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D]TKCKBRKE C]TK@E-HMV VECTFKL K@DMC

Dcmberm`e per Dgrl ^rdz ^rdz

Vecufkelds K`dmcds dl dqukckbrke cquk`e-vmper.

M `kgdrdlfkm `d cm sktumfkl qud sd prdsdltmbm dl dc fmptuce KKK, dl cm qud sddstu`kmrel `ksecufkelds fel secutes le vectkcds, moerm sd mber`ml mqudccms felsecutes vectkcds, ds `dfkr, qud feltrkbuydl tmjbkl m cm prdskl tetmc `dc skstdjmdl gmsd hmsdesm y tmlte dl cm gmsd cquk`m, feje dl cm hmsdesm dxkstdl mjbesfejpeldltds. Cm rdcmfkl qud humr`ml cms fejpeskfkelds `d fm`m fejpeldltd dlmjbms gmsds y dc dgdfte qud tkdld cm prdskl y/e cm tdjpdrmturm, sel dc ebidte `ddstu`ke `d dstd fmptuce.

^mrm sdr js dxpcfktes, les rdgdrkjes m cm gkhurm = dl cm fumc tdldjes dl gmsdcquk`m ulm secufkl bklmrkm fel fejpeskfkelds x=, x2 dl dqukckbrke fel ulm gmsdhmsdesm qud ds etrm secufkl bklmrkm fel grmffkelds jecmrds y

=, y

2

rdspdftkvmjdltd. Cm secufkl sd dlfudltrm `dltre `d ul jmtrmz qud m su vdz dst`dltre `d ul bme per `el`d fkrfucm ul gcuk`e qud cd pdrjktd jmltdldr sutdjpdrmturm P felstmltd, dstml`e dc jmtrmz feldftm`e m ul jmljdtre qudjmrfm cm prdskl felstmltd ^. Dstd skstdjm, qud felstm `d `es gmsds, ulm cquk`my etrm hmsdesm, dst dl dqukckbrke tdrje`kljkfe, ce fumc kjpckfm qud dc hms sddst fel`dlsml`e y dc cquk`e dvmperml`e `kljkfmjdltd.

Gkhurm =. Dqukckbrke Cquk`e-Wmper `d ul skstdjm bklmrke m prdskl ytdjpdrmturm felstmltds.

^

P

y=, y2

x=, x2

Gcuk`e d tdjpdrmturm felstmltd

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Dvk`dltdjdltd, dc skstdjm fujpcd fel cm fel`kfkl tdrje`kljkfm `d dqukckbrkefuml`e cm prdskl y cm tdjpdrmturm pdrjmldfdl felstmltds (vdr Dfumfkl KW-=>),ds `dfkr dc petdlfkmc qujkfe `d fm`m fejpeldltd dl fm`m gmsd m`qukdrd dc jksjevmcer9

=

H

: =C

(=)y

2H : 2

C (2)Vk ces fejpeldltds `d cm jdzfcm dl dqukckbrke sd pud`dl felsk`drmr k`dmcds, ds`dfkr skhudl cms cdyds `d Rmeuct y cm gmsd hmsdesm skhud cm Cdy `d @mctel,dltelfds cm prdskl tetmc ds9

p : p=+ p2 (6)

skdl`e p= y p2 cms prdskelds pmrfkmcds `d ces fejpeldltds (=) y (2)rdspdftkvmjdltd. Dltelfds, cm prdskl tetmc ds9

p : ^=x = + ^2x2 (6)

`el`d9

^=ds cm prdskl `d vmper `dc fejpeldltd (=) m cm tdjpdrmturm `dc skstdjm y^2ds cm prdskl `d vmper `dc fejpeldltd (2) m cm tdjpdrmturm `dc skstdjm.

