Das Pentagramma Mirificum von Gauß

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1 Wissenschaftliche Prüfungsarbeit gemäß 1 der Landesverordnung über die Erste Staatsprüfung für das Lehramt an Gymnasien vom 07. Mai 198, in der derzeit gültigen Fassung Kandidatin: Jennifer Romina Pütz Matrikel-Nr.: der Johannes Gutenberg-Universität in Mainz Thema: Das Pentagramma Mirificum von Gauß Fach: Erstgutachter: Zweitgutachter: Mathematik Prof. Dr. Manfred Lehn Prof. Dr. Duco van Straten Abgabedatum:

2 ts r3 s t är s r tr rs r s är ä r r tr r r r s är s tr r tr s ät3 ür s är s r t r rs s P t r r str t s ö t rst t s t ÿ s P t r r är s s P t s P t s t s t s P t r r r ÿ ss t3 P t r r t s r t r s t s t rü 3 P t r r r r t s t s t s P t r r r t ät3 3 r t s t r ss t rü 3 s P t r r 3 t r t 3 r tr tr s r s t3 s r t rs s t3 s t3 r t t r r tr tr s r r r r r r t s t s r3 s t r t r

3 t s r r r ÿ r st r s s ss 3 r t 3 3 rs st r t t r t r r t s s s t s r t 3 r s r rst r3 t r r ü r t t s r rö t t ÿ t 3 s t r r r t 3 r tü r r s t P t r r s ü r t r P t st r t r s r r t s ÿ t 3 r s s st 3 rt tt r t ss r t r t r ss t s t s P t r r r t s r ts s 3 s s r s är s r s 3 P t t t s r s s rt r r t s ü s t3 r 3 s t s t ÿ t t ss s s r s är s P t 3 s P t r r s t ässt s r s st t röÿ 1 t s t s r ö s r r t s r r tt t r s ÿ s t t r t s r r s röÿ s s t r s är s P t r ü s r r s s t s P t r r r ü s r t s 3 3 ö r r r ÿ r ä st r r s s st r s s t t s t r s är s tr s rs r s är s r tr rtr t s r t st ät3 s s t s 3 r r t rs t s r r s s ä st r tt t r s P t r r 3 str r s r t r t ss ÿ ü r s r 3 3 t r r s r tt t s r r ÿ s 3 P t t s t t r ss t s t s P t r r r s ÿ r r ÿ P t r r t 3 ä st ü r t t s 3 s ü s t s t 3 rst s 3 rr r r s 3 rst t P t s ÿ ss t3 s ä t t s t r t ü r s ÿ s t 3 s t r t3t r tt s rt t s r r ÿ t ss ss s t s t s P t r r r t st 3 ö s ÿ r r ü t t s 3 s ÿ t rs r t s är 3 r r r 3 ä st r t s s st r s s t t s ü r s r t r ss t s t r s ÿ r r r s r t s 3 r r t s r r t rst s t r s är s r tr 3 r rt

4 s ss r r ss r s r r s st s t3t r s r t s rt r rs P r P t r r t t s s r ss s t t rs t r r rs tät 3 s tät st s st t tq s r r t r t r ä rt r

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6 t 3 s r rü r r r tr r 3 st s rst P rs sst t s r r t s t3 3 s r ä r t r s r s t r t r r r r ü r rt r t r r 3 r s r t s r r s t r r 3 s str s t rt t rs r r ä r ts s ü rt 1 s sst 3 r r s r ä r r st r s t r s t r t r rst s t r t r r 3 t3 s r ä t s r t r 3 r r s ü rt s rt rü r s t3t r r r r s rts st r t r r r 3 r r 3 ü r r t st t r s s r t r ü rt r s r röÿ t rr s t r tr r r r s r st r t crd(θ) r chord(θ) s s tr t t 3 r ABC s s r M AB tt t s r s s ABC 3 t t θ r r tt t AM B t st r t rt s r (θ) := s r, s AB r s s r s s 3 t t r st t t 3 rs tt ts r sr 1 s sst s t r r s t r s r r t r r s ös t t r s t r st ür r t s r rü t r s 1 r s r s Pt ä s s r s r s rst r r r Pr t s t r t r r 3 s t3 s r sst s s ü r 3 r r s t r t s rü r s s r r s ä t st r s r Ü rs t3 r t r 3 r s är s r tr s st t s r ü r trä t r r s r t s r rs t 3 rst r s är s r s r ö s öst r str r ä t r st r tr s är t ä r r rts r tr t s t r st r t t s r tt st r s st r s st s r tr t s t t r r rts s 1 r t r st r t t s r r tt st r s st r s r s s t t r r rts s Pt t r st r t t s r r tt st r s st r s r s Pt t t r

7 r t t s t r r r str t r r t r s r rst r s r r rt r s r r ä r ür tr tr s rä t r s röÿt t s r t r s r t t r r r st ts 3 r t s s t t s r ÿ r st s är s r tr r t t3 r t r r s r tr r t t r r ts r s 3 rü s s r ä 3 t s r t s s r s är s r s r r tr s s rst Pt ä s t st r st s ür r rt ü r rt s s tr tr s ss s t rrs t s s t r r s r str s r r r t st r st r ts str s t rs t r t t s t s r tt 3 r t r t rt r ss s s Pt ä s r r 3 s st s r t s r t ätt s r ts s r ä r s rs r s s ss r s s s Pt ä s s r t s r r t ss t s st r 3 r s s r t s s r st t st s är s r r tr Pt ä s t r rs r t 3 ös r s r tr t t 3 rstr t s r tr tt t r t ÿst r s Pt ä s rrs t s r s ös s t t rt r Pr t s 3 s r r r s t r s ür s t t3t r Pt ä s t r t3t s s 1 s sst ür t t r s r 3 r s r ss rt t s s r rt rü r str s 3 r t r s s ü st t s r s ü r st s ss ässt s ü r r s r tr s st t s P r r r tr r str ü r äÿ t r t s r t s s t s sst r s r ä r s r s Pt ä s r s r ss s t 3 r s r st t ör rt r tr r t t ss s t s sst t r r s är s r tr 3 rü tr t rt r t r t 3 s ä ät3 s tt 3 r är öt r

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9 t P r P t t s r ÿ r s ss rt 3 r tr MP t s r r ÿ r s ÿt P r 3 P r t p 3 t r ÿ r s t s 3 s ü r tt t P 1 P r 3 r rt r r t r r ä r P 1 P s P 3 3 t r ÿ r s t s s 3 P 1 P ör P r p r P t r P r 3 r t r P r s t3t 3 s r ü r P P r t P 3 r 3 ö ü rt r t r r s r 3 P r t r s rs s r st t r t r P r s tr t t s r 3 ör P s s st s r t s r s s r P r s r 3 r ör P r s r st ts 3 r t s st ür r ö t s s t st ts 3 r t ä t t r r r t s t s r 3 ör P s r 3 r r r r t r P r t P r r t r P r t r P 3 r t r ür r st r r r t s t r 3 r P r ör t P s s st ts 3 r 3 r P r p ör P P str rt r ÿ 3 t r t 3 r s r ÿ r s s p r s rt 3 tt t t r r ä r P P 3 p st r tt t r s r st t r r t s p t r r s r ÿ r s s r P P t rt 3 r r P r p s t t ä t s r ÿ r s s tt t r s t ss tr MP r 3 r ÿ r s r P t t t tr MP rt 3 r p r ä t t M P t ä t rt 3 r p s s r s s r ÿ r s s r P r s ässt s r t r r ÿ r s rt r 3 p ör st t t ä t P s r t t t ss rt 3 r p 3 s MP t r ÿ r s rt 3 r r ÿ r s P t ä t s t s s 3 p ss ÿ 3 p t rt 3 r r ÿ r s P t ä t t r s P s P 3 P t s r P r p trä t r π π PMQ = r ür P t p

10 s r P t r P r p r t r r ÿ r s t t tr MQ r 3 p ör r r 3 p ör P P t t t rt 3 r r ÿ r s p t ä t t s s r rt 3 M Q r s t ss PMQ = r π t ABC s är s s r r 3 r t r t r3 rs r t r ÿ r s r s ss r s r ÿ r s s P r 3 ss P t s ö r r 3 ör P s 3 r t A B C A r 3 ör P s s s P r s är s s r ABC s P r r 3 ABC r t ABC 3 r t a b c s t α β γ s t α = π a, β = π b, γ = π c, a = π α, b = π β, c = π γ. tr t t ABC s s s r r t t s r r t s s P r r s s r t s r 3 π α = π (π a) = a r s t ss s 3 ABC ör P r r r ABC st r tr s ät3 ür s är s r s s t t s r tr r r ts r s är s r tr t t s t t r t rä t s s s s r t r r t s r r r r s r t t r str tr t r t s t ä r (α) = s r = sin α, röÿ 3 rst s Pt ä s s r r ts t s t3t r r s t s r 3 str t ü rt r s r t s t t rst r r r ärt r rst t str tr ä t st s t s str r t r r ts rt t t r s r röÿt t s r s str r t r t r ss r ü r r s t s s s st t 3 r 3 r s s r MAB 3 r t t tt ts α 3 r t

