IRURI CU UTILIZARE ÎN DESIGNUL DE PRODUS, OBINUTE PE SPIRALE LOGARITMICE TIP ( t ) erban BOBANCU. Universitatea Transilvania din Braov
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1 UNIVERSITATEA TRANSILVANIA DIN BRAOV Cda Desig de Pdus $i Rbic& Sipziul aial cu paicip ieaial PRieca ASIsa de Calcula P R A S I C ' 0 Vl III Desig de Pdus 7-8 Niebie Bav, Râia ISBN IRURI CU UTILIZARE ÎN DESIGNUL DE PRODUS, OBINUTE PE SPIRALE LOGARITMICE TIP ( ) eba BOBANCU Uivesia Tasilvaia di Bav uique cuves which have, ag hes, he eakable feaue visualize saigh lies, ccue i he asypic pi (he igi), seies f diesis (leghs f eighbu seges) which all i he glde ai F his eas, whe pceedig desig ew pducs aalyse he desig f se exisig pducs, hese spials (as well as hei deived spials) a visual guide f all igus haically ppis Absac: The lgaihic spials ype ( ) Cuvie cheie: #i, u$ul de au, spiala lgaiic$ Se uesc spiale cubele ale c aze vec su fucii uivce de ughiul : f ( ) cu vaiid îe i + ia ( ) p fi difei de ( + ) Spiala lgaiic ese defii de ecuaia î c csle a, R, a > 0, 0 (fig ) Deuiea de spial lgaiic pvie de la fa l c ese a echivale cu ecuaia de defiiie a spialei Dac se dau vali lui de la la + se bie ablul: Rezul g ' cg sau acg cs () a e a a e + Plul O ese u puc asipic peu spiala lgaiic Se cau acu ughiul fcu de az vec cu agea la cub î-u puc Fig Spial lgaiic
2 Ecuaia spialei lgaiice ai p fi scis cg Csecia elaiei () ese c spiala lgaiic p fi defii i ca fiid acea cub spial c aie azele vec sub u acelai ughi csa (fig ) Dac se bie, succesiv, g ' ' 0 cs Deci cecuile p fi cside ca fiid cazui paicul ale spialei lgaiice (spiale degee avâd ) Se cside acu ueele K ca fiid lugiile azel vec ale pucel spialei lgaiice siu pe aceeai deap, biue peu ughiuile + + K + k Se eaz acese puce cu P, P, P, (fig ) Îsea c d pe, + pe pe, Rezul ( + ) + e e e ( + ) ( + ) + e ad Di, se ajuge la e Dac se eaz e q se bsev c lugiile azel vec ale pucel succesive P, P, P, csiuie eeii uui i î pgesie geeic cu aia q Di ppia de ai sus ezul c feaz u i î pgesie geeic i difeeele K avâd aceeai aie q e q q q K Spialele lgaiice de ip ( ) su cubele uice c au, pie alele, ppiea eacabil de a puea vizualiza, pe depe ccuee î pucul asipic (de acuul), iui de diesiui (lugii de segee alu), afl î + apul de au Di aces iv, câd se pcedeaz la desigul u i pduse sau la aaliza desigului u pduse exisee, acese spiale (ca i spialele deiv di ele) csiuie u îdua vizual al uu ppiil igus aice Se p gsi acu acea spial lgaiic paicula, la c lugiile azel vec succesive K, especiv disle succesive deei de spial pe deap c ce ece pi igie K, s csiuie iul ( ) Auci, se defiee spiala lgaiic ( ) (fig ), ca fiid spiala lgaiic:, cu l () Aceas spial ppieile: Fig Ppiea de baz a spialel lgaiice 6 ) K
3 6 ) K Dac se eaz: K Aces i i ppiea de a fi u i ecue ip FIBONACCI,,,, 6 6,, auci, ppiea de ecue a uui i ip FIBONACCI ese: + + Aalg s-a defii i iul fiid +, sau ( ) ca ) ), +,, +,,,, +, 6, bie Peu iul ( ) K, luâdu-se csaa a, se e () Di (), se bie ca fiid: l l () ese exe de uil î desig Lucâd aaliic cu ecuaia () se p geea iui ( ) la ice sca alegâdu-se cespuz vala csi a Se p csidea (îlcuid pe di elaia () cu,, ec) i ale spiale lgaiice de ip ( ) ( ) Spiala lgaiic ( ) Î fig ese pezea spial lgaiic cu ecuaia Fig Spiala lgaiic ( ) Ecuaia () se p siplifica l ( e ) l a a () ese iul sic cesc cu eeii î pgesie geeic î c piul ee a, q Adic iul Se eaiee c iul ( ) c ppieile: ( ) K l cu, sau i a, K Î fig ese pezea spial lgaiic cu ecuaia a e l l cu