Vk mpckfmjes cm dfumfkl `d bmcmlfd, pmrm ul skstdjm bklmrke, tdldjes9

x=+ x2 : =.> (0)

e bkdl, pelkdl`e x2 dl gulfkl `d x=9

x2 : =.> - x= (0)

Vubstktuydl`e dstm dfumfkl dl (6), ebtdldjes9

p : ^=x = + ^2(=-x2)

e bkdl, `dsmrreccml`e, mhrupml`e y gmfterkzml`e9

p : ^=x = + ^2- ^2x= : ^2+ (^=- ^2) x=

p : ^2+ (^=- ^2) x= (4)

dstm ctkjm dfumfkl dxprdsm cm prdskl tetmc dl trjkles `d cm grmffkl jecmr `dule `d ces fejpeldltds (per felvdlfkl dc js vectkc), x=. Cms prdskelds `d

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vmper ^= y ^2 sel felstmltds fuml`e cm tdjpdrmturm ce ds. @d je`e qud cmdfumfkl WK-; tkdld cm gerjm `d ulm dfumfkl ckldmc `el`d ^2ds cm er`dlm`m mcerkhdl y (^= - ^2) ds cm pdl`kdltd `d ulm hrgkfm `d p dl gulfkl `d x=.

^er mlcksks `d dstm dfumfkl, dlfeltrmjes qud fuml`e x=: =.>, ce fumc efurrd

fuml`e sce omy fejpeldltd = (dc js vectkc), pure, sd mlucm per `kgdrdlfkm cmprdskl ^2, qud`ml`e qud p : ^=, dl tmlte qud sk x=: >.>, ce fumc kl`kfm qud leomy fejpeldltd =, skle sce fejpeldltd 2, dltelfds sd mlucm te`e dc sdhul`etrjkle, qud`ml`e sce ^2, ds `dfkr p : ^2.

Dc cuhmr hdejtrkfe `d dstm dfumfkl sd judstrm dl cm Hrgkfm =m9

Hrgkfm =m. Rdprdsdltmfkl hrgkfm cms prdskelds pmrfkmcds y tetmc dl gulfkl`d cm grmffkl jec dl gmsd cquk`m (x=) `dc fejpeldltd js vectkc, `d ul

skstdjm bklmrke m tdjpdrmturm felstmltd

Dl cm Hrgkfm =m sd judstrml cm prdskl pmrfkmc `dc fejpeldltd js vectkc, p =, cmprdskl pmrfkmc `dc fejpeldltd jdles vectkc, p2y cm prdskl tetmc `dc skstdjm p,qud ds cm sujm `d cms prdskelds pmrfkmcds, te`ms dl gulfkl `d cm grmffkl jec `dcfejpeldltd js vectkc x=. Cm prdskl pmrfkmc p= mujdltm ckldmcjdltd fel dcklfrdjdlte `d x=, ce feltrmrke qud cm prdskl pmrfkmc p2qud `ksjkluyd ckldmcjdltdfel dc klfrdjdlte `d x= (dvk`dltdjdltd mujdltm ckldmcjdltd fel dc mujdlte `dx2), dl tmlte qud cm prdskl tetmc p sd klfrdjdltm ckldmcjdltd mc mujdltmr x=, ce

fumc ds ulm felsdfudlfkm chkfm `d qud dc fejpeldltd (=) sdm dc js vectkc.Cm prdskl p `d cm Dfumfkl 4 y cm rdftm `d cm Hrgkfm 2m rdprdsdltm cm prdskldidrfk`m per cms jecfucms qud dsfmpml `d cm gmsd cquk`m. ^er etre cm`e, `dbdjesfelsk`drmr qud dl cms hrgkfms `d prdskl feje cm =m (y cms skhukdltds) cmer`dlm`m rdprdsdltm m cm prdskl dxtdrlm y dste les pdrjktd fejpmrmr m dstm felcm `d vmper, mltds rdgdrk`m. Dl dstd pulte ds felvdlkdltd rdtejmr dc felfdpte `dcdqukckbrke cquk`e-vmper pmrm ulm sustmlfkm purm, `ksfutk`e dl cms Hdldrmck`m`ds,

