Partitions of a Convex Polygon

Transcription

1 HAPTER 6 Partitios of a ovex Polygo 6 INTRODUTION I our survey we have come across some results o eumeratio of ocrossig cofiguratios o the set of vertices of a covex polygo, such as triagulatios ad trees I [9] exact formulae ad limit laws are determied for several parameters of iterest by Marc Noy, some results o the eumeratio of chord diagrams (pairigs of vertices of a covex polygo by meas of disjoit pairs) were preseted He also preseted limit laws for the umber of compoets, the size of the largest compoet ad the umber of crossigs The use of geeratig fuctios ad of a variatio of Levy's cotiuity theorem for characteristic fuctios eabled to establish that most of the limit laws preseted here are Gaussia Michael S Floater ad Tom Lyche [0] provided a ew way of eumeratig all partitios of a covex polygo of a certai type, ie, with a specified umber of triagles, quadrilaterals, ad so o, which icludes atala umbers as a special case We adopt the techiques of partitioig a polygo from [9,0] ad figure out some results o covex polygos 95

2 I Sectio 6 we state the defiitios of required termiology ad develop results o eumeratio of partitios of ovex Polygos which are partitioed ito co-iitial polygos ad also ito o co-iitial polygos i Sectio 6 Fially we preset scope for further work ad refereces The followig vocabulary o partitio of a polygo is adopted from the literature 6 DEFINITIONS AND NOTATION 6 Polygo: A polygo is a plae figure that is bouded by a closed path or circuit, composed of a fiite sequece of straight lie segmets 6 ovex Polygo: A covex polygo is a polygo whose iterior is a covex set 6 Simple Polygo: A simple polygo is a closed polygoal chai of lie segmets i the plae which do ot have poits i commo other tha the commo vertices of pairs of cosecutive segmets 64 Partitio of a polygo: A partitio of a polygo P is a set of polygos such that the iteriors of the polygos do ot itersect ad the uio of the polygos is equal to the iterior of the origial polygo P 65 Segmet of a polygo: Let V, V,, V be the vertices of a polygo P If there exists a lie VV such that the lie divides P ito two polygos P i j,p such that PU P = P ad PP VV i j, the the lie VV is called segmet of the i j polygo This partitioed polygo is deoted by i,j 66 r-partitioed polygo: If segmets divide the polygo ito r parts, the the polygo is called r-partitioed polygo The umber of r-partitioed polygos 96

3 of p is deoted by P r ( p ) If Vi V j, V,, i V j V ir V jr segmets divide the polygo ito r parts, the this r -partitioed polygo is deoted byi, j ; i, j ;; ir, jr Note: r - segmets divide the polygo ito r parts 67 o-iitial segmets: If all the segmets start from same vertex of the polygo, the these lies are called co-iitial segmets 68 o-iitial r-partitioed polygo: If segmets divide the polygo ito r parts ad all segmets have same iitial poit, the the polygo is called co-iitial r-partitioed polygo 69 No-co-iitial r-partitioed polygo: If segmets divide the polygo ito r parts ad o two segmets have same iitial poit, the the polygo is called o-co-iitial r-partitioed polygo 60 o-iitial r-sided polygo partitio of covex polygo: If a covex polygo is partitioed ito r-sided polygos ad all r-sided polygos have same vertex, the this partitioig is called co-iitial r-sided polygo partitio of covex polygo 97

4 6 ENUMERATION OF r partitioed POLYGONS Here we make a attempt to study the eumeratio of co-iitial r- partitioed polygos of a covex polygo We also prove certai results to eumerate the umber of o co-iitial r-partitioed polygos Vi Let P be a closed covex polygo with vertices V, V,, V If ji, V j is a segmet of P A collectio of segmets is said to be co-iitial if they have a commo vertex A segmet Vi deote this partitio by i, j V j partitios (=divides) P ito two parts We Result 6: With otatio as above, the umber of partitios of a -sided covex polygo formed by k co-iitial segmets is k where k Proof: For defiiteess let V be the commo vertex Ay set K of segmets with commo vertex V is obtaied by the - vertices V, V,, V These ca be chose i k ways Sice each of these vertices gives rise to a partitio alog with the vertexv, the umber of partitios is precisely k Similarly for ay commo vertexv i, the umber of partitios of a -sided covex polygo is k Hece the total umber of partitios of a -sided covex polygo formed by k co- k iitial lies is Theorem 6: The umber of partitios of a -sided covex polygo formed by oe partitioed lie is 98

5 Proof: Let V, V,, V be the vertices of a polygo P The possible -partitioed polygos with oe vertex of the partitio lie at V are,,4,5,6,-,- The umber is Therefore for ay vertex V i, the umber of partitios of a -sided covex polygo is But every partitio lie repeats twice like VV i jad VV j i Hece the total umber of partitios of a -sided covex polygo formed by oe partitio lie is Theorem 6: The umber of co-iitial partitios of a -sided covex polygo is Proof: From theorem 6 we observe that the umber of -partitioed polygos of -sided covex polygo is Also from theorem 6 we observe that co-iitial -partitioed, 4-partitioed,, (-)-partitioed polygos of -sided covex polygo are,,, respectively Therefore the umber of co-iitial partitios of a -sided covex polygo is Example: The umber of co-iitial partitioed polygos of a hexago is 4 Let A A A A4 A5 A 6 be a hexago 99

