Optimal Placing of Crop Circles in a Rectangle

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1 Optiml Plcing of Cop Cicles in Rectngle Abstct Mny lge-scle wteing configutions fo fming e done with cicles becuse of the cicle s pcticlity, but cicle obviously cnnot tessellte plne, no do they fit vey well in ectngle. Unfotuntely, most plots of lnd e ectngul. Additionlly, thee e occsionlly ext obstcles, like ods o houses. In this ppe, we use Geomety Expessions to investigte the optiml method of plcing cicles. Poblem In this ppe, we look t simple sitution: optimlly plcing two cicles in ectngle with spect tio less thn. In ddition to the size nd loction of the cicles, thee is nothe vible to conside. Insted of using full cicle, we cn use pc-mn shpe, o secto. Thus, we lso hve to detemine whethe the lge o smlle cicle should be the pc-mn. We divide the investigtion into two pts: one fo spect tios between nd.5, nd one fo spect tios between.5 nd. Fo simplicity, we ssume the shote side of the ectngle is, nd the longe side will be. Solution We conside few cses. The simplest solution is to simply plce two full cicles, ech tngent to two sides of the ectngle nd the othe cicle (.).. O, we cn hve eithe the smlle cicle (.) o the lge cicle (.3) be pc-mn.

2 [.] [.3] Thee e two vibles we chnge, spect tio nd dius, to lte the totl e, so we would gph in 3D. Howeve, due to the limittions of dwing 3D gph on D ppe, we simply tke coss-sections of the 3D gph nd look fo the mximum e fo the given spect tio. Lemm Fist, we show tht once the lge cicle hs been plced, it is lwys optiml to plce the smlle cicle in cone, tngent to two sides of the ectngle, the thn centeed, tngent to only one side of the ectngle (.). [.] VS Tke full smlle cicle with given dius, the tivil cse. The smlle cicle is clely the lgest when it is in the cone, tngent to two sides. Howeve, now conside the pc-mn cse. Tke one cicle with mximl dius, nd tke given dius fo the othe cicle. The totl e is thus only dependent on the ngle of the pc-mn s mouth. This ngle will be t minimum when the centes of the cicles e futhest pt. Thus, the smlle cicle should be in the cone, tngent to two sides nd the othe cicle. Pt : Aspect Rtio between nd.5 Once we detemine tht the optiml plcement of the smlle cicle is in the cone nd tngent to two sides, we detemine if nd when it is bette to hve the smll cicle s pc-mn (3.) o the big cicle s pc-mn (3.).

3 Using Geomety Expessions, we detemine the ngle of the fn in tems of spect tio () nd dius (). The two configutions e shown below with thei coesponding ngle fomule. A: Smll Cicle s Pc-mn B: Big Cicle s Pc-mn [3.] [3.] Angle Fomule: With the ngle fomul the e coveed by the cicle nd the pc-mn s pecentge of totl e possible cn be detemined. Pecentge of Ae Coveed (A): 8 8 ctn.5 π Pecentge of Ae Coveed (B): > < : 0.5ctn.5 : 0.5ctn.5 π π π π Note: Becuse the invese tngent hs limited nge, the fomul poduces negtive vlues fo ngles lge thn π. Configution A does not tke into ccount the pt of the eqution fo less thn π becuse ngles gete thn π e clely optiml. The piecewise function fo Configution B is split t =, whee is the ngle is equl to π.

4 Pecent e coveed fo configutions A nd B e plotted in 3D below (3.3). [[3.3] As seen in (3.) nd (3.), thee e two vibles tht detemine the e coveed by the cicle nd the pc-mn. The gph bove shows the pecentge of e coveed fo ll combintions of (.0-.5) nd ( ) fo the given nges. The yellow sufce is A nd the ed sufce is B. Wht we ce bout is the getest pecentge of e coveed fo ech given spect tio fom.0 to.5. Using the Optimiztion pogm fom Mple, we isolte t intevls of 0.0 nd find the mximum t ech step fo Configution A nd B. The outcome fo the mximum dius nd pecent e coveed is shown in the dt tble in the conclusion. The plot of the mximums t ech intevl of is shown below (3.). Once gin, the yellow cuve epesents Configution A nd the ed cuve epesents configution B. [3.]

5 Fom comping the lists poduced by clculting the locl mximum fo evey 0.0 intevl of, it is ppent tht configution A is bette until the spect tio is ppoximtely.6. At tht point on until =.5, configution B is optiml. Pt : Aspect Rtio between.5 nd The obvious solution is to plce two cicles of dius.5, nd mke one pc-mn (.). [.] But is it the optiml configution? Ae thee ny cses whee thee is bette configution? As it tuns out, it is indeed the best configution fo most spect tios, but not ll. We investigte some ltentives in moe detil. We look t thee cses, s descibed in the solution outline. Fist, we exmine the cse of two full cicles (.). - - [.] We denote the dius of one cicle. Geomety expessions then clcultes the dius of the othe cicle s shown, nd the totl e of the two cicles is. The totl pecent covege is thus.

6 Next, we look t the two pc-mn configutions, stting with the smlle cicle s the pc-mn. Thee e two cses: one with dii tngent to the lge cicle (.3) nd one with dii endpoints on the lge cicle (.). We now set the dius of the lge cicle s, nd the dius of the smlle s. πctn [.3] [.] Fo (.3), the pecent covege is. Unfotuntely, Geomety Expession s clcultion fo the ngle in (.) is too lge to fit on the pge. Howeve, we cn still input the expession into Mthemtic. Now, we look t the bigge cicle s the pc-mn. Agin, we hve two cses whee the dii e tngent to (.5) nd lying on (.6) the smlle cicle.

7 πctn [.5] [.6] Fo (.5), the pecent covege is fo (.6) is unfotuntely too big to fit on the pge. ( ). Agin, the eqution fo the ngle Now tht we hve the fomule fo ech cse, we plot them to detemine the optiml dii. Due to the impcticlity of plotting 3D gph s is necessy, we look t set of given vlues fo, nd detemine the optiml fo ech. The esults e summized in this plot.

8 Pecent Covege Optiml Cicle-plcing Aspect Rtio Full Cicle Lge Pc-mn Smll Pc-mn Conclusion Fo pcticl puposes, we include tble with the ppopite dt fo spect tios to the neest hundedth. The bolded column is the optiml sttegy, with the idel dius nd pecent covege. Note tht between tios of.53 nd.9, the optiml sttegy is to simply hve cicle nd pc-mn, both with mximl dius. Thus, thee is no bigge cicle, so sttegies A nd B e essentilly the sme. At.95, it becomes pefeble to hve no pc-mn t ll, nd two full cicles insted. A-Smll B-Big C-Full Cicle Aspect tio % Coveed Rdius % Coveed Rdius % Coveed Rdius

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11 This mteil is bsed upon wok suppoted by the Ntionl Science Foundtion unde Gnt No Any opinions, findings, nd conclusions o ecommendtions expessed in this mteil e those of the utho(s) nd do not necessily eflect the views of the Ntionl Science Foundtion.