Randomly Generated Triangles whose Vertices are Vertices of a Regular Polygon

Transcription

1 Radoly Geerated Triagles whose Vertices are Vertices of a Regular Polygo Aa Madras Drury Uiversity Shova KC Hope College Jauary, 6 Abstract We geerate triagles radoly by uiforly choosig a subset of three vertices fro the vertices of a regular polygo We deterie the expected area ad perieter i ters of the uber of sides of the polygo We use cobiatorial ethods cobied with trigooetric suatio forulas arisig fro coplex aalysis We also deterie the liit of these equatios to copare with a classical result o triagles whose vertices are o a circle Itroductio This paper is a exploratio ito a ethod of radoly geeratig triagles We start with a regular -go iscribed i a uit circle The out of possible subsets of vertices of size three, oe subset is chose uiforly The three vertices for our radoly geerated triagle Research ito this ethod of geeratig triagles has lead us to four ai questios: What is the average area of a triagle geerate by this process? What is the liit of this area as teds to ifiity? What is the average perieter of a triagle geerate by this process? What is the liit of this perieter as teds to ifiity? I this paper, we preset the aswers to the above questios I Sectio, we preset backgroud iforatio, icludig the solutio to a related proble dealig with three radoly chose poits o a uit circle I particular, we detail the followig classical result: Theore : Suppose that three poits are radoly chose uiforly ad idepedetly o the circuferece of a uit circle The the expected area of the triagle fored is ad the expected perieter of the triagle fored is I Sectio, we give exaples of fidig expected area ad perieter i our situatio for certai sall values of

2 I Sectio 4, we preset the ai theores ad proofs These are suarized i the followig result Theore : Let P be a regular -go iscribed i a uit circle If a set of three vertices of P is chose uiforly at rado, the the expected area A of the triagle fored is cot / A The expected perieter P of the triagle fored is P 6csc/ + cot/ Moreover, we have li A ad li P, which correspods to the results give by Theore I Sectio, we exted our ai result to deterie the expected area ad perieter of quadrilaterals, petagos, ad other polygos that are geerated radoly by choosig vertices of a regular -go We give two equatios that ca be used to fid both the expected area ad expected perieter of a -go geerated radoly fro the vertices of a regular -go Related Proble o a Circle Our research was otivated by the classical proble dealig with a uit circle [] Let P, Q, ad R be poits chose radoly, uiforly, ad idepedetly o the circuferece of a uit circle Two questios aturally arise: What is the expected area of P QR? What is the expected perieter of P QR? The solutio to the first questio ca be foud olie [] ad ca be suarized as follows: Theore If three poits are chose radoly, uiforly, ad idepedetly o the circuferece of a circle with radius oe, the the expected area of the triagle iscribed i the circle is Proof followig [] Oe of the poits ca be take to be, without loss of geerality Let θ ad θ be the polar agles of the other two poits, as see i Figure Due to syetry, we are able to assue without loss of geerality that θ is i the iterval [, ], while θ is allowed to take o ay value i the iterval [, The area of the triagle fored is give by A Aθ, θ si θ si θ si θ θ

3 Figure : Circle with Iscribed Triagle The expected area is give by A Aθ, θ dθ dθ This itegral ca be evaluated to give See [] for details of the itegratio A solutio for the average perieter ca be foud i a siilar way Theore If three poits are chose radoly, uiforly, ad idepedetly o the circuferece of a uit circle, the the expected perieter of the triagle iscribed i the circle is Proof The sae assuptios ca be ade as i the previous proof without loss of geerality Let θ ad θ be defied the sae as i the previous proof The legths of the sides, a, b, ad c, of the iscribed triagle are a si θ b si θ c si θ θ where θ [, ] ad θ [, the absolute value is used for c because θ ay be less tha θ Sice the perieter of the triagle is a + b + c, the itegral for fidig expected perieter is P [ a + b + c dθ dθ si θ + si θ + si θ θ si θ dθ + si θ dθ + + dθ dθ si θ θ dθ θ θ [ 4 si θ cos θ cos θ 8 cos θ + 6θ + 8 si θ 8 si θ si θ ] θ dθ dθ ] dθ