11 s r s ö r s t s st 3 s r s s r ü r ö t t r t r t r s s ät r s t t rs r ss str tr r ts 1 r s ss s t r P s r 3 är r s r s r t Pt ä s s s s t t r t r st ss r s s s r s t s rs s r s 1 cos rt st t3t r ür 3 r r t r s r r s r ä r s r t t r r r rst s t t ät3 3 ür s är s r r t s r s r s t t r st tt ss s r ss st t r rt r s Pr t s Pt ä s t3t tt ü rt s rt s r r r t t s üt 3 t r r ä t t r ss s s 3 ä st ss s t tü r r r r ä r 3 r Ü rs t 3 t r 3 t 3 tr s t r r st s Pt ä s r s tq r r r P r s r ür str s t r tr s r t3t r r r ss r r t s r r t s s t st r r s r ü t s s t s s s s t 3 t t r3 r t rs t t ä s 3 t s t s rt s s s s s r rs t s s r r t s s s rt s s r r ss r t P rsö t s s r st r r r t s rs üt3 t r s t r är 3 st t s är s t s t3 r 3 s t r t r s r t3 s s öst r r r rst r tr tr s ät3 sst t s 3 s st t s s r s s t t r r ü r3 ört s r r ts t st r r r t s s ä sin cos cos = tan sin = cot s r r s s r s s 3 r tr t t s s t3t r s s s 3 s t3 3 s r ä 3 r s 3 tr t s t ü st r s r s ss r r r s r rst r rt sst t s s t3t r t3 s r r ÿ rts r tt r s s s r ä r r s är s r tr 3 r tt s s 3 s t t r tr r t s t t ss t3t r r r r r r ür st r s r r r st t r ss r str s r r r ös t rt r s s r tr ä 3 s ss

12 r P rs r s r s s t r t r s är s r tr s ü r r öst r t s r stä r t s t t3 ss t3 rü r s t3t r s r s s P r r s rst t t r s ät r r t rs t t r s r st t 3 r t r s st r s s r ÿ r t ür s st är s r tr t r s öst r str s st3 r s ss t3 t r r r s t r t r r r 3 t r t t s r s t t rs tt s üt3 t r s t t t r s 3 ö t r rts s t s ü rs t3t s rt s tt t rs 3 ä t t s t s s r r r r t r str tr 3 tr ss t3 r s rst r r t st ät3 r s är s r tr ür r 3 ss t3 r r r t s t t3 s r r s s r ts s r t t s r s s s ä t t s r s st ts r t r 3 rü ü rt ts r st s ü rr s ss r 3 s ss t3 s ür r t r r s 3 st r r ä t r s r r s 3 t r rts st t ä st r s t t stä r t s 3 tr t s r ä t s r tt r s t r ät3 t3 s s s ts t3 r s är s r tr öst s r ss t3 s s t t r ä s rs t3t r 3 t r s t r r r t s r t s s s t3 s ts t st r s r r s s ässt s t är r r t r r s s t t3 s r K ür t r r s r r t ts t tt t M rs r s t3 ss t3 s är s r ABC t α β γ t a b c r t s s r t s r ü r sina sinα = sinb sinβ = sinc sinγ. ö t t s ss t3 s röÿ r sr ü rt s 3 r r t t 3 s r 3 s rt s t3 sinα = sin(π α). ö t s 3 röÿ s t ä r st t r s ü r

13 röÿ r s ässt s s t s ss t3 s t t ös s t Ü r ür t s s ss t3 s s t s t t s rt r s s r ür t r r r r t ss t s r s ABC s s r r π ä t s 3 s s ss t3 s 3ä s ö t s s t t3 s ö t 3 r rst s r r r s s s s t r tr s r t s s ss t3 s ABC r K r t ÿ t A MB r t ÿ t MC r t ÿ t A ä h A := AF t sinc= AD MA sinβ = h A AD MAD, ADF, sinγ AE sinb = MA = h A AE MAE, AEF. r ts t t MA = 1 r s t ür sinc = AD, sinb = AE. t t ür st h A h A = sinβsinc, h A = sinγsinb. s t3 s t sinβ sinγ r t r sinb sinβ = sinc sinγ. ü rt s ür P t B C s t r ss t3

14 s t r r t s s ss t3 s t t r ABC r K r tr t t r t BA t CA P t B 3 C rü r rt s t r t 3 r tba tca t s t r s r r t ä A A 3 P t MB MC t MA A MAA r t A 3 A t MA = MA + A A MA = MB MA + tba MA = MA + A A A A, MA = MC MA + tca A A. s r MA A MAA r ä t MA = cosc A A = sinc MA = cosb A A = sinb ÿ r n A rt s r t r r MBC t ä t3 r t 3 r s ÿ t t n A s r t r n A ( MB cosc+ tba sinc) = n A ( MC cosb+ tca sinb) n A MB cosc+ n A tba sinc = n A MC cosb+ n A tca sinb. MB MC r MBC 3 r n A r t r st t n A MB = A n MC = 0 ÿ r s ÿ t r n A tba β n A tba = cos( π β) = sinβ t n A tca = sinγ r s t sinβsinc = sinγsinb sinb sinβ = sinc sinγ. ü r r s ss t3 s r s ss tt r s s s üt 3 t r 3 t ä t s r 3 t r rts r 3 t tt st

15 r r s s t t r r st t s r t r t s t s ü rs t3t t st t r r tr r rt s t st st r s r 3 t Ü rs t3 s r ä t r s tt tt t r r t s t s s ür s r r ä s t t r r ss s s r s rt rü t r s str s tr tr s r s st s t3t s t ü r rt r s r s r s r r ss rt s t r r r r är t r ü r ts r r P r st r s r t r r t t ä s str r t3t s rä r r r ös r s r s rs tsst r rü t r t s t3t s ü r r t rt r ts t t r t t s ü r s t s s s r s r r r tr ä t t s r t s r tr s s r q q s 3 r ÿt r ts s r r rst rö t t r st ÿ 3 r t s tr tr s st s s r t t r r rst r 3 st r tr s stä r t t s öst r str t s r rst r ü r tr s ss ÿ ss t3 s r s s är s r t r t3 r s s s ss r ÿ 3 ä st r r sst r r t t r s ü rt s s s 3 ss t s t 3 r t r r rt t s ät3t s t rt t s t r r tr r r ä 3 s är s r 3 r ÿ rst s s s t r t r r s t r t ss r t 3 s r r st r r tr s r s r t r s rs s ss s t r s r r s r r r s s ä rt r s t s t rts ät3 r ä t s 3 s t r rü r s t s t 3 rst r t s ss t3 r s ss t3 ä st r s tt t s r r s tt s st r r P rs r t r ts s t 3 ö t r rt r 3 s s r t s ür t s s s t t s ür s s r s s 3 r tr t t ü t3t r r r r ts r3 r ä t s r r st s t s s s t3t st s t t t r t r s ässt s 3 s t s s s rü r rts ü r t s t r st r t t s r tt st r s st r s r s t s t t r

16 r rt t s r r rst r s sst t s r s s öst s tr t r t r st t r sst tr tr s t r s r t s r ä t s r ü s r t st t t s rü r s r 3 s s tr tr s t rö t t tr tr s t s r r s r t s r st s s r rs r s s r r t t r ä s r ä r t s ässt s s s s r r ss r t ö t r s t t3t s rst s s t st t ä 3 r s s ss t3 s rt r ür s r t ss r s ä t tt t s ss t3 ür ss t3 st st t ss r r r t s ss t3 t t s r st t t r r s s 3 r ös r s 3 r r trä ü r r str tr s t s ss t3 s rst t s s s s Pt ä s s t t r r s s ÿ s r r r r s s r t s r t s t3 s sst r r r tt s r r rt ä r s t r r t s ss t3 3 r 3 r r r t r s t r rt s t3 r s är s r tr s st t3t t s s rtr t r t r r tr str s r st r t s ss t3 r ä t t 3 rst t r t ü r rt r s tt t r rt t3t 3 r rt s t s ss t3 tt s r rts r ü t r s r tr s s s ss ÿ ss t3 r t s r ü t s 3 ss t3 r s ss t3 st t r s t s r t 3 rt r r 3 r r 3 st t t3 t s ss t3 ür s är s s r ABC t α β γ t t cosa = cosbcosc+sinbsinccosα, cosb = cosacosc+sinasinccosβ, cosc = cosacosb+sinasinbcosγ.