>.>=>.>2>.>6>.>0>.>4>.>7>.>8>.>?>.>;>.>=>>.>

==>.>

> >.2 >.0 >.7 >.? =

prdskl,

a^m

x=

^rdskelds ^mrfkmcds y Petmc

p= p2 p

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sdhl dc fumc fuml`e cm prdskl `d vmper `d ul cquk`e mcfmlzm dc jksje vmcer quddc `d cm prdskl dxtdrlm, sd mcfmlzm ul dstm`e `d dqukckbrke dl dc qud cms `es gmsdsdc cquk`e y dc vmper tkdldl cm jksjm dstmbkck`m` y dl felsdfudlfkm fedxkstdl.Dstes felfdptes sel mpckfmbcds tmjbkl m cms jdzfcms. Dltelfds, feje mltds sd`ksfutk, m prdskelds dxtdrlms khumcds m cm fmcfucm`m per cm dfumfkl, sd tdl`r

dqukckbrke `d dstms gmsds y sk cms prdskelds sel jdlerds qud cm `d dqukckbrke `m`mper cm dfumfkl y cm rdftm rdgdrk`ms, cm gmsd dstmbcd ds dc vmper dl tmlte qud mprdskelds jmyerds qud cm `dc dqukckbrke cm gmsd dstmbcd ds dc cquk`e. Dste sdrdprdsdltm dl cm Hrgkfm =b

Hrgkfm =b. Kltdrprdtmfkl `d ces dstm`es `dc skstdjm dl rdcmfkl m cmubkfmfkl `d cm prdskl dxtdrlm fel rdspdfte m cm prdskl `d vmper `d ul

skstdjm bklmrke m tdjpdrmturm felstmltd

Moerm sd `dsdm rdcmfkelmr cm prdskl tetmc fel cm grmffkl jecmr `d cm gmsdhmsdesm, pmrm ce fumc rdfurrkjes m cm prdskl pmrfkmc `m`m per cm Cdy `d Rmeuct y cm`m`m per cm Cdy `d @mctel dl cm fumc sd rdcmfkelm fel cm prdskl tetmc y cmfejpeskfkl `d cm gmsd hmsdesm. Cm prdskl pmrfkmc sd pud`d dxprdsmr dl trjkles`d cm prdskl tetmc y cm grmffkl jecmr dl cm gmsd hmsdesm9

p =: p y= (7)

y

p 2: p y 2 (8)

khumcml`e cms dfumfkelds WK-= y WK-== sd ebtkdld9

p

>.>

2>.>

0>.>

7>.>

?>.>

=>>.>=2>.>

> >.2 >.0 >.7 >.? =x=

p,jjOh

^dxt < p :< kquk`e

^dxt 3 p :< vmper

p

>.>

2>.>

0>.>

7>.>

?>.>

=>>.>=2>.>

> >.2 >.0 >.7 >.? =x=

p,jjOh

^dxt < p :< kquk`e

^dxt 3 p :< vmper

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^= x=: p y= (?)

@d `el`d9

y= : ^= x=/ ^ (?)

substktuydl`e cm dfumfkl 4, ebtdldjes9

y= : ^= x=/ (^2+ (^=- ^2) x=) (;)

`d cm fumc `dspdimjes x= `dspus `d mhrupmr9

y=^2 +y=(^=- ^2) x=) - ^= x= : >

x=(y=(^=- ^2) - ^= ): - y=^2

x=

: y=^

2/ (^

= - y

=(^

=- ^

2)) (=>)

Vubstktuydl`e dl cm dfumfkl 4, ebtdldjes 9

p : ^2/= + (^=- ^2) y=^2/ (^= - y=(^=- ^2)) :

p : (^2(^= - y=(^=- ^2)) + (^=- ^2) y=^2) / (^= - y=(^=- ^2))

p : (^2^= - y=^2(^=- ^2) + (^=- ^2) y=^2) / (^= - y=(^=- ^2))

ces `es ctkjes trjkles `dc lujdrm`er sel k`ltkfes, pdre tkdldl skhle feltrmrkeper ce qud sce prdvmcdfd dc prkjdre, qud`ml`e dc `dlejklm`er skl fmjbke mchule9

p : ^=^2 / (^= - (^=- ^2) y=) (==)

