Hiding disks in folded polygons

HidingDisksinFoldedPolygonsThereseC.BiedlyzErikD.DemaineMartinL.yAnnaLubiwyzGo dfriedT.ToussaintJune30,1998AbstractThis pap er considers the problem of nding a simple fold of a given p olygonPthat\hides" (covers b oth sides of ) the largest p ossible disk.We solve this problem by givingap olynomial-time algorithm to ndthelargestpairofequal-radiusnon-overlappingdisks in a p olygonP.The desired fold is then the p erp endicular bisector of the centersof these two disks.We also present some conjectures for the more general multiple-foldcase whenPis a square.1Intro ductionThe giftwrappingproblem asks whether a given p olyhedroncan b e wrapp ed up, or hidden,usingagivenpieceofpap er.Weconsiderthisproblemonedimensiondown:Supp oseyouare given a piece of pap er, in the shap e of some p olygonP.Given another shap eQ, can youhideQby foldingParound it?By \hiding,"we mean that b oth sides ofmust b e coveredbythepap er.Weallowpap ertostickoutb eyondQ,butinsistthatitb efoldedat.Thisdecisionproblemcanb egeneralizedtoanoptimizationproblem:whatisthelargestscalingofthe shap eQthat canb e hiddenby foldingParoundit?In thispap er we consider the case whereQis a disk.We consider onlysimplefoldsthatfoldthepap erPalongonestraightline.Ourstartingp oinissp ecialcasewhereweallowonlyone simple fold.The problem then b ecomes that of packing to non-overlappingcopiesofthelargestdiskp ossibleintoP.Toourknowledge,thisproblemhasnotb eenstudiedb efore.Relatedtothetwo-diskpackingproblemislargest-empty-circleproblem: ndlargest(single)diskthat tsinagivenp olygonP.Thisproblemistraditionall ysolvedbynotingthatthe center ofthe diskmust b eatavertex of themedialaxis[9].It can thusb eScho olofComputerScience,McGillUniversity,3480Street,Montreal,Queb ecH3A2A7,Canada, email:ftherese,godfriedg@cs.mcgill.ca.yDepartment of Computer Science, University of Waterlo o, Waterlo o, Ontario N2L 3G1, Canada, email:feddemaine,mldemaine,alubiwg@uwaterloo.ca.zSupp orted by NSERC.1