PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 45, Number 2, August, 1974
GLOBAL GEOMETRY OF POLYGONS. I:
THE THEOREM OF FABRICIUS-BJERRE
THOMAS F.BANCHOFF
ABSTRACT.
analogue
the numbers
closed
a finite
number
of the curve
each
the number
II
being
numbers
of this
result
use
curves
note
plane
Definition.
Presented
global
curve
that
two points
in this
of the line
side).
There
is also
analogue
received
April
by the
such
fields
class
the proof
to the case
of other
27, 1973 under
January
of mappings
in subsequent
global
in which
theory
theorems
these
finite
the title
Polygonal
17, 1973 and,
in [l].
of a closed
subdivision,
to the interval
inveson
techniques
as developed
X: [a, b] —> E
for some
X restricted
for polygonal
1 ).
a mapping
editors
result
will be used
a sense
that
a
conditions.
for vector
[3], but again
generalizes
of catastrophe
we mean
plane
leads
announced
be of differentiability
of this
(Theorem
paper
and
For a con-
regularity
points
being
C .
of a number
= b, the mapping
to the Society,
certain
proof
result
I
of a few examples
the curve
than
C of
intervals),
of the line,
sides
of critical
stronger
analogues
curves.
satisfying
into the plane
By a polygon
theory;
Examination
an elementary
developed
into the Euclidean
¿0 < Zj < • • • < t
number
containing
to the same
by Fabricius-Bjerre
As in [2], this
fit in with the polyhedral
interval
lines
to one side
techniques
slightly
of polygonal
and space
tangency
(or inflection
lie on opposite
curves
uses
and requires
1-manifold
The techniques
tigations
of double
with a finite
(i.e.
lying
are zero.
we present
in the plane.
of an arbitrary
lines
neighborhoods
was discovered
of regularity
In this
relating
C + ViF + I - II = 0, and in [2] Halpern
there
numbers
C . The result
both
for smooth
described
and winding
support
curve
points
the neighborhoods
all these
The proof
made
of double
that
of a polygonal
by Halpern
and lines
a closed
F of inflection
with
to the conjecture
proof
describe
the number
vex curve
proof
and
of inflections,
with a neighborhood
for which
a direct
curves.
number
and a finite
provide
by Fabricius-Bjerre
pairs
plane
X: \m, b] —>F
crossings,
methods
proved
of crossings,
for smooth
Let
Deformation
of a theorem
a =
t. < í < £.+ .
methods
in revised
in
form,
August 29, 1973.
AMS (MOS) subject classifications
(1970).
Primary 53A05, 57C35; Secondary
55A25.
Key words
and phrases.
Polygon,
inflections,
double
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tangency,
support
line,
deformations.
Copyright © 1974, American Mathematical Society
¿ó
i
238
T. F. BANCHOFF
is linear,
so in this
interval
í., , - /
Xit)=A±l-xit.)
',,,-i,
!+ 1
The points
gon.
X(z\)
, then
in general
is said
position,
polygon
edge
[X ., X.+ .].
[X., X+.]
and
such
X.+ 7 lies
dices
Let
vertices
F(X)
By a double
0 < i < f < m - 1 such
the four indices
the edge
edges,
of the line
support
that
pairs
i.e.
containing
of
th e
edges
the edge
line we mean a pair of in-
X . , and
X .+.
side of the line through
X.i and X.j and so do X.7—1, and X.,,.
°^
; +l
line through
0
i.e.
referred
[X., X.+ .] meets
of inflection
on one side
the number of double support lines with X.
and X
lie on the same
Let IIX)
denote
t
on opposite
X.i and X.; and let II,(X)
denote the number with X.i-1
t
the same side.
we consider
iX.|z' = 0, • • • , wz — 1} are
of crossings,
that
be the number
X.+ . lies
case
of the poly-
are collinear.
C(X) be the number
on the other.
{i, j) with
if the vertices
0 < i < j < m — 1 such
that
and in this
X, we may make precise
Let
(i, j) with
m-gon
the vertices
m.
to be general
to in the introduction:
indices
modulo
so no three
For a general
!+ 1'
X. and are called
X is a closed
to be reduced
A polygon
i
are abbreviated
If Xn = X
all indices
t - t.
+-L_X(i.xl).
'
t.,,-t_.
'i + 1
sides of the
and X.,,
on
7+ 1
We define the function
Mx) = c(x) + y2Fix) + iix)
n,(x)■r — - ~v
We may now state
the main result:
Theorem 1. // X is general,
This
theorem
are zero.
certain
dices
is clearly
We shall
prove
Definition.
family
throughout
By an elementary
of vertices
index
/ and some
deformation
point
deformation
of a general
of in-
polygon
< «t + ], of polygons
X deter-
is said
such
that
u if i' f: j and
(JL:lt\x'..
'
\uk+i-ukj
to be in general
'
position
ral for all but a finite
number
of the parameter
values
for these
values,
there
one triple
exceptional
X in a
the combination
X ., X (u) = X . for all
\uk+i-uk)
elementary
that
all indices
polygon
iX.|z' = 0, • • • , m - 1 j we mean a 1-parameter
x.iu),h^-%.+
This
for which
the given
the deformation.
Xiu) = \X i.u)\i = 0, • • ■ , m - 1 j, u,<u
for some
polygon,
by deforming
m-gon and by showing
unchanged
mined by a cycle
true for a convex
the theorem
way into a convex
remains
then H{X) = 0
is exactly
u,<u<
if X(u)
is gene-
"¿ + 1, and if
of collinear
vertices.