Mlmckzml`e dstm dfumfkl jmtdjtkfmjdltd, dlfeltrmjes qud fuml`e y=: =.>, cefumc efurrd fuml`e sce omy fejpeldltd = (dc js vectkc), pure, sd mlucml dl dc`dlejklm`er ces vmcerds `d ^=, qud`ml`e dl dc `dlejklm`er sce ^2, dc fumcfmlfdcm mchdbrmkfmjdltd cm jksjm vmrkmbcd dl dc lujdrm`er, qud`ml`egklmcjdltd, sce p : ^=, dl tmlte qud sk y= : >.>, ce fumc kl`kfm qud le omyfejpeldltd =, skle sce fejpeldltd 2, dltelfds sd mlucm dc sdhul`e trjkle `dc`dlejklm`er, qud`ml`e sce ^= y dstd m su vdz fmlfdcm mchdbrmkfmjdltd m cmjksjm prdskl, pdre dl dc lujdrm`er, qud`ml`e mc gklmc, sce qu p : ^2, ce fumcds chkfe y judstrm qud ds felskstdltd dc je`dce.

Dc fejpertmjkdlte hrgkfe `d dstm dfumfkl dst dxpudste dl cm Hrgkfm 2m. Pmltecm prdskl `dc skstdjm feje cm pdl`kdltd `d cm furvm sd klfrdjdltm mcklfrdjdltmrsd cm grmffkl jecmr `dc fejpeldltd js vectkc.

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Hrgkfm 2m. Rdprdsdltmfkl hrgkfm `d cm prdskl tetmc dl gulfkl `d cm

grmffkl jec dl gmsd hmsdesm (y=) `dc fejpeldltd js vectkc, `d ul skstdjmbklmrke m tdjpdrmturm felstmltd

Cm Dfumfkl == y cm furvm dl cm Hrgkfm 2m sd rdgkdrdl m cm prdskl `d dqukckbrke `dcskstdjm ferrdspel`kdltd m cm fejpeskfkl y=. Fuml`e cm prdskl `dc skstdjm khumcmcm prdskl `dc skstdjm, sd prdsdltm dc dqukckbrke `d gmsds C-W dl tmlte qud sk ds`kgdrdltd, ym sdm jmyer e jdler, ombr ulm secm gmsd dstmbcd. Ms, fuml`e cmprdskl `d vmper `dc cquk`e ds jdler qud cm prdskl dxtdrlm, cm gmsd dstmbcd sdrdc cquk`e, pdre fuml`e cm prdskl `d vmper ds jmyer qud cm prdskl dxtdrlm,fumcqukdr cquk`e dxkstdltd sd dvmpermrm dspeltldmjdltd y cm gmsd dstmbcd sdrmcm `dc vmper. Dste dst rdprdsdltm`e dl cm Hrgkfm 2b.

Hrgkfm 2b. Kltdrprdtmfkl `d ces dstm`es `dc skstdjm dl rdcmfkl m cmubkfmfkl `d cm prdskl dxtdrlm fel rdspdfte m cm prdskl `d vmper `d ul

skstdjm bklmrke m tdjpdrmturm felstmltd

>.>=>.>2>.>6>.>0>.>4>.>7>.>8>.>

?>.>;>.>

=>>.>==>.>

> >.2 >.0 >.7 >.? =

p

y=

p

p

>.>=>.>2>.>6>.>0>.>4>.>7>.>8>.>?>.>;>.>

=>>.>==>.>

> >.2 >.0 >.7 >.? =

y=

p

^dxt 3 p :< vmper

^dxt < p :< Cquk`e

p

>.>=>.>2>.>6>.>0>.>4>.>7>.>8>.>?>.>;>.>

=>>.>==>.>

> >.2 >.0 >.7 >.? =

y=

p

^dxt 3 p :< vmper

^dxt < p :< Cquk`e

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Dltelfds, `dc mlcksks `d cms dfumfkelds 4 y ==, ms feje `d cm ebsdrvmfkl `d cmsHrgkfms =m y 2m dlfeltrmjes qud dc jlkje `d prdskl efurrd dl mjbms furvmsdl x=:>.> y dc jxkje efurrd dl mjbms furvms fuml`e x=:=.>, mulqud ces vmcerdskltdrjd`kes pmrdfdl sdr `kgdrdltds. Dste gmfkcktm cm kltdhrmfkl `d mjbms furvms,

cuhmrds hdejtrkfes `d cms dfumfkelds mlmckzm`ms, pdre dl cm jksjm hrgkfm, tmcfeje sd kcustrm dl cm Hrgkfm 6m9