17 r t s tür r r r t s s s r 3 r3 r s t r tr s r t s s s ss t3 s ABC r K ä r P t C r tr MC s t3 MC =: d s ÿ ä r C s s t C A MA s A r t r B t C A = d tanb MA C, C B = d tana MB C, MA = d cosb MA C, MB = d cosa s ss t3 ür r t ÿ r MB C. A B = C B +C A C B C A cosγ, A B = MB +MA MA MB cosc. t3 r s r t r A B = d (tan a+tan b tanatanbcosγ), A B = d 1 ( cos a + 1 cos b cosc cosa cosb ). s t3 s ÿ s t d r s tan a+tan b tanatanbcosγ = 1 cos a + 1 cos b cosc cosa cosb. 1 cos = tan +1 sin = cos tan t r s tan a+tan b tanatanbcosγ = tan a+1+tan b+1 tanatanb cosγ = cosc cosacosb cosc cosa cosb

18 cosc = (cosacosb) (1+tanatanbcosγ) cosc = cosacosb+sinasinbcosγ. ü r r s ür r t s r t r s s t t s ss t3 s t r r t s s s ss t3 s t t r ABC r K tr t t r t AB t AC P t A rü r rt t r t 3 r tab tac t s t r s r r t ä B C 3 P t MA t MBB MC C r t B 3 C t MB = MB + B B MB = MA MB + tab MC = MC + C C B B, MC = MA MC + tac C C. s r MBB MC C r ä t MB = cosc, B B = sinc, MC = cosb C C = sinb ä r t a CMB st t ÿ r cosa = MB MC MB MC, MB = MC = 1 B C r ts t3 r s t cosa = MB MC = ( MA cosc+ tab sinc) ( MA cosb+ tac sinb) = ( MA) cosbcosc+ MA tac coscsinb+ MA tab cosbsinc

19 + tab tac sinbsinc. P r t s r r t s t ( MA) = 1 rü r s s t AB t AC st t s t r r A r t s t r rt 3 MA s t MA tab = MA tac = 0 r s t cosa = cosbcosc+ tab tac sinbsinc. r t AB t AC s ÿ P t A α = CAB t t cosα = r tab = tac = 1 t t AB tac tab tac, cosa = cosbcosc+sinbsinccosα. ü r r s t s ss t3 s r ä t t s P r täts r 3 s ss s t r rt tt r r rt r t tt r r ür üt 3 t r r tr t t s r r r s t s tt s s tr tr s t s t sät3 r s är s r tr t r t 3 3 s t r t t s r t tr t ÿ r s t s s t r tr r tr str s s rt r r s t s är s tr r tr 3 r rts st t rt rst r t t r ts s r s 3 t rts r tt t r r r r tr r rt t r t t r t t st rtr t r s s r r r 3 s r ç s èt r t s rt r s s t r s t r ü r ärt t s s3 t t r r tr t 3 s t r ss r ss s3 t r t s r r r r r r tr r ärt r s ö st r t t r s r r r r r ü 3 ss r t t r ÿ r st r tr s rs s är s s 3ä s s s rst sst t s s r t 3 r r r s är s r r t r r sä t r s s t stä s r t s är s r s s ÿ r 3 ö t r s r r ä st t t s r r r rts r ç s èt t r st r t t s r r tt st r s st r s t t t r

20 s s s s t t3t t rtr t r r s s r tr s r t r tt r t rs s r r r3 t r r r ss r st r t r s P r r s r ä s t r ts t ss r t rs s s P r r s st ÿ r s P r täts r 3 ss r ü t t r s ä 3 r r r r s ss 3 s 3 ät3 r är r r 3 täts 3 st t ss s r st t t s är s r r rt s r t r ss r r3 r 3 r ä t s s q 3 t r r s s rt s t t r r ts t 3 t r r ü t s r s r t r t s är s r s st t t t st s st r s ss t3 ä st r s tt ÿ r ü rt t r s r r r tr s r tr s 3 r t r t r ü t r r tr s t r t r t t rü r s t s ö r r s ststä tr tr t r r ü r ss t 3 r s r t ss ü r r tr tr s t ört r r t s s r ss t r s r rts r tt r ät3 r t r st s t t 3 r s s r s rstä s rü t ss s r t r t ss t t t r ür r r ü ö r s t s t t r t r r s r rst r s r rr s t rs rü rs s r t ss r r 3 s r r ür st r t t r ts ss t s st t s s r s t s r t ss s s s t s r r r t tsä rö t t r r t t r s r t r r t ss r ss t r s r st t rs r r r tr ÿ 3 tr t ss tr tr s t ss öt3 3 r ö r s t r s st r r P rs r r r ä 3 st ss s t P r t rr t tt s ss t3 s r t3 r s r r s ss t3 3 t r t r 3 s t 3 rü s r rö t t r r ts r 3 r t t tt ür r s r s t s ss t3 3 r r t s s P r täts r 3 s t r s t3 t s t t r3 r P t s s s ät r rt t r r r r t r t s täts r 3

21 t3 s ss t3 ür s är s s r ABC t α β γ t t cosα = cosβcosγ +sinβsinγcosa, cosβ = cosγcosα+sinγsinαcosb, cosγ = cosαcosβ +sinαsinβcosc. s tr t r 3 s är s r ABC K s 3 ör s P r r ABC s s s är s r K s t3t r t a = π α b = π β c = π γ α = π a β = π b γ = π c r t s ss t3 a r t r cosa = cosbcosc+sinbsinccosα cosα = cosβcosγ +sinβsinγcosa. r s t s ss t3 s b c r r P r t t r r r t s r s ss t3 t r rs r ü r r r t r r r s tt s t r t t t r ä r r s r t ü tr tr s 3 rst r 3 r ss r s tt 3 r ss s 3 t s r t s ä t t r t r s r r s t s 3 t r s t r s r tt r rü r rt t r r r r t t s r r r r s r r st r st st r r r st rt r t r s rs t t r s rt s ss t t r 3 s r t 3 t s s t s t r ss r r t r s t r t t tr r r s ä r s s s r ÿ r tt r 3 s r r s t r s r r r ts r t r ü r t t r s s t t s s ss ts r röÿt t s r r t s ü rst r r r ü r3 t r Pr t st t r t t s st st ts s r t s r t s r s s t st r t st r ö t st s rst s rs s t t s s r r

22 r t r s s r t r s r t ü rt r ts r s r r t s s ü rs t3t s t s r t t r r t s trä t rr t s t r s r ÿ r s t r rt s s r s r ärt r s t t ür ü r r r t r ss s s s s t ts äst r s ür t t s ät t ts ss r r äst t rt s t t s r t r3 3 r ÿ r r s r ü t s röÿt t s st r ü t ts r s r t r r t r s t t t s rt s s t rst r r rs rt st rö t t r t r r t r s str t r r ür r tr ür3 r s r s r r s t t r s r t r t s r st ts s s s r t r s s r r t s r röÿ s t r r r t s s s s r röÿ s t 3 r s q 3 ss r s r s r t t r t t ür s s s r ü r r r s ü r r t t r r r t s rt t 3 t rts r t r t st s t ss s ö st t r s s s ss r t s s rt r t st r 3 s s r s 3 ör r t s s tär s s r sst r 3 r t t t r 3 t r t s s 3 st t tär s s t r t s s s s s s s r t s s r r r tt t r ts r s r t tür sin( π α) = cosα t r s t t t s s tr s t t t t r t s r t t t r st r t rs t t t t s s s sq r 1tr t s r t rs s s t t s 1 t r r t st rt s t t s r rr rs

23 logsin( π α) = logcosα. s t s s rs t r t s s t r t s s s s rü r s ässt s s r r t r 3 s r t s s t r t s s t t rt t r r t s s t3 t t logsinα logcosα = log( sinα cosα ) = logtanα. r 3 3 t s r t s s s r r r r rt st tt s s s 3 t3 ässt s t t rt t3 tr tr s 3 r s s t r 3 t s s t s s rs s s t s r t st ü r st s s s s tt s t ü t r s t3t r st s r r ä r s 3 r s t s s ä t s r s 10 7 s t3t r r t s s s s s s r t s s s t s r r 3 r s st t s rst t s s rst s s t3 3 s r r r ss t r r t s 1 3 t s s 3 r r ü s r t ss s trä ä r s s s 1 e r r3 r s r r rü t r r s s s s s st s 3 ä s r r r s s t3t s r3 t s ät r rö t t ss r t t r r s s rs t t r s r s s r t t s s s r t s rü r t s t rs r r r r r tr ü st r Pr rt tr t t 3 sät3 t rt s r r rst r rt r s s r röÿ r t r tr r tr tr s ät3 r t r r r s rs t r r st r tr 3 r t r r t t tsät3 r s är s r tr 3 rü s r r t s r 3 ABC s s t γ = π t ss t3 sina = sincsinα, sinb = sincsinβ. r t s t s s t s s ss ätt s r t s t tür r rt