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Lemma
1.
For almost
all choices
of the vertex
X. the elementary
defor-
239
GLOBAL GEOMETRY OF POLYGONS. I
motion
described
Proof.
above
Consider
of the general
is in general
the finitely
polygon
X.
fail to be in general
position
of two lines
tary deformation
finite
from
By a piecewise
a 1-parameter
family
u,+l,
mation
Lemma
deformation
2.
if each
can
X. contains
a
X. in the complement
of the
points.
polygon
that for some
X we mean
finite
subdivision
X(zz.) = X and each
deformation.
family
Such a deformation
X(zz.) is general
and if each
X(u),
is said
elementary
to
defor-
position.
// X is general
X(zz), u
such
pairs
so we get an elemen-
of a general
space,
vertex
deformation
X . and
by choosing
X(zz) of polygons
position
distinct
collection,
intersection
of the parameter
is in general
mation
position
is an elementary
be in general
containing
containing
from this
X . to these
elementary
uQ < u. < • • • < u
u,<u<
is if a line
in general
set of lines
many lines
The only way that the elementary
point
of intersection
position.
<u
<u
then
there
, in general
is a piecewise
position
such
elementary
that
defor-
X(u ) is a convex
m-gon.
Proof.
Begin
the vertices
772-gon Y and proceed
inductively
X . near
Y . of Y so that
X . of X to vertices
7
successive
stage
with a regular
elementary
the polygon
with vertices
the vertices
;
deformation
is in general
X^, Xj,
to move
each
7
position
and so that
• • • , X'., ^+1,
at each
• • • , Y _ , is con-
vex.
We now need
position,
only
show
the changes
collineation.
This
for the basic
that
for an elementary
AC, AF, AL,
we show
collineations
simply
which
deformation
in general
and - All; add up to 0 as we pass
by exhibiting
can occur,
these
differences
and by observing
that
any
explicitly
for each
of these
AH = AC + M}/2)F + Ai£ - All Í = 0.
We consider
five different
types
(a) If no two of the three
none of the indices
double
tangency
involving
indices
(b) If precisely
adjacent
line,
vertices
then
indices,
of collineations:
collinear
vertices
crossings
stay
or inflections
vertex
vertices
if the vertex
does
not pass
then
one crossing
in each
(c) If precisely
index
through
of the third
at all and
is lost
of X but the
of the collineation
we have
no change
and one is gained
of
so
is zero.
two of the three
vertex
sides
the edge
edge
vertices
lie on an edge
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vertices
are affected
lie in an edge
are on opposite
and if it does,
the change
of X, then
the same.
two of the three
of the third
lie in an edge
are on the same
side
and the adjacent
of the collineation
line,
then
240
T. F. BANCHOFF
we have
eight
possibilities,
changes
in the four indices,
(d) If the three
then we have
which
as cases
vertices
schematically,
1 through
are consecutive
four possibilities,
(e) If the polygon
we list
cases
is a triangle,
with
the
8 in the table.
but the polygon
9 through
then
together
is not a triangle,
12 in the table.
all indices
are zero
before
and after
the deformation.
If X is a mapping
into the plane,
yield indices
1-manifold
into
I = 0, •••,«—
then
of a compact
the definition
1-manifold
(not
in the connected
necessarily
case
may be extended
C(X), F(X), I (X), and II((X). A polygonal
the plane
1, and
of all of the polygons
BEFORE
corresponds
X is said
X
to a finite
to be general
is in general
connected)
mapping
collection
to
X of a
of m.-gocxs
if the collection
X ,
of all vertices
position.
AFTER
AC ^AF Al( All,
1.
0
V-
-Z
0
0
0
0
0
0
0
1
v
-1
\
1
-1
3.
/ "\"
ooii
7.
-¿0
7.
0
10
10.
0
0
It.
10
1Z.
-110
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1
0
0
0
1
0
GLOBAL GEOMETRY OF POLYGONS. I
Theorem
1 . As before,
for a general
polygonal
241
mapping
of a compact
I-manifold X, H(X) = C(X) + ^F(X) + I/X) - II/X) = 0.
Proof.
Let
/ = 0, 1, >••,«
gion bounded
Y be a nested
— 1, such
by Y
collection
that
each
, and such
of disjoint
polygon
Y
Then
H(Y) = 0, and the theorem
as in the case
piecewise
elementary
mapping
follows
deformation
X
with vertices
77z;-gons Y ,
is contained
that the collection
gons in Y is in general position.
polygonal
convex
in the open re-
of all vertices
of poly-
C(Y) = 0 = F(Y) = I (Y) = llfY)
in general
near
of Theorem
position
those
from
so
1 by finding
a
X to be a general
of Y.
BIBLIOGRAPHY
1.
T. Banchoff,
Dynamical
2.
Systems,
B. Halpern,
76 (1970), 96-100.
3.
Fr.
Polyhedral
Academic
Global
catastrophe
Press,
theorems
theory.
I:
Maps
New York and London,
for closed
plane
of the line
to the line,
1973.
curves,
Bull.
of plane
closed
Amer. Math.
Soc.
MR 41 #7541.
Fabricius-Bjerre,
Scand. 11 (1962), 113-116.
On the double
tangents
curves,
Math.
MR 28 #4439.
DEPARTMENT OF MATHEMATICS, BROWNUNIVERSITY, PROVIDENCE, RHODE
ISLAND 02912,
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