Hrgkfm 6m. Rdprdsdltmfkl hrgkfm `d cm prdskl tetmc dl gulfkl `d cmsgrmffklds jecmrds dl gmsd cquk`m (x=) y dl gmsd hmsdesm (y=) `dc

fejpeldltd js vectkc, `d ul skstdjm bklmrke m tdjpdrmturm felstmltd

Cm rdftm `d rejbes ebsfures rdprdsdltm m cm dfumfkl 4 ds `dfkr tkdld fejevmrkmbcd kl`dpdl`kdltd cm grmffkl jec dl gmsd cquk`m `dc fejpeldltd js vectkc(x=) y feje vmrkmbcd `dpdl`kdltd m cm prdskl (p). M`kfkelmcjdltd cm furvm `dfum`rktes rdprdsdltm cm dfumfkl == y tkdld feje dltrm`m cm grmffkl jecmr `d ulfejpeldltd dl cm gmsd vmper (y=), pmrm cddr cm prdskl ferrdspel`kdltd, p. Cmkltdrprdtmfkl `d ces dstm`es qud rdprdsdltml ces pultes dl gulfkl `d suubkfmfkl rdspdfte `d dstms `es furvms sd `m dl cm Gkhurm 6b9

p-x=-y=

>.>

2>.>

0>.>

7>.>

?>.>

=>>.>

=2>.>

> >.2 >.0 >.7 >.? =

x=, y=

p,jjOh ^ WV x=

^ WV y=

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Hrgkfm 6b. Kltdrprdtmfkl `d ces dstm`es `dc skstdjm dl rdcmfkl m cmubkfmfkl rdspdfte `d cm cms furvms `d cms dfumfkelds WK-=> y WK-=7 pmrm ulskstdjm bklmrke m tdjpdrmturm felstmltd

Ms, sk mc hrmgkfmr dc pmr `d vmrkmbcds9 grmffkl jec `dc skstdjm y cm prdskl fejefeer`dlm`ms, dc pulte ferrdspel`kdltd sd ubkfm mrrkbm `d mjbms furvms, dl cmrdhkl jmrfm`m feje Cquk`e, skhlkgkfm qud m dsms fel`kfkelds dc skstdjm dsdstmbcd feje gmsd cquk`m, sk dc pulte fmd dl cm rdhkl `dbmie `d mjbms furvms, dlcm rdhkl jmrfm`m feje Wmper, cm gmsd dstmbcd `dc skstdjm sdr ul vmper ygklmcjdltd sk dstms feer`dlm`ms qud`ml `dltre `dc rdm qud sd dlfudltrm dltrdcms `es furvms, dl cm rdhkl jmrfm`m feje Cquk`e-Wmper, dstms gmsds sdrl cms

qud prdsdltd dc skstdjm `d je`e dstmbcd.Moerm, mlmckfdjes ce qud efurrd fuml`e ul fkdrte skstdjm m tdjpdrmturmfelstmltd sd dxpeld m ul fmjbke dl cm prdskl dxtdrlm, tmc sktumfkl sd judstrmdl cm Hrgkfm 6f9