4 Exaples ad Geeral Forulas I this sectio, we give exaples relatig to radoly choosig a subset cosistig of three vertices of a regular -go We also show how the ethods used i these specific cases lead to the forula for geeral values of that will be eeded i the ext sectio Area Exaples Give a regular -go R, we uber the vertices,,, By a, b, c, we will ea the triagle iscribed i R whose vertices are ubered a, b, ad c We will deote the area of a, b, c as Aa, b, c ad the perieter of a, b, c as P a, b, c We will assue for coveiece that a < b < c Figure : Petago As a exaple, we calculate the area of,, iside a regular petago iscribed i a uit circle, as see i Figure To fid the area of,,, we add up the areas of the three triagles withi the triagle, which gives us the equatio A,, si θ cos θ + si θ cos θ + si θ θ cos θ θ The usig the double agle idetity we ca siplify this equatio to A,, si θ + si θ + si θ θ where θ ad θ 4 Note that θ ad θ are both ultiples of the cetral agle of petago which is equal to So, the area of,, ca be writte as A,, [si + si + si ] This ca also be writte as A,, [ si + si + si + ] 4

5 This aalysis easily geeralizes to yield a geeral equatio for fidig the area of a, b, c i ay regular -go iscribed i a uit circle: Aa, b, c [ si b a + si c b + si a c + ] We ow tur our attetio to the case of the first geeral questio of iterest I particular, if R is a regular petago iscribed i a uit circle, ad a set three vertices of R are chose uiforly at rado, the we copute the expected area of the triagle fored I order to do this, we eed to fid the areas of all possible triagles a, b, c where a < b < c 4 We the calculate b a, c b ad a c + for all those triagles ad apply the forula derived above By addig the areas of all of these triagles ad dividig the result by the uber of all such triagles, we will fid the expected area The table below gives all possible triagles i petago alog with b a, c b ad a c + a, b, c b a c b a c +,,,,,,4,,,,4,,4,,,,4,,4,,4 Table : Triagles Geerated fro a Regular Petago The expected area of a radoly geerated triagle based o the vertices of a regular petago iscribed i a uit circle is A a<b<c 4 Aa, b, c By exaiig Table, we cout fiftee s, te s, ad five s that occur as either b a, c b, or a c + By collectig all like ters ad factorig out /, the equatio for the expected area reduces to A si + si + si We ow tur our attetio to carryig out the aalogous coputatio whe 6 I other words, we are geeratig rado triagles by choosig a set of three vertices

6 a, b, c b a c b a c + a, b, c b a c b a c +,, 4,, 4,,,,4,,4,,,, 4,,4,,,,,,4,4,,,,,4 4,,4,,,,,4,,4, 4,4, 4 Table : Triagles Geerated fro a Regular Hexago of a regular hexago iscribed i a uit circle Agai, we list all such triagles i order to collect like ters i the su which gives the expected area of the triagle fored I this case, there are possible triagles sice 6 I the above table, there are twety-four s, eightee s, twelve s, ad six 4 s that occur as b a, c b, or a c + Siilar to the case of the petago, we ca collect like ters i the su A 6 6 a<b<c Aa, b, c to obtai the result A 6 4 si si 6 + si si 4 6 The geeral equatio for fidig the expected area of a triagle fored by choosig three vertices of a regular -go iscribed i a uit circle is give by A [ si b a + si c b + si a c + ], where the su is take over all itegers a, b, c such that a < b < c I the ext sectio, we will explore ethods of siplifyig this su by collectig like ters as above, ad the applyig trigooetric suatio idetities Perieter Exaples We ow proceed to eploy siilar ethods to copute the expected perieter of a triagle radoly geerated fro the vertices of a regular petago or hexago iscribed i a uit circle As before, we use these results to idicate the proper for for the correspodig expressio ivolvig a regular -go 6

7 Let a, b, c be defied usig vertices of a regular -go iscribed i the uit circle as above Usig basic trigooetry alog with a double agle forula, it follows that the perieter of a, b, c is give by P a, b, c [ si b a + sic b + si + a c ] Let R be a regular petago iscribed i a uit circle The expected perieter of a triagle chose by radoly choosig a subset of three vertices of R is ow give by P a<b<c 4 P a, b, c Usig Table this ca be siplified to give P si + si + si Now let R be a regular hexago iscribed i a uit circle The expected perieter of a triagle chose by radoly choosig a subset of three vertices of R is ow give by P 6 P a, b, c 6 a<b<c Usig Table this ca be siplified to give P 6 4 si si 6 + si si 4 6 The geeral equatio for fidig the expected perieter of a triagle fored by choosig three vertices of a regular -go iscribed i a uit circle is give by P [si b a + si c b + si a c + ], where the su is take over all itegers a, b, c such that a < b < c We will siplify this expressio i the ext sectio 4 The Mai Theores Theore 4 Let R be a regular -go iscribed i a uit circle If a subset cosistig of three distict vertices of R is chose uiforly at rado, the the expected area of the triagle fored is give by A cot 7