24 r t s ss t3 rt cosc = cosacosb. s s ss t3 s r t r cosα = sinβcosa, cosβ = sinαcosb, cosc = cotαcotβ. t r sinα sinβ 3 cosa s t3 3 s t cosα = sinb sinc cosc cosb = tanbcotc, cosβ = sina sinc cosc cosa = tanacotc. r t t r t s s t3 r t s sinacosα = sinαsinctanbcotc sina cosα sinα = tanbcosc sinacotα = tanbcosacosb sinb = tanacotα r t r ür t sinbcosβ = sinβsinctanacotc sinb cosβ sinβ = tanacosc sinbcotβ = tanacosacosb sina = tanbcotβ s s r t r t r r ä t t s r rts r ts t ss s t r tär α = π α β = π β c = π c s r 3 sina= cosccosα, sina = tanbtanβ, t st s s t s s t r s r ä r3 tr t t s s ss t r s s r t r ös

25 sinb = cosccosβ, sinb = tanatanα, sinc = cosacosb, sinc= tanαtanβ, sinα= cosacosβ, sinα = tanbtanc, sinβ= cosαcosa, sinβ = tanatanc. s ür r t s är s r r 3 ä st r ü r s t s t ä t s 3 t ü r s t r r r r r rs r r r r t tr tr s r s rt r t t r s r t r t tr tr s Pr s r s t3t ä ÿ rst t r t s s 3 r t r t r r tr tr s r s r t t r 3 ä st r tr t s r r t s ÿ s är s r stä r r s rt ür ü rt r 3 ä st 3 tr t t s ü 3 r är s r t s är s r s t γ = π 3 t r ü röÿ α c β s r s ABC s r t r 3 t s t r t s r s r 3 s st s r s r t r r s r t r ä t s s r ü 3 t s s tt t t s 3 s r s 3 ü r 3 r 3 rs ür s r ABC s t tür 3 r s q 3 ss 3 s r t r ts t r s s t t ä π s t3 t r t rs r ss s r röÿ s r s s r t t t r r t rs t3t r s st r t r ts ÿ r r t rt s r tr t s s r r t tsä röÿ rs s ö t t r s r

26 3 r är s s r r r s P t r r ä st t s ü r r t3 tr t t r rt r s 3 r ä t sr t3 rs r s s tt t s st Pr t r t r r 3 Pr t r s s r ü r s r t s rs t r r ÿ t t s r t r r ts t s är s r s 3 ö ü t s ü r r t s 3 röÿ s r s s ü r ss s r t r st t r r t s s är s s r s t3t s t 3 r ts r rü r s r ärt r ü r s r r r t sä t ät3 ü r s r t s är s r r r s t t s t t r r rs st st t 3 r 3 t3 r t s ä t s t st3 t r t r s st t s t3 r st r tür t rs r rt t3 rt t s 3 t s rs r r r r t3 r r t r rs t s s s t r r t r t s t r r r ä r t r rs s s t äq t r s s tt t s st Pr t r s s r ü r Pr t r s r r 3 t3 r r t s s tt r st r 3 r 1tr r t r t r ü r 1tr r r t r st r s s s tü s Pr t r t s r tü Pr t r s r t tü r t t t 3ä t t t t s t3t r t r t s rü r t3 3 3 ä st r s sinm = tana 1 tana, sinm = cosg 1 cosg, r tt t t m 3 r 3 t a 1 a s ü r t g 1 g rs r r s r t t r t s t r q t r r t s r s t r 1tr r s t r t s s t r 1tr r

27 ö r s 3 ä st t äq t 3 st s r t tt r t t r 1tr t r s t r 1tr ü r 1tr t s t r 1tr s s st rstä r tt t r 3 3 ü r r ü 3 r är s t t 3 s s t s s r 3 t r t r t s r röÿ st ts r t s s s s r röÿ t t r t s r t s s s s t r r 3 r t s s s t ss ö t rs r r s s r ÿ s r r r t s s s s tt t s st r t r s r r 3 r r t r s s r ü r s s rü t r 3 s t3 t t s ss logsinm = logtana 1 +logtana, logsinm = logcosg 1 +logcosg. s ts r t s 3 s t3t r tt t r r t s s t3 t r logsinm = logtana 1 +logtana logsinm = log(tana 1 tana ) sinm = tana 1 tana, logsinm = logcosg 1 +logcosg logsinm = log(cosg 1 cosg ) sinm = cosg 1 cosg. ts r rs r r s r st t tsä äq t 3 t3 3 st äq t 3 r 3 r ts t rs 3 t3 st t r r 3 r är 3 r r röÿ s r s s r t s t3t r t3 s s r t s s r 3 s α β ä r t t s s s r t s t3t r s ü rt 3 ss 3 r är tär 3 s t3 s a α β b 3 r är s r s ABC s s α c β rü r s r s s s r 3 r rt r t s ss s 3 t s r t s cos( π m) = cot(π a 1)cot( π a ), cos( π m) = sin(π g 1)sin( π g ).

28 s s r s äq t s t cos( π m) = cot(π a 1)cot( π a ) sinm = tana 1 tana, cos( π m) = sin(π g 1)sin( π g ) sinm = cosg 1 cosg. s ässt s s st t ss ä r t r rs äq t s 3 s r rs s s t t3 s s r 3 r är r s tt t ä t t3 r t s r s r t t3 r r r rst t ÿ s s s r r ür r ä 3 r t r r s st t s s ss r s s s r ä t r st t s 3 ä st 3 ä ä r s tt t s tr t ö st t3 s ÿ ür s ä r s t ür 3 s s s s r ärt r ts r r r r t ü r r r r 3 tät s ü 3 r är s r t r s t r ö r ü r r t s s t r s t r stä t s r tr t r 3 ä r r t rt r ÿ s BPS r t s s är s s r t t P P BSP, SPB, PBS = π t t ü 3 r är BPS π BSP π P π SPB P r s t ss s s s t ü r s ä t ä r s tt t s tr t ö ür rst r s s, P, π BSP; π BSP,, π P ; π P π, BSP, π SPB; π SPB, π P, P ; P, π SPB,. s r ä rst röÿ s tt t m ä t r r s r s r 3 a 1 a t t r ss ü ä t r s s r r 3 r ä t t t t r rts r tr t s t q st t r r rts s st r t r t t ss s s s t3 s r r tr t

29 s 3 s t s s tt t s r s t s 3 s s r s tana 1 sin t t =. sin r m tana Pr s t s 3 t t s s rst s tt r r ts 3 t ss s ür r röÿ t a b = c d logb+logc = loga+logd. r röÿ s r ü ss ür s t logsin r m+logsin t t = logtana 1 +logtana. r rü r s s r t ss logsin t t = 0 t t s logsin r m = logtana 1 +logtana. s st r ts s 3 r r s r äq t 3 rst r rs r s t s s st 3 t s r st r r t r t tr t t r 3 ä st ü ö ä, π π SPB, π P ; BSP, P, π SPB; π P, π SPB, π, P ; BSP, ; P, π P, π BSP. r s r ä rst röÿ r s tt t ä t r s t r s ü r r r t rt ss s r ü ä ss t r r t s st st s r 3 tät r ü ä r ä t s r s t s s t s s r ü r 3 s t s s tt t s r s t s 3 s t s s t s s r ü r s sin r ( π g 1) sin r m = sin t t sin r ( π g ). ö t r ss s s r t s t ü r s ür r t s t t t t 1tr s s t t r t s t s t s s t t t t r 1tr r st r s ü r s ss t s r s r ür s r t s r ür ür r t s t r t s t t t 1tr s s t t r t s t s t s st t t r t s t t t t r 1tr

30 s st r ts t r röÿ s r s s r t ä ss r 3 t s s t t r t3t r t r s t logsin r m+logsin t t = logsin r ( π g 1)+logsin r ( π g ). r t r logsin t t = 0 r s r r t logsin r m = logsin r ( π g 1)+logsin r ( π g ). s st s 3 r r 3 t äq t 3 r 3 t ss t3 r s t stä s st s r ü r s är s r tr r r s ä st t s ü r t r r s P t r r s rs t3 ss r rs