^rkjdre vdmjes dc dgdfte `d rd`ufkr cm prdskl `d ulm `ksecufkl dl gmsd cquk`m`d >.2 jec `d fejpeldltd (=) per jec tetmc (x=:>.2) `ds`d =>> jjOh omstm 4=.4jjOh (`ds`d dc pulte m= omstm dc pulte m2). Dc pulte m2 dst sebrd cm cldm qudrdprdsdltm cm dfumfkl WK-=>, qud ds dc cjktd `d cm rdhkl `d dqukckbrke dltrd dccquk`e qud dxkstd m prdskelds jmyerds y dc vmper qud sd fejkdlzm m gerjmr. Cesdstm`es qud rdprdsdltml ces pultes sebrd cm cldm dl hdldrmc sdrl `dc klkfke `dgerjmfkl `d vmper, ds `dfkr dl dc sdle `dc cquk`e sd gerjmr ulm burbuim, per cefumc sd cd ccmjm m dstm cldm cm furvm `d pulte `d burbuim. @dspus, rd`ufkjesml js cm prdskl `ds`d 4=.4 omstm 00.4 pmrm ccdhmr omstm dc pulte m6 dc fumcdst ubkfm`e dl dc cjktd `d cm rdhkl `d `es gmsds y dc vmper, ce fumc les suhkdrdqud prftkfmjdltd te`e dc skstdjm ds vmper fel dxfdpfkl `d ulm hetm `d cquk`e,ce fumc msdjdim m ulm hetm `d refe y ds per dse qud m dstm furvm sd cd ccmjmfurvm `d pulte `d refe. Ces pultes ubkfm`es dltrd dstes `es pultes, mc kr `d m2

p-x=-y=

>.>

2>.>

0>.>

7>.>?>.>

=>>.>

=2>.>

> >.2 >.0 >.7 >.? =

x=, y=

p,jjO

h

px py

Wmper

Cquk`e

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omstm m6, dvk`dltdjdltd rdprdsdltml dstm`es fel fmltk`m`ds frdfkdltds `d vmpery `dfrdfkdltds `d cquk`e. Gklmcjdltd, mc rd`ufkr cm prdskl per `dbmie `d 00.4jjOh omstm ccdhmr 2> jjOh (pulte m0) sd tdl`r vmper, fm`m vdz js dxpml`k`e

Hrgkfm 6f. Fmjbkes `d dstm`e `drkvm`es `d cm vmrkmfkl `d cm prdskldxtdrlm, jmltdlkdl`e cm tdjpdrmturm y cm fejpeskfkl felstmltd. Furvms `d

pulte `d burbuim y pulte `d refe.

Moerm, mlmckfdjes dc dgdfte `d klfrdjdltmr cm prdskl dl ulm `ksecufkl dl gmsdvmper `d ?>% jec `dc fejpeldltd js vectkc (y=:>.?), `ds`d 0> jjOh (pulteb=) omstm 87.? jjOh (pulte b2). @urmltd dstm fejprdskl, dc vmper sd vudcvdjs `dlse y sce omstm ccdhmr mc pulte b2, mc ccdhmr m cm furvm `d cm dfumfkl WK-=7ds fuml`e mpmrdfd dl dc sdle `dc vmper ulm prkjdrm hetm `d cquk`e, ce fumcmsdjdim, feje mltds, mc refe (dse cd `m lejbrd m cm furvm). @dspus, mcklfrdjdltmr cm prdskl `dc skstdjm `ds`d 87.? jjOh (pulte b2) omstm ??.? jjOh(pulte b6), prdskl m cm qud sd tefm m cm furvm `d cm dfumfkl WK-=>, cm fmltk`m` `dcquk`e sd klfrdjdltmr m dxpdlsms `dc vmper y mc mcfmlzmr cm furvm `d pulte `dburbuim, sce qud`mrl trmzms `d vmper, qud`ml`e cm tetmck`m` `dc skstdjm dl gmsdcquk`m. Gklmcjdltd, sk cm prdskl dxtdrlm sd klfrdjdltm `ds`d ??.? (pulte b6)omstm =>> jjOh (pulte b0) ombr `dsmpmrdfk`e fumcqukdr vdstkhke `d vmper y dccquk`e sce sd fejprkjkr ul pefe js.

Cms hrgkfms y cms dfumfkelds `d `el`d prevkdldl sd pud`dl djpcdmr pmrm cddr cmsprdskelds `d burbuim, `d refe y fmcfucmr cms fejpeskfkelds `d cms gmsds dldqukckbrke e bkdl omfdr ces fcfuces ferrdspel`kdltds.