8 Proof Fro the precedig sectio, it follows that [ si b a + si A c b + si a c + ], where the su is take over all itegers a, b, c such that a < b < c For each k such that k, let p, k be defied to be the uber of ties k occurs as b a,c b, or a c + i the above su, so that A p, k si k Clai p, k k for all ad k Proof of Clai : We divide the proof ito three cases I k > II k III k ad k All operatios o triples are doe odulo i each coordiate Case I: k > Start with the triple a, b, c, k, j where k + j Note that there are k choices for j Every triple of the for i, k + i, j + i where i has exactly oe of b a, c b, or + a c equal to k, ad these are the oly triples with oe of the differeces equal to k Sice there are k choices for j ad choices for i, we have p, k k Case II: k We divided the coputatio ito two parts, which will be added together: i a, b, c, k, k ad ii a, b, c, k, j where j k Part i: a, b, c, k, k I this case, every triple of the for i, k + i, k + i where i < / has all three of b a, c b, ad + a c equal to k These are the oly triples with all three of b a, c b, ad + a c equal to k Sice there are / choices for i ad k appears ties i each triple, it follows that k occurs ties i all Part ii: a, b, c, k, j Here, k + j ad j k Note that there are k choices for j Every triple of the for i, k + i, j + i where i has exactly oe of b a, c b, or + a c equal to k, ad these are the oly triples with oe of the differeces equal to k Sice there are choices for i, k occurs k ties i all Addig the results of parts i ad ii, we get p, k k Case III: k ad k Siilar to Case II, this case is doe i two parts: i a, b, c, k, k ad ii a, b, c, k, j where j k 8

9 Subcase i: a, b, c, k, k I this case, every triple of the for i, k + i, k + i where i has exactly two of b a, c b, or + a c equal to k These are the oly triples with two of the differeces equal to k Sice there are choices for i ad k occurs ties i each triple, it follows that k occurs ties i all Subcase ii: a, b, c, k, j Here k + j ad j k Note that there are k choices for j Every triple of the for i, k + i, j + i where i has exactly oe of b a, c b, or + a c equal to k, ad these are the oly triples with precisely oe of the differeces equal to k Sice there are choices for i, k occurs k ties i such triples Addig subcase i ad ii, we get p, k k Hece, i all three cases, we have show that p, k k This copletes the proof of Clai Fro Clai, it follows that A k si k 4- Equatio 4- iplies that A si k k si k 4- Let fx coskx ad gx sikx Euler s Forula gives that e ikx cos kx + i si kx, where i It follows fro the partial su forula for a geoetric series that fx + i gx e ikx ei x e ix Siplifyig this right had side, ad coparig real ad iagiary parts yields the trigooetric suatio idetities ad fx coskx gx sikx Also, by differetiatig, we obtai f x k sikx d dx cos + x cosx + cos x cosx 4- six + si x si + x 4-4 cosx [ cos + x cosx + cos x cosx ] 4-9

10 Equatio 4- iplies that A g/ + f / A stadard yet tedious siplificatio usig trigooetric idetities ow yields A cot The questio cocerig perieter ca be aswered by a siilar process Theore 4 Let R be a regular -go iscribed i a uit circle If a subset cosistig of three distict vertices of R is chose uiforly at rado, the the expected perieter of the triagle fored is give by P 6csc + cot Proof As this proof is early idetical to the proof of Theore 4, we provide oly a sketch Fro the precedig sectio, we have P [si b a + si c b + si a c + ], where the su is take over all itegers a, b, c such that a < b < c Usig the Clai fro the proof of Theore 4, we fid that P k si k The by usig the Euler s forula ad the fuctios fx ad gx as i the proof of Theore 4, we ca siplify this expressio to the closed for as desired Now that we have derived closed for expressios for the expected area ad perieter, we ca deterie the liit as teds to ifiity ad copare with the aswers for the proble o the uit circle fro Sectio Theore 4 Let A ad P be as i Theores 4 ad 4 The li A ad li P

11 Proof We rearrage ters ad apply L Hôpital s rule li A li cot > li > li > li P 6csc li + cot > li > li > 6 cos si cos 6 + cos si + cos 6 si si We fid that the liits of A ad P correspod precisely with the aswers for the proble o the circle foud i Sectio Extesios I this sectio, we study the effects of icreasig the uber of vertices chose, thus costructig radoly geerated polygos based o the vertices of a regular polygo iscribed i a uit circle The aalysis is very siilar to the case of rado triagles detailed i the last sectio, ad therefore, we will oly sketch the ai details here Further details ad extesios are left to the iterested reader We use A, ad P, to represet the expected area ad perieter, respectively, of a -go geerated radoly by uiforly choosig a subset of size of the vertices of a regular -go iscribed i a uit circle Theore Let R be a regular -go iscribed i a uit circle Suppose that a subset cosistig of vertices of R is chose uiforly at rado The the expected area of the covex -go fored is give by A, si a j+ a j, j where the first su is over all a,, a with a < a < < a, ad

