ti0polygonsize1i10i1polygonsizei0i1fVector2RealxTJxF30 x697x38 Txf 6x065 x0 Tdx 00originpolygoni0Vector2RealxTJxF30 x697x38 Txf 6x ID: 608080
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Minimum-AreaRectangleContainingaSetofPointsDavidEberly,GeometricTools,RedmondWA98052https://www.geometrictools.com/ThisworkislicensedundertheCreativeCommonsAttribution4.0InternationalLicense.Toviewacopyofthislicense,visithttp://creativecommons.org/licenses/by/4.0/orsendalettertoCreativeCommons,POBox1866,MountainView,CA94042,USA.Created:May17,2015LastModied:September11,2020Contents1Introduction22TheExhaustiveSearchAlgorithm23TheRotatingCalipersAlgorithm33.1ComputingtheInitialRectangle..................................53.2UpdatingtheRectangle.......................................73.2.1DistinctSupportingVertices................................73.2.2DuplicateSupportingVertices................................73.2.3MultiplePolygonEdgesAttainMinimumAngle.....................83.2.4TheGeneralUpdateStep..................................104ARobustImplementation114.1AvoidingNormalization.......................................114.2IndirectComparisonsofAngles...................................124.3UpdatingtheSupportInformation.................................124.4ConversiontoaFloating-PointRectangle.............................131 Thepreviousversionofthisdocument(February9,2008)discussedthesimpleO(n2)algorithmforcomputingtheminimum-arearectanglethatcontainsaconvexpolygonofnvertices.Thecurrentversionincludesadiscussionoftherotatingcalipersalgorithm,analgorithmthatcanbeusedtocomputetheminimum-arearectangleinO(n)time.Likemostalgorithmsincomputationalgeometry,arobustimplementationusing oating-pointarithmeticisdiculttoachieve.DetailsoftheGTEimplementationareprovidedhere,whereweuseexactrationalarithmetictoguaranteecorrectsignclassicationsforsubproblemsthatariseinthealgorithm.1IntroductionGivenanitesetof2Dpoints,wewanttocomputetheminimum-arearectanglethatcontainsthepoints.Therectangleisnotrequiredtobeaxisaligned,andgenerallyitwillnotbe.Itisintuitivethattheminimum-arearectangleforthepointsissupportedbytheconvexhullofthepoints.Thehullisaconvexpolygon,andanypointsinteriortothepolygonhavenoin uenceontheboundingrectangle.Lettheconvexpolygonhavecounterclockwise-orderedverticesVifor0in.Itisessentialtorequirethatthepolygonhavenotripleofcollinearvertices.Theimplementationissimplerwiththisconstraint.Anedgeoftheminimum-areaboundingrectanglemustbecoincidentwithsomeedgeofthepolygon[1];insomecases,multipleedgesoftherectanglecancoincidewithpolygonedges.Inthediscussionweusetheperpoperator:Perp(x;y)=(y;�x).If(x;y)isunitlength,thenthesetofvectorsf(x;y);�Perp(x;y)gisright-handedandorthonormal.2TheExhaustiveSearchAlgorithmTheinputisasetofmpoints.Theconvexhullmustbecomputedrst,andtheoutputisasetofnpoints.Theasymptoticbehaviorofthehullalgorithmdependsonm,wherepotentiallymismuchlargerthann.Oncewehavethehull,wecanthenconstructtheminimum-arearectangle.Thesimplestalgorithmtoimplementinvolvesiteratingovertheedgesoftheconvexpolygon.Foreachedge,computethesmallestboundingrectanglewithanedgecoincidentwiththepolygonedge.Ofallnrectangles,choosetheonewiththeminimumarea.Tocomputetheboundingrectangleforanedge,projectthepolygonverticesontothelineoftheedge.Themaximumdistancebetweentheprojectedverticesisthewidthoftherectangle.Nowprojectthepolygonverticesontothelineperpendiculartothepolygonedge.Themaximumdistancebetweentheprojectedverticesistheheightoftherectangle.Forthespeciedpolygonedge,wecancomputetherectangleaxisdirectionsandtheextentsalongthosedirections.PseudocodeforthealgorithmisprovidedinListing1. Listing1.TheO(n2)algorithmforcomputingtheminimum-arearectanglecontainingaconvexpolygon.structRectanglefVector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;center,axis[2];Realextent[2],area;//area=4*extent[0]*extent[1]g;RectangleMinAreaRectangleOfHull(Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;const&polygon)f2 