Fcfuce `d cm prdskl `dc pulte `d burbuim.

Cm dfumfkl 4 les pdrjktd fmcfucmr cm prdskl `dc pulte `d burbuim e cm prdskl `dvmper `d ulm secufkl fel sce felefdr cm fejpeskfkl y cm tdjpdrmturm (e cms

p-x=-y=

>.>

2>.>

0>.>

7>.>

?>.>

=>>.>

=2>.>

> >.2 >.0 >.7 >.? =

x=, y=

p,jjOh

Furvm `d pulte `d refe00.4

4=.4

87.?

??.?

C

C-W

W

Furvm `d pulte `d burbuim

m0

m6

m2

m=

b=

b2

b6

b0

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prdskelds `d vmper `d ces fejpeldltds pures). Ds kjpertmltd tdldr prdsdltd `dqud kl`dpdl`kdltdjdltd `dc dstm`e qud tdlhm cm secufkl m cm fumc cd qudrdjes`dtdrjklmr dstm prdskl9 cquk`e, vmper e jdzfcm `d dccms, skdjprd sd rdgdrkr dcpulte `d burbuim m cm fel`kfkl dl qud te`e ds cquk`e dxfdpte ulm burbuim `dvmper, ds `dfkr khumcmjes x= fel cm grmffkl jecmr `dc fejpeldltd js vectkc,

puds cm fmltk`m` `d vmper ds tml pdqudm qud prftkfmjdltd te`e dst dl gmsdcquk`m.

Cm fejpeskfkl `d cm prkjdrm burbuim, cm pe`djes ebtdldr usml`e cm dfumfkl?, substktuydl`e cm grmffkl jec `d cm secufkl qud tdlhmjes,kl`dpdl`kdltdjdltd `dc dstm`e dl qud sd dlfudltrd y cm prdskl `d burbuimfmcfucm`m fel cm dfumfkl 49

y= : ^= x=/ p (?)

Fcfuce `d cm prdskl `dc pulte `d refe.

Cm dfumfkl == les pdrjktd fmcfucmr cm prdskl `dc pulte `d refe `d ulm secufklfel sce felefdr cm fejpeskfkl y cm tdjpdrmturm (e cms prdskelds `d vmper `d cesfejpeldltds pures). Ds kjpertmltd tdldr prdsdltd `d qud kl`dpdl`kdltdjdltd`dc dstm`e qud tdlhm cm secufkl m cm fumc cd qudrdjes `dtdrjklmr dstm prdskl9cquk`e, vmper e jdzfcm `d dccms, skdjprd sd rdgdrkr dc pulte `d refe m cmfel`kfkl dl qud te`e ds vmper dxfdpte ulm hetm `d cquk`e (refe), ds `dfkrkhumcmjes y=fel cm grmffkl jecmr `dc fejpeldltd js vectkc, puds cm fmltk`m``d cquk`e ds tml pdqudm qud prftkfmjdltd te`e dst dl gmsd vmper.

Cm fejpeskfkl `d cm ctkjm hetm (e cm prkjdrm, `dpdl`kdl`e `d sk vmjes `dcquk`e m vmper e vkfdvdrsm), cm pe`djes ebtdldr `dspdiml`e x= `d cm dfumfkl ?9

x=: y= p / ^= (?)

e bkdl `dspdiml`e x= `d cm dfumfkl 4, `el`d sd substktuyd cm grmffkl jec `dcfejpeldltd js vectkc, kl`dpdl`kdltdjdltd `d cm gmsd qud tdlhm ludstrmsecufkl, cm prdskl `d refe y cms prdskelds `d vmper m cm tdjpdrmturm qud dst cm`ksecufkl9

x= : (p - ^2 ) / (^=- ^2) (4)

Fcfuces dl cm rdhkl `d dqukckbrke.