4 sau a a, c ppieile K, i K ( ) Î fig 6 ese pezea spial lgaiic cu ecuaia l l l cu sau a a c ppieile K, i K Î fig 7 su peze seie de depughiui (depughiui aice) pi cluile ca ece spial lgaiic ( ) (aceas spial u ese age ci seca la lauile depughiului) Se csa c î-u u desul de de luci, ecuaia spialei lgaiice ese uiliza sub fa: Fig Segee deei de spial pe az vec cg, fiid evide c g sau cg O îeag seie de aui [], [], [], [], [], [6], [7], [8], [9], 0], [] c uilizeaz aceas f a ecuaiei, defiesc failie de spiale (deuie echiagul), fucie de ule vali ale lui : Aceas failie de spiale apxieaz de fap spiale lgaiice ip ( ) dup cu ueaz: Fig Spiala lgaiic ( ) Fig 6 Spiala lgaiic ( ) spiale avâd 0 apxieaz spiale 6 lgaiice ( ) (avâd 9,7 ) cu ea pceual, 998 % spiale avâd 0 apxieaz spiale lgaiice ( ) (avâd 9,887 ) cu ea pceual 0, 6 % spiale avâd 60 apxieaz spiale 8 lgaiice ( ) (avâd 8,0 ) cu ea pceual, 7 %
5 spiale avâd 70 apxieaz spiale lgaiice ( ) (avâd 69,06 ) cu ea pceual, 8 % spiale avâd lgaiice ( ) 80 ea pceual, 89 apxieaz spiale (avâd 8,9 ) cu % spiale avâd 8 apxieaz spiale lgaiice ( ) (avâd 8,60 ) cu ea pceual 0, 7 % Se p csidea i spialele lgaiice ( ), N, a, < < Ele au ecuaiile ( ) :, N, < < Acese spiale lgaiice au ceeile ivese Câd ughiul fa de spialele lgaiice ( ) pla ese pziiv i cesc, lugiile azel vec descesc, ia câd ughiul pla ese egaiv i descesc, lugiile azel vec ale pucel de pe spial descesc Mai ul, gaficul spialei lgaiice ( ) ese sieicul fa de axa pla (fig 8) a gaficului spialei lgaiice ( ) Aceas afiaie ese jusifica de fapul c lugiea azei vec a pucului abia P cespuz ughiului pla + k de pe spiala lgaiic ( ) ese egal cu lugiea azei vec a pucului P ~ cespuz ughiului pla k de pe Î-adev spiala lgaiic ( ) ( + k ) ( k ) + l Fig 8 Spial lgaiic ip ( ) sieica sa Se p bie lag geealiz siila, csideâd spialele lgaiice ( ), R, 0 O asfel de spial lgaiic, î cd pl, ecuaia i Fig 7 Spia iabillis Spiala lui BERNOULLI Spiala lgaiic ( ), R, 0, < < Peu 0 spiala degeeeaz î cecul cu
6 ceul î pl i aza egal cu Dac > 0, auci, csideâd seideap abia cu igiea î plul O, se deseaz, la fel ca î cazul spialel lgaiice ( ), N, c lugiile azel vec ale pucel de iesecie ale seidepei cu ( ) cesc de la u puc de iesecie la succesul su î pgesie geeic, aia fiid q > Dac expeul ese, > 0, ezul c lugiile azel vec ale pucel de iesecie, auci câd cee (î ses igeic), descesc î pgesie geeic, aia fiid q, 0 < < Fiecei spiale lgaiice ( ), R, 0 îi cespude (fig 8) spiala lgaiic ( ), c ese sieica spialei lgaiice ( ) fa de axa pla Reveid acu la failia de spiale lgaiice cu 0, 0, 80, 8, 60, 70, se csa c fiec die ele ese geea de spial lgaiic de ip, R (vala lui da cu ujie la a eia zecial) dup cu ueaz:,6 0 spiala lgaiic ( ), 0,96 0 spiala lgaiic ( ), 7,8 60 spiala lgaiic ( ),,7 70 spiala lgaiic ( ),,0 80 spiala lgaiic ( ),, 8 spiala lgaiic ( ) Bibligafie Ebas, A K The Fibacci Seies, The Lucas Seies ad Glde Rai hp:// jwilsceugaedu/emt668/emat6680f 99/Ebas/KURSATgeeyp/fib%6luc 6phi/AKEwie-uphl Ebas, A K MATH 700-Fudais f Geey (Fall '99 Pjec) hp://jwilsceugaedu/emt668/emat6 680F99/Ebas/KURSATgeeyp/glde %0spial/lgspial-his yhl K, R Fibacci Nubes ad Naue hp://wwweesueyacuk/pesal/rk /Fibacci/fibahl K, R, Glde Seci Rai: Phi he ube ad is geey hp:// www cs suey ac uk / Pesal / R K / Fibacci / phi hl Pees, G The Equiagula Spial: Spia iabilis hp:// lie edwds cc ca us / isuc / dald / CalcPj / Sp98 / GabeP / Spial h 6 Tully, D Equiagula Spial, Lgaiic Spial, Beulli Spial hp://lieedwdscccaus/isuc/da ld/calcpj/fall98/dt/equiagulasp ialh 7 Xah, L Equiagula Spial hp:// www xahleeg / SpecialPl Cuves_di / EquiagulaSpial_di/equiagulaSpialh l 8 Beulli s spial, hp:// libay hikques g / 8 / / sy / page / beul h 9 hp://wwweasuevesc/ah/glderaihl 0 hp://wwwahsfc/aslve/csa/g ld/gldhl Se Glde Geey hp:// galaxy cau edu / sih / KW / gldegehl
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