12 a + a The expected perieter of this -go is P, j si a j+ a j The precedig result shows, for exaple, that if a set of 4 vertices of a regular -go iscribed i a uit circle is chose uiforly at rado, the the expected area ad perieter of the quadrilateral fored are give by A,4 sib a + sic b + sid c + sia d +, 4 ad P,4 4 sib a + sic b + sid c + sia d +, where the su is take over all itegers a, b, c, d such that a < b < c < d We ca collect like ters i the suatios i Theore to prove the followig result Theore Let R be a regular -go iscribed i a uit circle Suppose that a subset cosistig of vertices of R is chose uiforly at rado The the expected area of the covex -go fored is give by A, + k si k The expected perieter of this -go is give by P, + k si k We ow sketch the details of a ethod of siplifyig the above suatios to closed for Sice the ethods for A, ad P, are early idetical, we will focus siply o A, ad leave the aalogous results cocerig P, for the iterested reader Fro Euler s forula ad Theore, we see that A, I g,, where g, + k α k, with α e i/

13 We ote that α g, [ + [ + k [ + [ + k k k α k+ α k k + + k k k α k α k [ ] g, + g, + + ] α k ] α k ] ] This iplies that g, g, α + + Recall that α e i/ cos/ + i si/ It follows that α ᾱ cos/ i si/ We ca ake the deoiator of the expressio for g, real by ultiplyig uerator ad deoiator by ᾱ, resultig i g, ᾱ + α ᾱ g, + Substitutig i for α ad coputig real ad iagiary parts yields g, cos + i si 4 + si g, + + i cot [ ] g, i cot Re g, + i I g, + + [ Re g, + + I g, cot +i I g, + Re g, cot + cot ] Recall that the A, is equal to the iagiary part of g, Therefore, A, I g, + Re g, cot + + cot

14 Write g, a + ib, ad ote that a ad b are both fuctios of ad that A, b Let a b The above coputatios iply the recurrece a b [ a + b cot [ a cot b + ] + cot ] I vector for, this recurrece becoes [ a cot b + cot a b + + cot ], with iitial coditio a b Usig this recurrece, oe ca copute b A, b cot cot cot + 9 cot 4 cot + 7 cot 4 cot 4 +6 cot +9 cot 6 cot ++ cot cot 4 8 cot +9 + cot 87 4 cot +9+4 cot4 4 6 Table : Expected Area A, The vector recurrece ca also be solved usig diagoalizatio of the atrix M cot cot The eigevalues of M are i cot ad i cot ad the respective eigevectors are ad This ca be used to show that i i b I j j i cot j 4

15 We ay siplify this further to yield A, l jl+ j+l j j j l + cot l+ The above ethods ay be applied directly to copute P, This is left to the reader The iterested reader ay also like to pursue the geeralizatio to the circle proble fro Sectio for >, ad its relatioship to the results give here 6 Ackowledgeets We would like to ackowledge our woderful advisor, Dari Stepheso of Hope College, for guidig us through the research process We would also like to thak Hope College ad the Natioal Sciece Foudatio for allowig us to participate i this research progra ad for fudig We thak Chris Sith of the Grad Valley State Uiversity REU for help i uderstadig the hypergeoetric suatios that arise i Sectio We are thakful to our colleague Daiela Bau for helpig us with the proof of Theore 4 Fially, we would like to thak all of our brilliat colleagues, who ot oly helped us out with our research but also ade coig to work a joyful ad eorable experiece The coputer software that ade this research possible icludes: MAPLE, Adobe Photoshop/Reader, Geoeter s Sketchpad, ad Wiedt We appreciate all those who helped us to effectively use these progras, especially Mark Pearso, Dari Stepheso, ad Airat Beketjev Refereces [] Hogg R ad Tais E Probability ad Statistical Iferece, 6th Editio, Pretice Hall, [] Weisstei, Eric W Circle Triagle Pickig, fro MathWorld A Wolfra Web Resource [] Zwilliger, D CRC: Stadard Matheatical Tables ad Forulae, st Editio, Chapa ad Hall,