can
be
described
and
computed
in
real
time.
Key
words,
polygons,
computational
algebraic
topology,
computational
geometry,
Whitney-Graustein
theorem,
winding
number
AMS(MOS)
subject
classifications.
68Q20,
55M25
1.
Why
a
circle
differs
from
a
figure-of-eight.
First
consider
closed
planar
curves
that
are
smooth.
Intuitively,
a
"kink"
on
such
a
curve
is
a point
without
a
unique
tangent
line.
It
seems
obvious
that
there
is
no
continuous
deformation
of
figure-of-eight
to
a
circle
in
which
all
the
intermediate
curves
remain
kink-free
(see
Fig.
1).
Figure
2
shows
another
curve
that
clearly
has
a
kink-free
deformation
to
a
circle.
Let
us
make
this
precise.
By
a
(closed
planar)
curve
we
mean
a
continuous
function
C
:[0,
1]-->
E
2,
C(0)=
C(1),
where
E
2
is
the
Euclidean
plane.
The
curve
C
is
regular
if
the
first
derivative
C’(t)
is
defined
and
not
equal
to
zero
for
all
t[0,
1],
and
C’(0)
C’(1).
Let
h
:[0,
1
[0,
1
-->
E
2
be
a
homotopy
between
curves
Co
and
C1,
i.e.,
h
is
a
continuous
function
and
each
Cs
:[0,
1]-->
E
2
(0
-<
s-
<
1)
is
a
curve,
where
we
define
Cs(t)=
h(s,
t)
(t[0,
1]).
The
homotopy
is
regular
if
each
C
is
regular.
Two
regular
curves
are
regularly
equivalent
if
there
is
a
regular
homotopy
between
them.
A
classical
result
known
as
the
Whitney-Graustein
theorem
[5],
[1]
says
that
two
curves
are
regularly
equivalent
if
and
only
if
they
have
the
same
winding
number
(up
FIG.
1.
Transforming
a
figure-of-eight
to
a
circle"
kink
appears.
*
Received
bythe
editors
February
3,
1988;
accepted
for
publication
(in
revised
form)
September
12,1990.
?
Informatik,
Universitaet
des
Saarlandes,
D-6600
Saarbruecken,
Federal
Republic
of
Germany.
The
work
of
this
author
was
supported
by
Deutsche
Forschungsgemeinschaft
grant
Me6-1.
Courant
Institute
of
Mathematical
Sciences,
New
York
University,
251
Mercer
Street,
New
York,
New
York
10012.
The
work
of
this
author
was
supported
by
National
Science
Foundation
grants
DCR-84-
01898
and
DCR-84-01633.
603