Kinkfree
deformations
of polygons
Gert Vegter
Dept. of Computing
Science,
University
of Groningen,
P.O.Box
800, 9700AV Groningen,
The Netherlands
Abstract
We consider
a discrete
version
of the
Whitney
Graustein
theorem concerning
regular equivalence of
closed curves.
Two
regular
polygons
P and P’,
i.e. polygons without
overlapping
adjacent edges, are
called regularly
equivalent
if there is a continuous
oneparameter
family
Pa, 0 5 s 5 1, of regular poly
gons with
PO =
P and PI
=
P’.
Geometrically
the oneparameter
family
is a kinkfree
deformation
transforming
P into P’.
The winding
number of a
polygon is a complete invariant
of its regular equiva
lence class. We develop a linear algorithm
that deter
mines a linear number of elementary
steps to deform
a regular polygon into any other regular polygon with
the same winding
number.
1
Introduction
The WhitneyGraustein
theorem
states that
up to
regular
(‘kinkfree’)
deformation
a regular
closed
curve in the plane is completely
determined
by its
winding
number.
In this paper we consider the dis
crete version of this theorem in which closed curves
are replaced with
polygons.
A setting for this prob
lem is given in Section 2. In Section 3 we show how
to reduce a polygon
to isothetic
form in which every
edge is parallel
to one of the coordinate
axes. Sec
tion 4 contains the algorithm
that reduces an isothetic
polygon to isothetic
normal form that is uniquely
de
termined
by its winding
number.
In Section 5 we
Permission to copy without fee a11 or part of this material is granted
provided that the copies are not made or distributed for direct com
mercial advantage,
the ACM
copyright notice
and the title of the
publication and its date appear, and notice is given that copying is by
permission of the Association for Computing Machinery. To copy
otherwise, or to republish, requires a fee and/or
specific permission.
0 1989 ACM 0897913183/89/0006/0061
$1.50
indicate
possible extensions
of this work to the clas
sification
of polygonal
curves on piecewise flat sur
faces.
This
paper was inspired
by similar
work of
Mehlhorn
and Yap, cf. [MY].
They
derive an algo
rithm
that computes a quadratic
number of elemen
tary steps to transform
between regularly
equivalent
polygons.
The sequence of steps can be determined
in linear
time.
Their
normal
form as well as their
method is however completely
different
from ours.
Acknowledgements
The author is grateful to Chee
Yap for communicating
the work [MY]
to him, and
to Jan Tijmen
Udding for helpful
criticism.
2
Equivalence
of Regular
Poly
gons
A closed curve C in the plane is called regular if it has
a smooth parametrization
f : [0, l] + R2 such that
f(0)
= f(l),
S(O) = f’(1)
and f’(t)
# 0 for 0 5 t 2 1.
The winding
number w(C)
of C is the degree of the
map f*
: S1 +
S1 defined by f+(t)
= f’(t)/lf’(t)l
The regularity
of f allows us to consider t as a pa
rameter on the unitcircle
9.
Geometrically
speaking
the winding
number of C is the number of full turns

counted with sign 
of the tangent of C as we go
around C. The figureofeight
has winding
number 0,
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Polygons, equivalence class, Regular polygon, cyclic signature, chainsequences

Click to edit the document details