RectangleminRect;minRect.area=maxReal;//largestfinitefloating=pointnumberfor(size ti0=polygon.size()=1,i1=0;i1polygon.size();i0=i1++)fVector2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;origin=polygon[i0];Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;U0=polygon[i1]=origin;Normalize(U0);//lengthofU0is1Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;U1==Perp(U0);Realmin0=0,max0=0;//projectionontoU0=axisis[min0,max0]Realmax1=0;//projectionontoU1=axisis[0,max1],min1=0isguaranteedfor(size tj=0;jpolygon.size();++j)fVector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;D=polygon[j]=origin;Realdot=Dot(U0,D);if(dotmin0)min0=dot;elseif(dot&#x]TJ/;ག ;.97;8 T; 10;&#x.599;&#x 0 T; [0;max0)max0=dot;dot=Dot(U1,D);if(dot&#x]TJ/;ག ;.97;8 T; 10;&#x.599;&#x 0 T; [0;max1)max1=dot;gRealarea=(max0=min0)*max1;if(areaminRect.area)fminRect.center=origin+((min0+max0)/2)*U0+(max1/2)*U1;minRect.axis[0]=U0;minRect.axis[1]=U1;minRect.extent[0]=(max0=min0)/2;minRect.extent[1]=max1/2;minRect.area=area;ggreturnminRect;gRectangleMinAreaRectangleOfPoints(std::vectorVector2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;const&points)f//Assumptionsforinput://1.points.size()&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;=3//2.points[]arenotcollinear//Assumptionsforoutput://1.polygon.size()&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;=3//2.polygon[]arecounterclockwiseordered//3.notripleofpolygon[]iscollinearstd::vectorVector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;polygon;ComputeConvexHull(points,polygon);returnMinAreaRectangleOfHull(polygon);g ThetwoloopsmakeitclearwhythealgorithmisO(n2).Ifthenumberofverticesnissmall,thisisaviablealgorithmtouseinpractice.However,thechallengingpartofimplementingthealgorithmusing oating-pointarithmeticistocomputetheconvexhullofthepointsrobustly.Theminimum-arearectangleconstructionforaconvexpolygonisasubproblemforcomputingtheminimum-volumeboxcontainingaconvexpolyhedron[2].Inthiscontext,theO(n2)behaviorcanbenoticeable,soitisworthwhiletohaveafasteralgorithmavailableforthe3Dsetting.3TheRotatingCalipersAlgorithmTherotatingcalipersalgorithmiscreditedto[4]|althoughthatnamewasnotused|forlocatingantipodalpointsofaconvexpolygonwhencomputingthediameter.Thealgorithmmaybefoundinasourcemorereadilyavailable[3,Theorem4.18].Thenameofthealgorithmisdueto[5],anarticlethatdescribesmanyusesfortherotatingcalipers.3 Appliedtotheminimum-arearectangleproblem,therotatingcalipersalgorithmstartswithaboundingrectanglehavinganedgecoincidentwithapolygonedgeandasupportingsetofpolygonverticesfortheotherpolygonedges.Therectangleaxesarerotatedcounterclockwisebythesmallestanglethatleadstotherectanglebeingcoincidentwithanotherpolygonedge.Thenewsupportingsetofverticesisbuiltfromtheprevioussetandfromthenewpolygonedgevertices.Intheremainderofthissection,apolygonedgehVi;Vi+1iwillbereferencedbyitsindicesinstead;thus,wewillrefertotheedgehi;i+1i.Theindexadditioniscomputedmodulon,thenumberofverticesofthepolygon.ThesupportingverticesforarectangleformamultisetS=fVi0;Vi1;Vi2;Vi3g,eachvertexsupportinganedgeoftherectangle.Theyarelistedintheorder:bottom,right,top,andleft.Onevertexcansupporttwoedges,whichiswhyScanbeamultiset(duplicateelements).ThesupportingindicesareI=fi0;i1;i2;i3g.Figure1showsthetypicalcongurationforaboundingrectanglewithedgecoincidenttoapolygonedgeandwiththreeadditionalsupportingvertices. Figure1.Thetypicalcongurationofaboundingrectanglecorrespondingtoapolygonedge.ThepolygonverticesareVifor0i8;theyarelabeledinthegureonlywiththesubscript.Thecurrentrectangleisdrawninred,hasaxisdirectionsU0andU1,andissupportedbyedgeh0;1i.Theminimumrotationisdeterminedbyedgeh5;6i,whichmakesitthenextedgetoprocess.ThenextrectangleisdrawninblueandhasaxisdirectionsU00andU01. InFigure1,theinitialrectanglehasitsbottomedgesupportedbypolygonedgeh0;1i;considerthecounterclockwise-mostvertexV1tobethesupportingvertex.TherightedgeoftherectangleissupportedbyV3,thetopedgeoftherectangleissupportedbyV5,andtheleftedgeoftherectangleissupportedbyV7.ThesupportingindicesfortheaxisdirectionsfU0;U1gareI=f1;3;5;7g.Thecandidaterotationanglesarethoseformedbyh1;2iwiththebottomedge,h3;4iwiththerightedge,h5;6iwiththetopedge,andh7;8iwiththeleftedge.Theminimumanglebetweenthepolygonedgesandtherectangleedgesisattainedbyh5;6i.Rotatethecurrentrectangletothenextrectanglebytheanglebetweenthetopedgeofthecurrentrectangleandh5;6i.Figure1showsthatthenextrectanglehasedgecoincidentwithh5;6i,buttheotherpolygonedgeswerenotreachedbythisrotationbecausetheiranglesarelargerthantheminimumangle.Thenamingofthenextrectangleedgesisdesignedtoprovidethecongurationwehadfortheinitialrectangle;thenewaxisdirectionsaredrawinblueandprovidetheorientationthat4 goeswiththenames.Thenewbottomedgeish5;6i,anditscounterclockwise-mostvertexV6ischosenasthesupportpointforthebottomedge.ThenewaxisdirectionsareU00=(V6�V5)=jV6�V5jandU01=�Perp(U00).ThesupportingindicesfortheaxisdirectionsfU00;U01gareI0=f6;7;1;3g.Asingleloopisusedtovisittheminimum-areacandidaterectangles.TheloopinvariantisthatyouhaveaxisdirectionsfU0;U1gandsupportingindicesI=fb;r;t;`g,orderedbytheedgestheysupport:bottom(b),right(r),top(t),andleft(`).Thepolygonedgehb�1;biiscoincidentwiththerectangle'sbottomedge.Vbisrequiredtobethecounterclockwise-mostvertexonthebottomedge.Inthepreviousexample,52Iisselectedbecauseoftheminimum-angleconstraint.TheremainingelementsofIsupporttherotatedrectangle,sotheybecomeelementsofI0.ThesuccessorofV5isV6,so6becomesanelementofI0.3.1ComputingtheInitialRectangleLetthepolygonverticesbeVifor0in.Theinitialrectanglehasitsbottomedgecoincidentwiththepolygonedgehn�1;0i.Bystartingwiththisedge,weavoidusingthemodulusargumentintheloopindexing.TheinitialaxisdirectionsareU0=(V0�Vn�1)=jV0�Vn�1jandU1=�Perp(U0).ThepolygonverticesareconvertedtothecoordinatesystemwithoriginV0andaxisdirectionsU0andU1.Wemustsearchfortheextremevaluesoftheconvertedpoints.Anextremevalueistypicallygeneratedbyonevertex,butitispossiblethatitcanbegeneratedbytwovertices.Inparticular,thishappenswhenapolygonedgeisparalleltooneoftheaxisdirections.Becausewehaveassumedthatnotripleofverticesiscollinear,itisnotpossibletohavethreeormoreverticesattaintheextremevalueinanaxisdirection.Whenanextremevalueoccurstwicebecauseapolygonedgeisparalleltoanaxisdirection,thesupportingpointforthecorrespondingedgeoftherectangleischosentobethecounterclockwise-mostvertexoftheedge.Thisisrequiredbecausetheupdatestepcomputesrotationanglesforedgesemanatingfromthesupportingvertex,andthoseedgesmusthavedirectionpointingtotheinterioroftherectangle.Forexample,Figure1showsthath0;1iisparalleltoU0.WechooseV1asthesupportvertex,andtheemanatingedgeusedindeterminingtheminimalrotationangleish1;2i.Listing2showshowtocomputethesmallestrectanglethathasanedgecoincidentwithaspeciedpolygonedge.TheRectanglestructurestorestheaxisdirectionsandthesupportingindices.Theareaoftherectangleisalsostoredforuseintherectangleupdatewhenapplyingtherotatingcalipersalgorithm.Thisfunctionisusedtocomputetheinitialrectanglefortherotatingcalipersmethod,butintheimplementationitisalsousedfortheexhaustivealgorithm. Listing2.Pseudocodeforcomputingthesmallestrectanglethathasanedgecoincidentwithaspeciedapolygonedgehi0;i1i.structRectanglefVector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;U[2];intindex[4];//order:bottom,right,top,leftRealarea;g;RectangleSmallestRectangle(intj0,intj1,std::vectorVector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;const&vertices)fRectanglerect;rect.U[0]=vertices[j1]=vertices[j0];Normalize(rect.U[0]);//lengthofrect.U[0]is1rect.U[1]==Perp(rect.U[0]);rect.index=fj1,j1,j1,j1g;5 Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;origin=vertices[j1];Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;zero(0,0);Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;d ;� Td;&#x [00;support[4]=fzero,zero,zero,zerog;for(size ti=0;ivertices.size();++i)f//Convertvertices[i]tocoordinatesystemwithoriginvertices[j1]and//axisdirectionsrect.U[0]andrect.U[1].Theconvertedpointisv.Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;diff=vertices[i]=origin;Vector2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;v=fDot(rect.U[0],diff),Dot(rect.U[1],diff)g;//Theright=mostvertexofthebottomedgeisvertices[i1].The//assumptionofnotripleofcollinearverticesguaranteesthat//rect.index[0]isi1,whichistheinitialvalueassignedatthe//beginningofthisfunction.Therefore,thereisnoneedtotest//forotherverticesfarthertotherightthanvertices[i1].if(v[0]&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;support[1][0]jj(v[0]==support[1][0]&&v[1]&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;support[1][1]))f//NewrightmaximumORsamerightmaximumbutclosertotop.rect.index[1]=i;support[1]=v;gif(v[1]&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;support[2][1]jj(v[1]==support[2][1]&&v[0]support[2][0]))f//NewtopmaximumORsametopmaximumbutclosertoleft.rect.index[2]=i;support[2]=v;gif(v[0]support[3][0]jj(v[0]==support[3][0]&&v[1]support[3][1]))f//NewleftminimumORsameleftminimumbutclosertobottom.rect.index[3]=i;support[3]=v;gg//Thecommentintheloophastheimplicationthatsupport[0]=zero,sothe//height(support[2][1]=support[0][1])issimplysupport[2][1].rect.area=(support[1][0]=support[3][0])*support[2][1];//width*heightg Figure2showsacongurationwheretwopolygonedgesareparalleltorectangleedges,inwhichcasetwoextremevaluesoccurtwice. Figure2.Thesupportingverticesofthepolygonaredrawninred.h0;1isupportsthebottomedgeoftherectangleandV1isasupportingvertex.h3;4isupportstherightedgeoftherectangleandV4isasupportingvertex. 6 3.2UpdatingtheRectangleAftercomputingtheinitialrectangle,weneedtodeterminetherotationangletoobtainthenextpolygonedgeanditscorrespondingrectangle.Thetypicalcongurationofonecoincidentedgeandthreeothersupportingverticeshastheessenceoftheideafortherectangleupdate,butothercongurationscanarisethatrequirespecialhandling.Inthediscussion,arithmeticontheindicesintothepolygonverticesisperformedmodulothenumberofvertices:theindicesi+j,i+j+n,andi+j�nallrefertothesamevertex.3.2.1DistinctSupportingVerticesThetypicalcongurationofFigure1isthesimplesttoupdate.Asinglepolygonedgeiscoincidentwiththerectangleandhasuniqueverticessupportingtheotherthreeedges.ThesupportingindicesareI=fi0;i1;i2;i3g.Lethij;ij+1ibetheuniqueedgethatformstheminimumanglewiththerectangleedgeitemanatesfrom,wherej2f0;1;2;3g.InFigure1,thevertexfromwhichtheedgeemanatesisV5(j=2).Thenewcoincidentedgeishij;ij+1i.ThenewsupportingindicesI0areobtainedfromIbyreplacingijwithij+1andbypermutingsothatijistherst(bottom-supporting)vertex:I0=fij+1;ij+1;ij+2;ij+3g.Theadditionsinthej-subscriptsarecomputedmodulo4.3.2.2DuplicateSupportingVerticesItispossiblethatonepolygonvertexsupportstworectangleedges,inwhichcasethevertexisacorneroftherectangle.Figure3showssuchaconguration.Thecurrentrectanglesaredrawninredandthenextrectanglesaredrawninblue. Figure3.