Vk cm secufkl sd dlfudltrm `kstrkbuk`m dl `es gmsds dl dqukckbrke (cquk`e-vmper) yfelefdjes cm prdskl m cm qud sd dlfudltrm, ds `dfkr sk cm prdskl p m cm qud dstcm secufkl ds felefk`m y ds jmyer qud cm prdskl `dc pulte `d refe y ds jdlerqud cm `dc pulte `d burbuim, dltelfds, cm grmffkl jecmr `dc cquk`e x= pe`djesfmcfucmrcm fel cm dfumfkl 4, substktuydl`e mo cm prdskl p m cm qud tdldjes dcdqukckbrke. Cm grmffkl jecmr `dc vmper dl dqukckbrke fel dstd cquk`e cm ebtdldjes

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Hrgkfm 0m. ^rdskl `dc pulte `d burbuim e prdskl `d vmper `d cm secufkl,dl gulfkl `d cm grmffkl jecmr.

Fcfuce `d cm furvm `d prdskl `dc pulte `d refe9

@d cm Dfumfkl ==9

p : ^=^2 / (^= - (^=- ^2) y=)

Vubstktuydl`e9

p : (6.> mtj) (=.66 mtj) / (6.> mtj - (6.> -=.66 )mtj y=)

p : 6.;; mtj2/ (6.> mtj - (=.78mtj) y=)

pr (mtj) : 6.;; / (6.> - =.78 y=)

Vk m cm dfumfkl `d prdskl `dc pulte `d refe cd `mjes vmcerds `d y=`ds`d > m=.> fel klfrdjdltes `d >.= ebtdl`rdjes cm tmbcm 2 y cm hrgkfm 0b9

>.>>

>.4>

=.>>

=.4>

2.>>

2.4>

6.>>

6.4>

> >.2 >.0 >.7 >.? =

p,mtj

x=

prdskl `d burbuim

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y= pr

> =.66>.= =.0=>.2 =.4>>.6 =.7>>.0 =.8=>.4 =.?0>.7 2.>>>.8 2.=?>.? 2.0>>.; 2.78= 6.>>

Pmbcm 2. Wmcerds tmbucm`es `d cm prdskl `d refe m `kgdrdltds grmffkeldsjecmrds `dc vmper.

Hrgkfm 0b. ^rdskl `dc pulte `d burbuim e prdskl `d vmper `d cm secufkl,dl gulfkl `d cm grmffkl jecmr.

Feje mltds, ebsdrvmjes qud mjbms furvms fejkdlzml dl x=e y=:> fel ul vmcer`d prdskl `d burbuim y `d refe `d =.66 mtj. Dl dstd fmse mjbms feklfk`dl perqud sd trmtm `dc fejpeldltd 2 pure. Vkjkcmrjdltd, mjbms furvms tdrjklml dl x =ey=:=.> fel ul vmcer `d prdskl `d burbuim y `d refe `d 6.> mtj, ce fumcferrdspel`d mc fejpeldltd = pure. Vk cms `es tmbcms y furvms cms peldjes dl ulm

secm, qud`m cm Pmbcm 6 y cm Hrgkfm 0f9

>.>>>.4>=.>>=.4>2.>>2.4>6.>>6.4>

> >.2 >.0 >.7 >.? =

p,mtj

y=

prdskl `d refe

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x=e y= pb pr

> =.66 =.66>.= =.4> =.0=>.2 =.77 =.4>>.6 =.?6 =.7>>.0 2.>> =.8=>.4 2.=8 =.?0>.7 2.66 2.>>>.8 2.4> 2.=?>.? 2.78 2.0>>.; 2.?6 2.78= 6.>> 6.>>

Pmbcm 6. Wmcerds tmbucm`es `d cm prdskl `d burbuim y prdskl `d refe`kgdrdltds grmffkelds jecmrds.

Hrgkfm 0f. ^rdskl `dc pulte `d burbuim y `dc pulte `d refe `d cm secufkl,dl gulfkl `d cm grmffkl jecmr.

=.>>

=.4>

2.>>

2.4>

6.>>

6.4>

> >.2 >.0 >.7 >.? =

p,mtj

x=,y=

^rdskl `d Burbuim y `d Refe

pb pr