6 The partitios of a hexago formed by oe segmet are,,4,5,4,5,6,5,6 4,6 The umber of partitios of a hexago formed by segmet lie is The -partitioed polygos of hexago are,;,4,;,5,4;,5,4;,5,4;,6,5;,6,5;,6,5;,,6;, 4,6;4, 4,6;4, 4,;4, 5,;5, 5,;5, 5,;5, 6,;6, 6,;6,4 6,;6,4 Therefore the umber of co-iitial -partitioed polygos of a hexago is The umber of co-iitial partitioed polygos of a hexago Theorem 64: The umber of partitios of a -sided covex polygo formed by k o-co-iitial lies is k Proof: Let V, V,, V be the vertices of a polygo P Let k 00

7 From theorem 6, the umber of partitios at V is of partitios is ad the total umber We assume that the umber of partitios of a -sided covex polygo formed by m o-co-iitial segmets is m Let k m ad cosider the segmets Vi V j, V,, i V j V i V m jm FixV i, V i The each of the remaiig k segmets divides the polygo with vertices V, V,, V except Vi at V i m ways Fix V i, V i The each of the remaiig k segmets divides the polygo with vertices V, V,, V except Vi, V at i V i m ways Proceedig this way up to the ed, we ultimately get that the umber of partitios of a -sided covex polygo formed by m+ o-co-iitial segmets at V is 4 5 k m m m m There fore the umber of partitios of a -sided covex polygo formed by m+ o-co-iitial segmets is m Hece the umber of partitios of a -sided covex polygo formed by k o-coiitial segmets is k Theorem 65: The umber of o-co-iitial partitioed polygos of a -sided covex polygo is 5 Proof: From theorem 64 we observe that o co-iitial -partitioed, - partitioed,, -partitioed polygos of -sided covex polygo are,, 5 respectively Hece the umber of o-co-iitial partitioed polygos of -sided covex polygo is 0

8 Example: The umber of o-co-iitial partitioed polygos of a 9-sided polygo is 4 Solutio: Let A A A A4 A5 A6 A7 A8 A 9 be a 9-sided polygo The -partitioed polygos with repetitio beig rouded are Therefore the umber of o-co-iitial -partitioed polygos is The -partitioed polygos are

9 Hece the umber of o-co-iitial -partitioed polygos is The 4-partitioed polygos are

10 (repetitios are rouded off) Hece the umber of o-co-iitial 4-partitioed polygos is Hece the umber of o-co-iitial partitioed polygos of 9-sided covex polygo is Theorem 66: If r, the the umber of co-iitial r-sided polygo partitios of a -sided covex polygo is ( ) r Proof: Let P be a -sided covex polygo ad r N such thatr The -sided covex polygo ca be partitioed as r-sided polygos The left ad right r-sided polygos have oe segmet as a side ad remaiig sides are sides of P ad remaiig r-sided polygos have two segmets as sides ad remaiig sides are sides of polygo P The left ad right r-sided polygos have r- sides which are sides of polygo P ad remaiig r-sided polygos have (r-)-sides that are sides of polygo P 04

11 Hece the umber of co-iitial r-sided polygo partitios of -sided covex polygos is ( ) r Theorem 67: If r the umber of r -partitioed polygos of a -sided covex polygo partitioed by a closed chai coected with segmets is r r r Proof: Let V, V,, V be vertices of a polygo P Let the closed chai coected with segmets Let r V V, V V,, V V be i i i i ir i (, ;,5 ; 5,7), (, ;,5 ; 5,8), (, ;,5 ; 5,9),, (, ;,5 ; 5,-) (, ;,6 ; 6,8), (, ;,6 ; 6,9),, (, ;,6 ; 6,-) (, ;,- ; -,-) (,4 ; 4,6 ; 6,8), (,4 ; 4,6 ; 6,9),, (,4 ; 4,6 ; 6,-) The umber of above partitioed polygos are thrice 4 Hece the umber of above partitioed polygos are Whe r r r (,4 ; 4,- ; -,-) (,-5 ; -5,- ; -,-) But every polygo repeats k we ca similarly prove that the umber of partitioed polygos are But every polygo is repeated r times Hece the umber of partitioed polygos are r 4 r r Hece the umber of r partitioed polygos of -sided covex polygo partitioed by the closed chai coected with segmets is r r r 05

12 SOPE FOR FURTHER WORK Motivated by the above out come of our ivestigatio o eumeratio of iteger partitios ad partitios of covex polygos, we fid iterestig to cotiue our pursuit alog the followig lies Extesio of these results to overpartitio pairs Relatio betwee r-partitios of differet positive itegers for which umber of r-partitios is maximum Developig programs for the eumeratio of r partitios of from the ew results i MATLAB Preparatio of programs for LS (List Scheduligs) algorithm by employig built i fuctio 'fmico' i MATLAB for r partitio optimizig some objective fuctios like maximizig the sum of the smallest k parts, miimizig the ratio of the largest to the smallest part i which k r omputatioal aspects of r partitios of covex polygos with the aid of NEURAL NETWORKS aimig at traffic cotrol problems L metric performace o the LS (List Scheduligs) algorithm Eumeratio of co-iitial ad o co-iitial r-partitioed polygos, partitioed polygos i a secod layered polygo [9] Search for ay possible lik betwee iteger partitios ad partitios of covex polygos 06

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