(a)Acongurationwhereapolygonvertexsupportstworectangleedges.V3supportstherightedgeandthetopedge.ThesupportingindicesforthecurrentrectangleareI=f1;3;3;5g.ThesupportingindicesforthenextrectangleareI0=f6;1;3;3g.(b)Theduplicate-vertexcongurationcanbethoughtofasthelimitingcaseofadistinct-vertexconguration,whereVT!V3.ThesupportingindicesforthecurrentrectangleareI=f1;3;T;5g.ThesupportingindicesforthenextrectangleareI0=f6;1;3;Tg.NowmoveVTtotherightsothatitcoincideswithV3.Thecurrentrectanglesdonotchangeduringthisprocess,butthetopedgeofthenextrectangleofthedistinct-vertexcasebecomesthetopedgeofthenextrectangleoftheduplicate-vertexcase. (a)(b) 7 Theselectionofthenewcoincidentedgeandtheupdateofthesupportingindicesareperformedjustasinthecaseofdistinctsupportingindices,exceptthereisnominimum-anglecandidateedgeemanatingfromtherstoftheduplicatevertices.IntheexampleofFigure3(a),I=f1;3;3;5g.h1;2iisacandidateedge.ThereisnoedgeemanatingfromtherstoccurrenceofV3.Oryoucanthinkoftheedgeash3;3i,whichisdegenerate.Ineithercase,thatsupportingvertexisskipped.TheedgeemanatingfromthesecondoccurrenceofV3,namely,h3;4i,isacandidateedge.Finally,h5;6iisacandidateedge.Asillustrated,theminimum-angleedgeish5;6i.I0isgeneratedfromIbyreplacingV5byV6,thecounterclockwise-mostvertexoftheedge,andtheverticesarecycledsothatV6occursrst:I0=f6;1;3;3g.3.2.3MultiplePolygonEdgesAttainMinimumAngleItispossiblethattheminimumangleisattainedbytwoormorepolygonedges.Figure4illustratesseveralpossibilities.8 Figure4.(a),(b),and(c)arecongurationswheretwopolygonedgesattaintheminimumangle.(d)isacongurationwherethreepolygonedgesattaintheminimumangle.Thediscussionafterthisgureisaboutthevariouscases. (a)(b) (c)(d) InFigure4(a),thesupportingindicesforthecurrentrectangleareI=f1;3;4;6g.Twopolygonedgesattaintheminimumrotationangle,h1;2iandh4;5i.Choosingh1;2iasthenewbottomedge,thecorrespondingsupportisV2.h4;5isupportsthenewtopedgeandthesupportisV5.ThenewsupportingindicesareI0=f2;3;5;6g.ToobtainI0fromI,theoldsupportV1isreplacedbytheotherendpointV2andtheoldsupportV4isreplacedbytheotherendpointV5.Nocyclingisnecessary,becauseV2isthebottomsupport9 and2isalreadytherstelementinI0.InFigure4(b),thesupportingindicesforthecurrentrectangleareI=f1;2;3;5g.Twopolygonedgesattaintheminimumotationangle,h1;2iandh3;4i.Choosingh1;2iasthenewbottomedge,thecorrespondingsupportisV2.h3;4isupportsthenewtopedgeandthesupportisV4.ThenewsupportingindicesareI0=f2;2;4;5g.ObservethatV2isaduplicatethatsupportstwoedgesoftherectangle.ToobtainI0fromI,theoldsupportV1isreplacedbytheotherendpointV2andtheoldsupportV3isreplacedbytheotherendpointV4.Nocyclingisnecessary,becauseV2isthebottomsupportand2isalreadytherstelementinI0.InFigure4(c),thesupportingindicesforthecurrentrectangleareI=f1;2;3;4g.Twopolygonedgesattaintheminimumrotationangle,h1;2iandh3;4i.Choosingh1;2iasthenewbottomedge,thecorrespondingsupportisV2.h3;4isupportsthenewtopedgeandthesupportisV4.ThenewsupportingindicesareI0=f2;2;4;4g.ObservethatV2isaduplicatethatsupportstwoedgesoftherectangleandV4isaduplicatethatsupportstheothertwoedges.ToobtainI0fromI,theoldsupportV1isreplacedbytheotherendpointV2andtheoldsupportV3isreplacedbytheotherendpointV4.Nocyclingisnecessary,becauseV2isthebottomsupportand2isalreadytherstelementinI0.InFigure4(d),thesupportingindicesforthecurrentrectangleareI=f1;3;4;6g.Threepolygonedgesattaintheminimumrotationangle;h1;2i,h4;5i,andh6;0i.Choosingh1;2iasthenewbottomedge,thecorrespondingsupportisV2.h4;5isupportsthenewtopedgeandthesupportisV5.h6;0isupportsthenewleftedgeandthesupportisV0.ThenewsupportingindicesareI0=f2;3;5;0g.ToobtainI0fromI,theoldsupportV1isreplacedbytheotherendpointV2,theoldsupportV4isreplacedbytheotherendpointV5,andtheoldsupportV6isreplacedbytheotherendpointV0.3.2.4TheGeneralUpdateStepThecurrentrectanglehasbottomedgecoincidentwiththepolygonedgehVi0�1;Vi0i.ThecorrespondingaxisdirectionisU0=(Vi0�Vi0�1)=jVi0�Vi0�1j.TheperpendicularaxishasdirectionU1=�Perp(U0).ThesupportingindicesareI=fi0;i1;i2;i3g.Thenextrectanglehasbottomedgecoincidentwiththepolygonedgehi00�1;i00i.ThecorrespondingaxisdirectionisU00=(Vi00�Vi00�1)=jVi00�Vi00�1j.TheperpendicularaxishasdirectionU01=�Perp(U00).ThesupportingindicesareI0=fi00;i01;i02;i03g.WeneedtodetermineI0fromI.DenethesetMf0;1;2;3gasfollows:ifm2M,thenim2Iandthepolygonedgehim;im+1iformstheminimumanglewiththerectangleedgesupportedbyVim.Theindexadditionim+1iscomputedmodulon.WhenMhasmultipleelements,theminimumangleisattainedmultipletimes.IntheexampleofFigure1,I=f1;3;5;7gandh5;6iattainstheminimumangle,soM=f2g.IntheexampleofFigure3(a),I=f1;3;3;5gandh5;6iattainstheminimumangle,soM=f3g.IntheexamplesofFigure4(a,b,c),M=f0;2g.IntheexampleofFigure4(d),M=f0;2;3g.StartwithanemptyM.Processeachik2Iforwhichik+16=ik,wheretheindexadditionk+1iscomputedmodulo4.Computetheanglekformedbyhik;ik+1iandthesupportedrectangleedge,wheretheindexadditionik+1iscomputedmodulon.Itisnecessarythatk0.Ifk�0,storethepair(k;k)inasetAsortedontheanglecomponent.AfterprocessingallelementsofI,ifAisnonempty,thenitsrstelementis(kmin;kmin).IterateoverAandinsertalltheindexcomponentskintoMforwhichk=kmin.ItispossiblethatAisempty,inwhichcasetheoriginalpolygonmustalreadybearectangle.TogenerateI0fromI,foreachm2Mreplaceimbyim+1,wheretheindexadditioniscomputedmodulo10 n.TheresultingmultisetoffourelementsisJ.Letm0bethesmallestelementofM.Thenextrectangleisselectedsothatitsbottomedgeissupportedbyhim0�1;im0i;thatis,i00=im0.CycletheelementsofJsothatim0occursrst,whichproducesthemultisetI0.4ARobustImplementationThecorrectnessofthealgorithmdependsrstontheinputpolygonbeingconvex,evenwhentheverticeshave oating-pointcomponents.Experimentshaveshownthatformoderatelylargen,generatingnrandomanglesin[0;2),sortingthem,andthengeneratingvertices(acos;bsin)onanellipse(x=a)2+(y=b)2=1canleadtoavertexsetforwhichnotallthegeneratedverticesarepartoftheconvexhull.Ofcourse,thisisaconsequenceofnumericalroundingerrorswhencomputingthesineandcosinefunctionsandtheproductsbytheellipseextents.Ifaconvexhullalgorithmisnotappliedtothesepoints,theexhaustivealgorithmandtherotatingcalipersalgorithmcanproduceslightlydi erentresults.Thecorrectnessofthealgorithmisalsothwartedbytheroundingerrorsthatoccurwhennormalizingtheedgestoobtainunit-lengthvectors.Theseerrorsarefurtherpropagatedinthevariousalgebraicoperationsusedtocomputetheanglesandareas.Toobtainacorrectresult,itissucienttouseexactrationalarithmeticintheimplementation.However,normalizingavectorgenerallycannotbedoneexactlyusingrationalarithmetic.Thenormalizationcanactuallybeomittedbymakingsomealgebraicobservations.Theminimum-areaboxcanbecomputedexactly.Ifyouneedanoriented-boxrepresentationwithunit-lengthaxisdirections,youcanachievethisattheveryendoftheprocess,incurring