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Convexifying Star-Shaped Polygons Hazel Everett, Sylvain Lazard, Steve Robbins, H

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Convexifying Star-Shaped Polygons Hazel Everett, Sylvain Lazard, Steve Robbins, H 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 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 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 threelink chain The partition can b e done such W J Lenhart and S H Whitesides Re that the threelink chain is contained in a small conguring Closed Polygonal Chains in Eu wedge The strategy is to transform this chain clidean dspace Discrete and Computa while leaving the others unchanged so that either tional Geometry 13 the middle edge is aligned with o or one of the .
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