31 s P t r r s r s r t s r t r s r r s r ö t r äÿ t r ü rt 3 r är s s är s r s rs t s P t r r r r r ä ü r t r t ä t s s ü r ÿ r s 3 s s t3t r rst 3 t r 3 t r tt r r tt rt r rt ü t r ü t r rst s r t s t t r r str t s ö t rst t s t r t r ss s s K ts t tt t M t ABC r t s r t γ = π α β a b c < π ss s str t r r r s rt t s s 3 t s t r str t s P t r r s s t s r str t s 3 ä st 3 r ÿ r s ö s B r t tr str rt r E s t r r ÿ r s r E s r t s t s r t r r C t s r ÿ r s s r F t r ÿ r s r t r t s ÿt s ü r ÿ r s ö s s s 3 r t B C D E 3 F rü r s s s 3 r ÿ r s ö t r P t P Z Q O 3 S 3 s s är s s P t r s tt tst ÿ r ü r t r P ZC ZQD QOE OSF SP B r t s r t s P t s ts r r r s 3 tt t B C D E F P Z Q O S t r t s s s P t r r rst t s t ss t s r ts r t r s r str t s t ä t ü r t r s s ü r ÿ r s ö s 3 r t s 3 s s t3 s P t r r r s r r t r t r r s s rä t st r s t är

32 st r tät s r ärt tür t rs st r s r r t r r r s r t r r t s P t s r s ÿ r r t rs st t s r s ss ss t r P s r t s tr t r t s P t s P ZQOS s s s r r 3 r ü r ÿ r s s P t r r ä st t str t s rs r t rt r r 3 r ss s s P r 3 s rä r s s P t s st r t s P r t ür r r t s P t s r t r s P r r ü r t t s P r ü 3 s r P t P ZQOS s P t s s t P r t s s s t s s t3t ü rt s r s ü PZQOS r t s st P r t s s P t r r st s 3 s s st r s t s t ä t 3 r t r t str t s r t s s r t s r t r t rs str t A T A T C A T C S S s P D S R t Q s P E B B r str t s P t r r r r t r AST t r t A s s s t s r s π s t s r s s s 3 r ÿ r s r ä rt 3 r P r s 3 P t S s tst r tt t rst s r tt t s t r ÿ r s r A S s r s t 3 t 3 t s r tt t s t r ÿ r s r S T r C r tt s r tt t s t r ÿ r s r A T r t P 3 ss r r ür P t T r str r s P r t s r r tt t rt rst tt t D t 3 t E t t r R ür s str t s r s P t r r s s r st tt s r s t r P P t r P t r t r

33 t 3 t3t tt t Q t r P r s P t A B C D E P Q R S T s P t r r s r r t r s st r tät r r s t3t s 3 str r t s P r r ü r t s P t s P QRST st rstä s s r 3 r ÿ r s ö A B C D E r t s s tt t r P r t r 3 ör P r r ÿ r s s s s s st r tt t r r ÿ r s P P r P r 3 st r r s s s r t r är r r r t3 s tst s r t r r P t r r r st r tät s P t s s s r s ss r s r t s r r s t rs r r t ü r t r s s r t rs st t s t3 s 3 r är t r r s 3 tr s t t ss 3 r r tt str t s r t s s s tt s r ä t r r ä st r r 3 rü 3 3 r är s t tt r r r s s ü 3 r är s r t r s s rst r r r s t r s r ABC r s r τ := (a,b, π α, π c, π β). r r τ r t rt r tr P s t r r rst tr t3t P s t s r t s n(τ) := τ = (a,b,( π α),...) = (b, π α, π c, π β,a). s r τ s r t s s är s s r s s γ r t st r röÿ r r s r t rt s r t t a = n(a) = b b = n(b) = π α c = n(c) = β α = n(α) = c β = n(β) = π b s γ = γ = π 3 t t s rs t t s st s t 3 s n 5 (τ) = τ 3 n 5 = id. r t t r s r ä t g(τ) := n(τ ) = n (τ) = (a,b,( π α),...) = ( π α, π c, π β,a,b). s r s r t t r s s är s s r s γ r t s t3t rü r s s t3t s t g(a) = π α g(b) = π c g(c) =

34 π a s g(α) = β g(β) = π b s s r t s β t s s r s ÿ t r t r s r t P s t r r r t st t st s s r ss 3 s s B = A r 3 rü 3 r str t s P t r r t r 3 r är r r t s är s r P 1 P Q 4 t t P Q 4 =: π p 5 Q 4 P 1=: π p 3 P 1 P =: p4 Q 4 P 1 P =: p 1 P 1 P Q 4 =: p P Q 4 P 1 = π s r s r p i < π ür i t r r s t3 r p i := π p i ür i s 3 r P 1 P Q 4 ör 3 r är τ 1 = ( p 5, p 3, p 1, p 4, p ) = g 0 (τ 1 ). r r τ 1 s r t r r t r r t r P i P i+1 Q i+3 i =,3,4,5 r 3 s t 3 r är τ i+1 = g i (τ 1 ) = ( p i, p i+3, p i+1, p i+4, p i+ ). r P i P i+1 Q i+3 i = 1,,3,4,5 t P i+1 Q i+3 P i = π, P i P i+1 = pi+3. Q 4 P Q 5 p 4 p 5 P 1 P 3 Q 3 p 3 p 1 p P 5 Q 1 P 4 Q s r P t r r t 3 t s ÿs r ür st r t ts ss r s t β s t3 s s t t st s r r r s rr ü r r ür t g = n 4 id = n 5

35 s ü r t r r r s r ss s 3 s t rü r ss s ss tt tst t r r P t P 1 P P 3 P 4 P 5 s är s s P t s ÿ s 3 s t Q 1 Q Q 3 Q 4 Q 5 s P t r r t t s r r t s P t s r ts r t s r r ÿ r s r P i P i+1 r P r 3 P P i+3 3 r P t P i ts r t 3 r P r r P i+ P i+3 ör P r r t rs t ss s P t P 1 P P 3 P 4 P 5 s st r st rü r s s r P i P i+1 Q i+3 i = 1,,3,4,5 s t r ä t P t Q i r t s P t s s r 3 tr s t s t3t s P t r r t r r t r st s s t π P i P i+ =, i = 1,...,5. s t t ts r s s ss tt ts P i MP i+ ür i = 1,...,5 r t s s t P i P i+ P t ä π s t 3 ss s s s r st 3 s P P t s r P r r π trä t s r s r P t P 1 P P 3 P 4 P 5 s P 1 P 3 P1 P 4 P r 3 P P1 st r P 3 P 4 P3 P 4 s s P t r P r 3 P 1 r s t ss r r t P 1 s ä π ss s ÿ r s rä s t s t t t r r r t t st π t p i 3 p i r ü r P t r r üss r s π s s s r r t t Q i ü 3 r är p i r ü r r r s r 3 r är r r s t s r r t p i 3 r t s r t s P t s s s r p i r r ss t3 r s π s s s ss t s s r s r röÿ s s s p 1 =P 3 P 4 är π ür P 3 t Q 5 3 s s P 1 s P s P 3 ür r P r 3 P 4 p 5 =P P 3 ür s t s r ÿ r s p 4 ä r P r 3 P 4 s ätt 3 r ss s s P t r t s P t s ür ü P t P 1,P,P 3,P 4,P 5 ür r r s är s r s r r r r p i röÿ r s π ä 3 r r P t r s s är r t r 1 s s r r ür t r r r r t ss p i < π, i = 1,...,5 t s s s r P t P 1 P P 3 P 4 P 5 s t 1

36 r t ÿ s P t r r ÿ s rr s t r ts t t r r r r ÿ r r s r ür s s tt t r ÿ s rä t t t s r ts s t r r t s r r ür3 st r t tür s rt ÿ tt r t ss s s P r st 3 s r r r s t ÿ 3 ä st s s s ät r s r s t s s s ö rs r3 r s rs tät ött ÿ r r s ät r r ÿ ÿ ött r rt s ss r s 3 rü r rts r är ss s r3 är ÿ 3 ü r3 s t r rs tät st t 3 s ÿ s t t ÿ 3 r t r ss rt t ü r t s t3 r r s ÿ t ÿ 3 ä st s t t r r s r sst s rst s r ÿ s r s q s t s r t t s r r s s s s ä t t s röÿt t s t t r s r t st r str t s r äÿ s t r t r r rü st s r t t r ts ä r s s t s r rts t ss s s s r str t röÿt rt s r tt s t r s t r üt 3 t r r s r s r rst 3 t r s s ÿ ö rs s ät r tr t ÿ s r r t s t r s ött r s r t r s r3 r rst r s t r s r s 3 t s r s ät r r t t ÿ r t rö t t s 3 t s s 3 r r s ör r rt 3 tr rt r s r t r t s r str s r r t t s s s s s r s rs r tr s t ä r s s r t r t s t r P t t t r s rs äs s ä rt s t r ss ür t s r t tr t r t t s ss rü r ÿ r t s s st s rs s 3 rt s r sq s t s r s r s r s r 3 s r r P t t t r tt ÿ r ts r t 3 r 3 rö t t r 3 tr rt s r r