oating-pointroundingerrorsonlyintheconversionofrationalvaluesto oating-pointvalues.Theseoccurwhencomputingtheextentsintheaxisdirectionsandwhennormalizingvectorstoobtainunit-lengthdirections.4.1AvoidingNormalizationConsiderthecomputationofthesmallestrectangleforaspeciedpolygonedgehj0;j1i.LetW0=Vj1�Vj0andW1=�Perp(W0),whicharetheunnormalizedaxisdirectionswithjW1j=jW0j.Theunit-lengthaxisdirectionsareUi=Wi=jW0jfori=0;1.LetthesupportverticesbeVb,Vr,Vt,andV`whosesubscriptsindicatetheedgestheysupport:bottom(b),right(r),top(t),andleft(`).Thewidthwandheighthoftherectanglearew=U0(Vr�V`);h=U1(Vt�Vb)(1)andtheareaaisa=wh=(W0(Vr�V`))(W1(Vt�Vb)) jW0j2(2)Boththenumeratoranddenominatorofthefractionfortheareacanbecomputedexactlyusingrationalarithmeticwhentheinputverticeshaverationalcomponents.Theconstructionofthesupportverticesusescomparisonsofdotproducts.Forexample,Listing2hasaloopwithcomparisonsforthecurrentright-edgesupport.Thecomparisonv[0]�support[1][0]isanimplementationofU0(Vi�Vi1)�U0(Vr0�Vi1)(3)whereVi1istheoriginofthecoordinatesystem,r0istheindexintothevertexarraythatcorrespondstothecurrentsupportvertex,andiistheindexofthevertexunderconsideration.TheBooleanvalueofthe11 comparisonisequivalenttothatusingunnormalizedvectors,W0(Vi�Vi1)�W0(Vr0�Vi1)(4)Theexpressionsinthecomparisoncanbecomputedexactlyusingrationalarithmetic.4.2IndirectComparisonsofAnglesWemustlocatetheedgeemanatingfromasupportvertexthatpointsinsidetherectangleand,ofallsuchedges,formstheminimumanglewiththerectangleedgecorrespondingtothesupportvertex.LetDbetheunit-lengthrectangleedgedirection(usingcounterclockwiseordering).ThatdirectionisU0forthebottomedge,U1fortherightedge,�U0forthetopedge,and�U1fortheleftedge.ForanemanatingedgeoriginatingatsupportvertexVsandterminatingatVfinsidetherectangle,thevectorformedbytheverticesisE=Vf�Vs.TheanglebetweenDandEisdeterminedbycos=DE jEj(5)Wecanapplytheinversecosinefunction(acos)toextracttheangleitself.DisobtainedbynormalizingoneofU0orU1,whichwealreadyknowcannotgenerallybedoneexactlywithrationalarithmetic.Similarly,wemustcomputethelengthofE,e ectivelyanothernormalization.Andtheinversecosinefunctioncannotbecomputedexactly.Observethat2(0;=2].Theanglemustbepositivebecausetheedgeisdirectedtowardtherectangleinterior.Iftheangleattainsthemaximum=2,thenthesupportvertexmustbeacorneroftherectangle.Unlessthepolygonisalreadyarectangle,theremustbeatleastoneotherboundingrectangleedgethathasanemanatingedgewithanglesmallerthan=2.Toavoidtheinexactcomputationstoobtain,wecaninsteadcomputethequantitiesjU0j2sin2,whichisapositiveandincreasingfunctionof.SortingofandjU0j2sin2leadtothesameordering.UsingsomealgebraandtrigonometrywemayshowthatjU0j2sin2=(WiPerp(E))2 jEj2(6)whereD=Uiforsomechoiceofsignandofindexi.Thefractionontheright-handsideoftheequationcanbecomputedexactlyusingrationalarithmetic.Thequantitiesofequation(6)arecomputedforallemanatingedgesandstoredinanarrayastheyareencountered.Weneedtosortthevaluestoidentifythosethatattaintheminimumvalue.Althoughanycomparison-basedsortwillwork,swappingofarrayelementshasanassociatedcostwhenthevaluesarearbitraryprecisionrationalnumbers.Ourimplementationusesanindirectsortbymaintaininganarrayofintegerindicesintothearrayofvalues,andtheintegerindicesareswappedasneededtoproduceanalintegerarraythatrepresentsthesortedvalues.4.3UpdatingtheSupportInformationThisisastraightforwardimplementationoftheideasofSection3.2.4.Theonlyissuenotmentionedyetisthatastherotatingcalipersalgorithmprogresses,thepolygonedgesarenotvisitedsequentiallyin-order.12 