Convexifying Star-Shaped Polygons Hazel Everett, Sylvain Lazard, Steve Robbins, H. Schröder, Sue Whitesides

To cite this version:

Hazel Everett, Sylvain Lazard, Steve Robbins, H. Schröder, Sue Whitesides. Convexifying Star- Shaped Polygons. 10th Canadian Conference on Computational Geometry (CCCG’98), 1998, Mon- treal, Canada. pp.10-12. ￿inria-00442788￿

HAL Id: inria-00442788 https://hal.inria.fr/inria-00442788 Submitted on 23 Dec 2009

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Convexifying StarShap ed Polygons

H Everett S Lazard S Robbins H Schroder S Whitesides

Extended Abstract

Intro duction

The reconguration problem for chains is to de

termine whether a chain of n links can b e moved

from one given conguration to another The

o

links have xed lengths and may rotate ab out

their endp oints Previous work on the recong

uration of chains eg has allowed links to

pass over one another so that the links act as

distance constraints but not as obstacles

We study a variant of this problem which we

call polygon convexication the initial congura

Figure Partition into wedges for n even

tion of the chain forms a simple p olygon the nal

conguration is a convex p olygon and the links

b

are not allowed to cross It is unknown whether

every p olygon can b e convexied

A p olygon is starshaped if it contains a p oint

a

o p ossibly lying on the b oundary that can see

c

all the other p oints in the p olygon more pre

cisely for any p oint p in the p olygon including

o

its b oundary the line segment o p do es not in

tersect the exterior of the p olygon A p olygon

Figure A radial expansion applied to a two

is in general position if no three of its vertices

link chain

are collinear Our main result is that every star

shap ed p olygon in general p osition can b e con

vexied

wedge angle is smaller than it is called big

otherwise We present a recursive algorithm that

straightens one by one the vertices of P until P

The Algorithm

is a convex p olygon or a p entagon We show how

to convexify p entagons indep endently

Let P b e a starshap ed simple p olygon whose n

Supp ose for the moment that n is even and

vertices are in general p osition A wedge is a

n The rst step is to partition the plane into

closed region delimited by two rays that share

wedges by rst nding a p oint o in the kernel of

their endp oint A wedge is called smal l if its

P and then constructing rays from o through ev



This research was conducted at the International

ery second vertex of P See Figure The result

Workshop on Wrapping and Folding We would like to ac

is a set of wedges each of which contains a two

knowledge the other participants T Biedl E Demaine

link chain a p olygonal chain of vertices and

M Demaine A Lubiw J ORourke M Overmars I

Streinu and G Toussaint for useful discussions as well as

edges whose extremities lie on the rays forming

NSERC and FCAR for nancial assistance

the wedge b oundary and whose intermediate ver

1

Universite du Queb ec a Montreal Montreal Canada

2

tex lies inside the wedge Now we apply a radial

McGill University Montreal Canada

3

Loughb orough University UK expansion motion see Figure in which the ex

c

e

a

c

a

d

a

b

d

d

c

b

o

o

o

b

Figure Reconguring a threelink chain Ver

Figure Closing a big wedge Vertex c is rotated

tex c is rotated counterclo ckwise around d

clo ckwise ab out o

other two edges lies on a wedge b oundary The

transformation is accomplished by rotating one of

tremities of each twolink chain move along their

the two intermediate vertices around the extreme

resp ective rays at equal constant sp eed away from

vertex to which it is adjacent Figure shows

o until some intermediate vertex straightens The

an example In the case that the middle edge is

straight vertex is kept straight for the remainder

aligned with o we repartition the p olygon so that

of the algorithm eectively reducing the num

this edge lies on a wedge b oundary Thus af

b er of links by one and the algorithm is called

ter reconguring the threelink chain each wedge

recursively

contains a twolink chain and exactly one edge is

We show that during a radial expansion the

on a wedge b oundary Then during the radial

p olygon remains starshap ed To show this we

expansion this edge stays on the wedge b ound

prove that at no time during a radial expansion

ary and all other wedges are treated as b efore

do es an intermediate vertex leave its wedge In

fact this will only b e true when the wedge angle

We have shown that using the motion de

is less than so b efore applying the radial expan

scrib ed ab ove we can recongure a starshap ed

sion we rst close the big wedge if there is one

p olygon that has n vertices into a convex

since n there exists at most one To close

p olygon in which case we stop or into a p en

the big wedge see Figure we rotate around

tagon The case n diers from the case n o dd

o toward the interior of the wedge an endp oint

and greater than b ecause when n all the

of the twolink chain contained in that wedge

threelink chains may b e contained in big wedges

all the other vertices of P that lie on the rays

In order to circumvent this diculty we cho ose

emanating from o do not move The only ver

the p oint o in the kernel of P to b e a vertex of

tices that move are the vertex we rotate and its

P Using the transformation describ ed ab ove we

two adjacent intermediate vertices The motion

close one big wedge until one vertex straightens

stops when an intermediate vertex straightens or

or b oth wedges are at Then a simple motion

when the big wedge angle b ecomes smaller than

can recongure the p entagon into a quadrilateral

by some small We show that during this mo

which is easy to convexify

tion the p olygon remains starshap ed by proving

For the time complexity note that since O n

that the two moving intermediate vertices remain

radial expansions are needed each of which takes

inside their resp ective wedges

O n time to compute the entire motion descrip

We need to do something slightly dierent in

2

tion can b e computed in O n time

the case that n is o dd and n Firstly

we partition the plane into wedges as b efore

but this time we have one wedge containing a

References

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