37 r t s r P s rö t t r s r t r t 3 r t s s t r s t P s Pr ss r r rs t ÿ st r ÿ r t 3 P t r r Ü r s ts trä 3 r t t P s s r ÿ r rst r s t s r t r t P t r r s r s t3t t r r ärt s P t r r 3 t ÿ r ts r s t rs ü r s r t s är s r t3t s ü s r 3 P r s t rst t r ü r t r s s s t t r t r ss t3 s t s P t r r r r ts t t r ä t rt t ss 3 P t r r ÿ s st 3 rü t ü r s t r s r t ss rü r s st rö t t s s r t s s t r rs s ss r r t t ss s s s r rs t st ts r s 3 s s s t st 3 t s P t r r s ÿ rst r t r t s P t r r ÿ r r t ÿ t P t r r 3 s r t p i s P t s P 1 P P 3 P 4 P 5 s r ÿ rt p i = P i+ P i+3, i = 1,...,5. t s t3 r P i P i+1 Q i+3 i = 1,,3,4,5 3 r är ( p i, p i+3,

38 p i+1, p i+4, p i+ ) t rs ür s s r r sin p i = tan p i+ tan p i+3. s ö r t t r tr tr s r s t3 r ÿ r s t3t ÿ sin p i =tan p i+ tan p i+3 cosp i =cotp i+ cotp i+3 1 cosp i = tanp i+ tanp i+3 sec p i =tan p i+ tan p i+3. α:= tan p 1, β := tan p, γ := tan p 3, δ:= tan p 4, ǫ := tan p 5. s st r röÿ ür s s ÿ Ü r ür t r r r r t ö r ss α,β,γ,δ,ǫ > 0 t r st t tt ss 0 < p i < π ür i t t st tan p i r röÿ r s r s t r s sec p 1 = γδ, sec p = δǫ, sec p 3 = ǫα, sec p 4 = αβ, sec p 5 = βγ. tr tr s s t3 t ür t ss 1+tan = sec ür r s r t s s t 1+α = γδ, 1+β = δǫ, 1+γ = ǫα, 1+δ= αβ, 1+ǫ = βγ. s s ä ör 3 t st s s s 3 s r tt r t s r r t r r s r r t r r s r s t st ts ss t s r r röÿ r t r rs r s 3 röÿ r st r st r sät3 s 3 ä 3 t rs r ss röÿ r 3 r r ss s s t t s s s r röÿ α β ö t γ δ ǫ r ÿ r t γ = 1+α, δ = αβ 1, ǫ = 1+β δ δ

39 = 1+α αβ 1, = 1+β αβ 1. 3 t r röÿ 3 s α γ s ö t ü r t s r β = δǫ 1, δ = 1+α, ǫ = 1+γ γ α = 1+α 1+γ γ α 1 = 1+α+γ. αγ s s t s r ü t ä p i,i = 1,...,5 s s är s P t s P 1 P P 3 P 4 P 5 s r ö s r t t ä r s s är s r s t3t s s P t r r 3 r ts r r r s s ss s t p i,i = 1,...,5 s P t s P 1 P P 3 P 4 P 5 t r t r s s s t sec p 1 = γδ cos p 1 = 1 γδ 1 p 1 = arccos( ), γδ ür ü r t p = arccos( 1 δǫ ), 1 p 4 = arccos( ), αβ 1 p 3 = arccos( ) ǫα 1 p 5 = arccos( ). βγ t r t ss s p i r arctan s s r st t r r s r tt r t s t ÿ s rs s ö r r s t ä t3 ö ür α, β, γ, δ, ǫ t 3+α+β +γ +δ +ǫ = αβγδǫ = (1+α)(1+β)(1+γ)(1+δ)(1+ǫ). s t r r s t αβγδǫ = αβγ( 1+α )( 1+γ γ α ) = β +αβ +βγ +αβγ = β +1+δ +1+ǫ+α(1+ǫ)

40 = +α+β +δ +ǫ+1+γ = 3+α+β +γ +δ +ǫ. 3 t r s ö 3 s t3 r s (1+α)(1+β)(1+γ)(1+δ)(1+ǫ) = γδ δǫ ǫα αβ βγ (1+α)(1+β)(1+γ)(1+δ)(1+ǫ) = α β γ δ ǫ (1+α)(1+β)(1+γ)(1+δ)(1+ǫ) = αβγδǫ. rü r s t t ÿ s r t t r s 3 s s röÿ t3 ür α,β,γ,δ,ǫ t ǫ((1+β)α (1+γ)δ) = ǫ((αβ δ 1) (γδ α 1)) = (1+γ)(1+β δǫ) (1+β)(1+γ ǫα). s ür s s s t3 s öt t s ä s t t ǫ((1+β)α (1+γ)δ) = ǫ(α+αβ δ γδ +1 1) = ǫ((αβ δ 1) (γδ α 1)) s ǫ((1+β)α (1+γ)δ) = ǫα(1+β) δǫ(1+γ) ǫαδǫ+δǫǫα = ǫα(1+β δǫ) δǫ(1+γ ǫα) = (1+γ)(1+β δǫ) (1+β)(1+γ δǫ). ÿ s s s r r t 3 P t r r r t ÿ r r st t s s 3 rü ä t r röÿ s är s P t r ÿ α = 9, β = 3, γ =, δ = 5, ǫ = 1 3. ür r t s s r t p i s r t s t sec p 1 = γδ = 10, sec p = δǫ = 5 3, sec p 3 = ǫα = 3, sec p 4 = αβ = 6, sec p 5 = βγ = 4 3.

41 t ä s är s P t P 1 P P 3 P 4 P 5 tr s 1 1 p 1 = arccos( ) = arccos( ) 71,6 arctan( α), γδ 10 p = arccos( 1 3 ) = arccos( δǫ 5 ) 39,3 arctan( β), 1 p 3 = arccos( ) = arccos( 1 ) 54,7 arctan( γ), ǫα 3 1 p 4 = arccos( ) = arccos( 1 ) 65,9 arctan( δ), αβ p 5 = arccos( ) = arccos( βγ ) 30,0 arctan( ǫ). rü r s r ü s röÿ tür r s s ät3 s ässt s r s t3 r t ü r rü ür αβγδǫ s r s ö r ä t αβγδǫ = 0. s r äÿ P t st s P t P 1 P P 3 P 4 P 5 r äÿ t s p 1 = p = p 3 = p 4 = p 5 s t tür α = β = γ = δ = ǫ. t r r s s t 3 s s r ässt s α t r 1+α = α. α = 1+α 1 = α α 5 4 = (α 1 ) α = 1± 5. r t α = 1 5 t s ös r r st t ss α > 0 t ss r är t s α = 1+ 5, s s tt ts r t s t r r t s ür α = β =... ss α = sec p i = 3+ 5, i = 1,...,5.