Moreover,ifthepolygonhaspairsofparalleledge,itispossiblethattheminimum-arearectangleiscomputedwithoutvisitingallpolygonedges.Inourimplementation,wemaintainanarrayofnBooleanvalues,oneforeachcounterclockwise-mostvertexofapolygonedge.TheBooleanvaluesareallinitiallyfalse.Afterthesmallestrectangleiscomputedforapolygonedgehi0;i1i,theBooleanvalueatindexi1issettotrue.Whenthesupportinformationisupdatedduringeachpassofthealgorithm,theBooleanvalueistestedtoseewhethertheedgehasbeenvisitedasecondtime.Ifithasbeenvisitedbefore,thealgorithmterminates(otherwise,wewillhaveaninniteloop).4.4ConversiontoaFloating-PointRectangleOurimplementationoftherotatingcalipersalgorithmtrackstheminimum-arearectangleviathedatastructurestructRectanglefVector2ComputeType&#x]TJ/;ག ;.97;8 T; 5.;݉ ;� Td;&#x [00;U[2];std::arrayint,4�index;//order:bottom,right,top,leftRealsqrLenU0;//squaredlengthofU[0]Realarea;g;ThisisslightlymodiedfromthatofListing2;thesquaredlengthofU0isstoredforusethroughoutthecode.TheComputeTypeisarationaltype.Therectanglecenterandsquaredextentscanbecomputedexactlyfromtheminimum-arearectangleminRect,Vector2ComputeType&#x]TJ/;ག ;.97;8 T; 5.;݉ ;� Td;&#x [00;sum[2]=frvertices[minRect.index[1]]+rvertices[minRect.index[3]],rvertices[minRect.index[2]]+rvertices[minRect.index[0]]g;Vector2ComputeType&#x]TJ/;ག ;.97;8 T; 5.;݉ ;� Td;&#x [00;difference[2]=frvertices[minRect.index[1]]=rvertices[minRect.index[3]],rvertices[minRect.index[2]]=rvertices[minRect.index[0]]g;Vector2ComputeType&#x]TJ/;ག ;.97;8 T; 5.;݉ ;� Td;&#x [00;center=mHalf*(Dot(minRect.U[0],sum[0])*minRect.U[0]+Dot(minRect.U[1],sum[1])*minRect.U[1])/minRect.sqrLenU0;Vector2ComputeType&#x]TJ/;ག ;.97;8 T; 5.;݉ ;� Td;&#x [00;sqrExtent;for(inti=0;i2;++i)fsqrExtent[i]=mHalf*Dot(minRect.U[i],difference[i]);sqrExtent[i]*=sqrExtent[i];sqrExtent[i]/=minRect.sqrLenU0;gThearraynamerverticesstorestherationalrepresentationofthe oating-pointverticesreferenceinpreviouspartsofthisdocument.Theconversiontothe oating-pointrepresentationoftheboxhassomelossofprecision,butthisoccursattheveryend.OrientedBox2Real&#x]TJ/;ག ;.97;8 T; 6.;e ;� Td;&#x [00;itMinRect;//input=typeminimum=arearectangle;for(inti=0;i2;++i)fitMinRect.center[i]=(Real)center[i];13 itMinRect.extent[i]=sqrt((Real)sqrExtent[i]);//Beforeconvertingtofloating=point,factoroutthemaximum//componentusingComputeTypetogeneraterationalnumbersina//rangethatavoidslossofprecisionduringtheconversionand//normalization.Vector2ComputeType&#x]TJ/;ག ;.97;8 T; 5.;݉ ;� Td;&#x [00;axis=minRect.U[i];ComputeTypecmax=max(abs(axis[0]),abs(axis[1]));ComputeTypeinvCMax=1/cmax;for(intj=0;j2;++j)fitMinRect.axis[i][j]=(Real)(axis[j]*invCMax);gNormalize(itMinRect.axis[i]);g//Quantitiesaccessiblethroughget=accessorsinthepublicinterface.supportIndices=minRect.index;area=(Real)minRect.area;Ifyouneedtocontinueworkingwiththeexactrationaldatageneratedbytherotatingcalipersalgorithmratherthanthe oating-pointrectangle,ourimplementationallowsyouaccesstoit.ThesourcecodefortheimplementationisintheleMinimumAreaBox2.h.References[1]H.FreemanandR.Shapira.Determiningtheminimum-areaencasingrectangleforanarbitraryclosedcurve.InCommunicationsoftheACM,volume18,pages409{413,NewYork,NY,July1975.[2]JosephO'Rourke.Findingminimalenclosingboxes.InternationalJournalofComputer&InformationSciences,14(3):183{199,1985.[3]FrancoP.PreparataandMichaelIanShamos.ComputationalGeometry:AnIntroduction.Springer-Verlag,NewYork,1985.[4]MichaelIanShamos.Computationalgeometry.PhDthesis,YaleUniversity,1978.[5]GodfriedToussaint.Solvinggeometricproblemswiththerotatingcalipers.InProceedingsoftheIEEE,Athens,Greece,1983.14