42 ÿ r t t ür t ä s r äÿ P t s p i = 1 = α 1+ 5 = = arccos( ) 51,8 arctan( α). 4 ür αβγδǫ s r s ö t3 t t r t r αβγδǫ = α 5 = α α α = (1+α) (1+α) α = α+α +(1+α) α = α++α+(1+α) = 5α+3 = r r stä t r r3 t3 3 s r r rs r äÿ r är s s P t t r s s r tr t s r t s P t s 3 tr rt t rs r s t t s r 3 t r ÿ ü t s r t s r r t r t 3 ä st r t t r ärt P i MP i+ π = P i P i+ =, i = 1,...,5. s t t s s s ssmp 1 rt 3 MP 3 t P s t s P t r r r s t s r s P t s P 1 P P 3 P 4 P 5 t st t st s t3 r P 1 = (0,1,0), P 3 = (1,0,0). P 1 P 3 s t3 s t st tt t s t r rü r s t MP 5 MP 3 s MP 4 MP 1. tr t r 3 ä st MP 5 MP 3 P 3 r t r t s t3t ss ür P 5 t r t (x 5,y 5,z 5 ) r r t t ss x 5 = 0 s t3t r r P t P 5 t s t r 3 ä r s t P 5 s s s tst t r t s r t t s MP 5 = 1

43 t t P 1 MP 5 =P 1 P 5= p3 ür r t y 5,z 5 P 5 r t s s t y 5 = cosp 3, z 5 = sinp 3. r ä rt ür r t P 4 s r t r s s t P 5 = (0,cosp 3,sinp 3 ), P 4 = (cosp 1,0,sinp 1 ). s P t s t3 s t st s t r ts s ÿ tr t r P t P t r t (x,y,z ) tr MP t rt 3 MP 4 MP 5 r s t 3 ä st ss s r r t MP MP 4 s ss 0 = x cosp 1 +z sinp 1 z = x cosp 1 sinp 1 z = x cotp 1. t ss MP rt 3 MP 5 t y cosp 3 +z sinp 3 = 0. r s t3 r ä t 0 = y cosp 3 x cotp 1 sinp 3 y = x cotp 1 sinp 3 cosp 3 y = x cotp 1 tanp 3. P s t3t s r t (x,x cotp 1 tanp 3,x cotp 1 ) x s ä t r ss ss r st P 3 st s ss s t x +x cot p 1 tan p 3 +x cot p 1 = 1. r s t t ür x x (1+cot p 1 tan p 3 +cot p 1 ) = 1 x tan p 1 +tan p 3 +1 tan = 1 p 1 tan p 1 tan p 1 +sec = x p 3 tan p 1 tan p 1 +tan p 5 tan = x p 1

44 1 1+tan p 5 = x cosp 5 = x. t3t s rt ür x s r t s y = x cotp 1 tanp 3 = cosp 5 cotp 1 tanp 3 = tanp 3 = 1 = cosp 4, tanp tanp 3 tanp 1 secp 4 z = x cotp 1 = cosp 5 tanp 5 = = sinp 5 = cosp 3 sinp 5. tanp 1 secp 5 tanp 5 tanp 1 secp 3 s r t s ür r t P s s t s P = (cosp 5,cosp 4, cosp 3 sinp 5 ). t r r t r t s P t s ä t t ä rr t s r r ss ts r ÿ ü t r t t r ö r tt t r 3 ss t t r s r r 3 t ä p i s P t s ä s s ü t s s 3 s ÿ r ür P s r ts r t r t ä t r 3 rs t ä ä p 1 p r t P ss s t r r s s s r s s ä t p 1 p 3 s t ä cosp 5 = 1 secp 5 = 1 βγ 1 = 1+α+γ γ αγ α = 1+α+γ = = tanp 1 1+tan p 1 +tan p 3 tanp 1 sec p 1 +tan p 3, cosp 4 = 1 secp 4 = 1 αβ = 1 α 1+α+γ αγ

45 = = γ 1+α+γ tanp 3 sec p 1 +tan p 3. sinp 5 = 1 cos p 5 tan p 1 = 1 sec p 1 +tan p 3 1+tan p 3 = sec p 1 +tan p 3 secp 3 =, sec p 1 +tan p 3 s r r ss ö r r t P s t3 r t s tanp 1 tanp P = ( 3 1,, ). sec p 1 +tan p 3 sec p 1 +tan p 3 sec p 1 +tan p 3 s rst r r t s t sp st 3 r 3 s 3 rt r s s 3 ÿ st t r ts r r ss s P t r r 3 r ts r s t3t s ss ö r s r s st t P 1 = (0,1,0), P 4 = (cosp 1,0,sinp 1 ) P 3 = (1,0,0), P 5 = (0,cosp 3,sinp 3 ), tanp 1 tanp P = ( 3 1,, ). sec p 1 +tan p 3 sec p 1 +tan p 3 sec p 1 +tan p 3 r Ü rs t t r st r r t r t P i t r röÿ s r s r r t r r r s 1 1 P 1 = (0,1,0), P = (,, 1 ), βγ αβ αβγ 1 P 3 = (1,0,0), P 4 = (,0, γδ α γδ ), P 5 = (0, 1 γ, αǫ αǫ ). t s r r t ö r s s r t P i s P t s r r t t3 t s P t s P 1 P P 3 P 4 P 5 t z β αγxy αxz γyz = 0 t3 P t M = (0,0,0)

46 s s st t s s r s ü P t P 1,P,P 3,P 4 P 5 r ü ür st t a,b,c,d,e,f R ax +by +cz +dxy +exz +fyz = 0. s t ä t s t r r tr s t t s ss r t 3 s r t rs r s t rs st ss s t3 rs r 3 tt t r t r s t t tsä st t3 r 3 ä st P t P 1 P 3 r s rt t ss t ss a = b = 0. s t t ss r 3 t c z t s r s s s st t r s t s s r r ür s ö r r r z + d c xy + e c xz + f c yz = 0 : z +pxy +qxz +ryz = 0. t3 r P t P,P 4 P 5 r s s r s r t r r p,q,r s ös ss 3 p = sec p 1 +tan p 3 tanp 1 tanp 3, q = tanp 1, r = tanp 3. s rt ö r s t3 s t ä t s r 3 r ts r ä p 1 p 3 s ss tür s s s P t r r r 3 r ts r s t3t r ü rs t r r s r rt ür p, q r t s r r t r p = β αγ, q = α, r = γ. t3 r p,q,r s r t r s r P i,i = 1,...,5 r ü t r z β αγxy αxz γyz = 0. t 3 t 3 ü P t ässt s r s ä 3 t ö t s s r s t rt t

47 ÿ s s ür ÿ s s α, β, γ, δ, ǫ 3 ä s r t s ts r 0 = z 3 9 xy 9xz yz = z xy 3xz yz. r t r P i ö r t ür s s s P 1 = (0,1,0), P = ( 1 P 4 = (,0, ), P 5 = (0, 3, 1 6, 1 1 3, 3 ), P 3 = (1,0,0), ). 3 z P 4 P 5 x P 3 P 1 y P t s s s är s P t s r ts z P 4 P P 1 P 5 P 3 x P 3 P P 1 y t s s s är s P t s

48 s r äÿ P t t α = β =... t α = 1+ 5 s t t 0 = z 1+ 5 = z 3+ 5 xy (1+ 5) 1+ 5 xy xz xz yz. r t r P i s s r 1+ 5 yz P 1 = (0,1,0), P = (,, ( 5 3) + 5 ), P 3 = (1,0,0) P 4 = (,0, ( 5 1) ), P 5 = (0,, ( 5 1) + 5 ). 4 4 z P 4 P 5 x P 3 P 1 y P t s s s är s P t s r ts P 4 P 5 P 3 P 4 P 5 P 3 P 1 P P 1 P t s s s är s P t s

49 t s tr s r t s s r s t3t r s rst ts r s r s r ä s s t ä r r tt s r r tt t r s 3 r t3 s s st s r tt s st t s r ts ss s t3 s t r s t3t ässt s r ss s s r s s t t s 3 r 3 r t s s s st 3 r3 r r s r t sst ts r s s t t r t s tr s r t s r t ü s s st t ÿ ä Ü r r r t s tr s r t t s r r tt 3 ür st r 3 ä st t r tr 1 r q r t s r tr 1 s rü 0 = n i,j=1, i=j a ij x i x j + n i,j=1, i<j 0 = x T A x + b T x +c, a ij x i x j + n b i x i +c x 1 x = Rn, b 1 b = Rn, A = (a ij ) R n n,a ji = a ij, c R x n b n s r st n = 3, x = (x,y,z) T, b = c = 0 A = (a ij ) R 3 3 a 1 = a 1 = p, a 13 = a 31 = q, a 3 = a 3 = r ür tr 1 A t s r s 0 p A = p q r 0 q r 1 ür s r s t = 0 β αγ β αγ 0 α i=1 γ 1 α γ, 0 = z β αγxy αxz γyz β αγ α 0 x β 0 = (x,y,z) αγ y. 0 α γ γ 1 t s tr s r t 3 3 ss tr 1 A tr 1 A r t r trä A s r rt A st s r t r st s P s A st r s 3 ä st s r t r st s P r 3 t I 3 ts tr 1 t s det(t I 3 A) = t (t 1) pqr 8 pqr 8 (p 4 q r (t 1)+ t+ 4 4 t) z

50 = t 3 t p t p q 4 4 = t 3 t p +q +r 4 r pqr t t 4 4 t+ p pqr 4 = t 3 t αβ γ +α+γ t+ αβγ +αβ γ 4 4 = t 3 t (1+δ)(1+ǫ)+α+γ (1+δ)γ +(1+δ)(1+ǫ) t+ 4 4 = t 3 t 1+δ +ǫ+1+β +α+γ γ +1+α+1+δ +ǫ+1+β t+ = t 3 t = t 3 t αβγδǫ α+β +γ +δ +ǫ t+ 4 t+ αβγδǫ α+β +γ +δ +ǫ 4 t3 r r t r ω := αβγδǫ s r t s s ür s r t r st s P p 3 A p(t) = t 3 t ω 1 t+ ω 4 4. s t t ss ω > 0 α,β,γ,δ,ǫ > 0 r ö t s ä st s st s P s p st 0 = t 3 t ω 1 t+ ω 4 4 = 4t 3 4t +(1 ω) t+ω. ös s r ts r rt A s ür s r r t 3 rt r t ö 3 t r st s ös 3 st r 3 ä st r r r st s P p ü r t s t3t r s 3 ω ä t s t rt t s r t3 ä ω 1 st r 1 r st t st s t 1 = G < 0 ür t r st t st ω röÿ r s st t r r t s r rt ω 0 s t s 3 t r r s t st t = G,t 3 = G st ω < ω 0 s t s t 1 = G t r r st st r s ω = ω 0 s t s t r t st t = t 3 = G > 0 r r t s rt ω 0 ts r t α 5 t α = 1+ 5 s s t3 3 s r r 1tr st s r t r st s P s t rs s r P s t 3 r P s t r st r 3 ö r s 3 ä st rst t s P s p (t) = 1t 8t+1 ω. ö 1tr st 3 st s t3 r rst t

51 ös t 0 = 1t 8t+1 ω t 1 = 1+3ω, 6 t = + 1+3ω. 6 s t3 r s rt ür t 3 t t st3 st s s t tsä 1tr st t ü r rü t t ss s st s s t 1 t t t t s ä st s t rs r rt r 1tr st r t 1 t p(t) s t3 s ässt s t r ss p(t 1 ) ür ω > 0 t röÿ r st ä r p(t ) ä r ω s r s s röÿ r s t p(t ) = 0 ω = = ( 1+ 5 ) 5 = α 5 t t ä 3 t rs p(t 1 ) > 0, p(t ) > 0 s t3t st ω < α 5, p(t 1 ) > 0, p(t ) = 0 s t3t 3 st ω = α 5, p(t 1 ) > 0, p(t ) < 0 s t3t r st ω > α 5. ω > 0 t ss 1 st rt r ä t st t = G 3 t st t rü r s s t s st t st s t = G = G ä r s r tt s r 3 t r st t röÿ r s s üss t = G > 0 t = G > 0 s r t r s st ω s ss röÿ ω 0 = α 5 r r t s rt ss ω α 5 st s t3t s r t r st s P A s r r st r t 3 s 3 s s st s t3t s r 3 1 st t s r st 3 r rt A r t G,G,G 3 t3 s G < 0 t ässt s tr 1 A s tr 1 rst G 0 0 A = 0 G G s tr 1 s r t tr 1 A r t rs st r ss tr 1 A r t sst s r t ss s s ts r ä r A t r r t s st t 1 3 s t r t r (1,0,0) T,(0,1,0) T 3 (0,0,1) T s r t s s r t sst s s t s r r tür t r t r t r s s t r s r t r 3 r rt s rt r rt t r 3

52 rt G,G,G ör r 3 s s t x,y z s s s r t s 3 r rs r s r t sst s t s t rä rt s r r tr 1 A 3 t s tr s r t s r s s t ür s q 3 0 = x T A x 0 = x T A x G = (x,y,z ) 0 G G 0 = G x +G y +Gz. x 0 = G ( G ) y + G ( ) G x y z z 1. s r r t s P t s t ä t t s 3 s r t r t sst s s s r t s s x y z s ä r s r s ss r ö z = 1 G ts r r rt r 3 t x y s G G G r 3 rx 3 y s z s ts r t r t s r t3 M r ä t s ss ässt s s r s st t ss s r s är s s P t P 1 P P 3 P 4 P 5 t ss r t s tr s r t r s ts r t tür t s s s r t ss r t r ss s P t s r P t s r s s är P t s P t s r r ä t r r t r ö t s r P t t 1 s s r r ss t3 p i ss s t 3 ÿ s s ä r α, β, γ, δ, ǫ r t s r ω = 0 > α 5 s ss s t3 r r st p t p(t) = t 3 t 19 4 t+5 s t r rt G,G,G A ür s ss s ä r s s rt r G,197, G 1,069, G,18.

53 s r t sst ss s s s s tr s r rt r ss s s t r t r s r rt r s s ä r s s r t r r rst r 3 G ör t r st r 3 t r 3 G r r tt r 3 G ör st 0,680 0,351 0,644 0,581, 0, 793, 0, ,447 0,498 0, 743 t s r t r ss s t r t r t P i s P t s r s 3 r t P i P 1 = ( 0,181, 0,793,0,581), P 3 = (0,644,0,351,0,680), P 5 = ( 0,711, 0,05,0,701). P = (0,699, 0,164,0,697), P 4 = ( 0,501,0,583,0,639), t r t s ür r rt ä r s s 0 x (,055) + y ( 1,03) z 1. t s s r s s ts r t s r r 3 s ä r s r s s s z = 1 t r ä t s t rs (1,0,0) T t r ä t (0,1,0) T 3 r r t r s r r t s tr s r t s r äÿ P t s r äÿ P t t α = β =... = 1+ 5 ω = α 5 rt r r 3 t ω = ω 0 s t r 3 rs

54 rt G < 0 G = G > 0 t ür t rt G G = 1+ 5 = α. t ss ässt s r 3 t t rt t r G = G = = α. r ss s r t sst s s tr s r rt r ss r t s t r rt s rt r rt t r 3 s rt r s t ä r s s 0 0,743 0,669 0,437, 0, 60, 0,669, 0, 899 0, 9 0,35 rst s r t r 3 t rt G ör r r tt t r 3 rt G r t r t P i s P t s t r t sst P 1 = (0,437, 0,60,0,669), P = (0,707,0,30,0,669), P 3 = (0,0,743,0,669), P 4 = ( 0,707,0,30,0,669), P 5 = ( 0,437, 0,60,0,669). r t s tr s r t t t r äÿ 0 = x ( 5 1) + y ( 5 1) z 1. s r r t s tr s r t

55 ür s t t s s r s z = 1 ä 5 1 r t r x y s r tt s ts r t r r 3 s ts ss s s r s t t s q 3 r ä s r s s st t r s s r s r r s s t s s r r äÿ s r s s P t s 13 tr s s rü r t r r x r s x y r s y z r s z s t s t r z s s r s r 3 r x y s P y P a b y φ b x a φ x φ 13 tr s t s t x a + y = 1, a > b, b C r r s t tt t rs r s a r r s s r st A = (a,0) r r rü r t E C 3 P t P = (x,y) r s 3 ör P t P = (x,y ) r s C st 13 tr s P rt s r φ = AOP ür r t x,y R P t x = a cosφ, y = b sinφ. s r ä 3 ä st s t P = (x,y ) r P = (x,y) 1 s tt t s t s t r 1 s 3 r tf r t r t F = (x,0) tr t r s r t r FP O t r t F FOP = AOP = φ s r t cosφ = OF OP

56 cosφ = x a x = a cosφ. t3 r s rt ür x r s ös y s r t r (a cosφ) a + y b = 1 b cos φ+y = b y = b sinφ. t r s r t r s ss ü t s s är s P t s P 1 P P 3 P 4 P 5 t r r ts s r ss s r t s tr s r t r z s s r t sst s ts r t ä st r tt r t E ts P t (0, 0, 1) rü rt s r rü r t st 3 r tt t r s t r ÿ 3 t s t r E st r E : z = 1 r 3 r r s s är s P t s E r t P i s t r r r tt t M r ä t E rs r Pr t s t s P i r E 3 r t R i ss r t 3 r t R i := (x i,y i,1) r t r s s r s är s P t P 1 P P 3 P 4 P 5 r ts s P t R 1 R R 3 R 4 R 5 r E ÿ r R i r Pr t s t r s r r ä s s ts r t r r s t r tt ä s s t r E s P t st s r s s s r t s t G G 3 s s r s ä G ö G x y r t r R i t s r s s rü t r x i = G G G cosφ i, y i = G sinφ i. r 3 t r φ i,i = 1,...,5 r rt 13 tr s r t R i,i 1,...,5 R i r t r E : z = 1 t ür r t r R i G G R i = ( G cosφ i, G sinφ i,1), i = 1,...,5. s Ü r t ÿ s t r t s r r t 3 P t r r 3 r s s s t r t st s r ür3 st

57 st t s t 3 s 13 tr s r t R i r rs s r t r 3 r r t r 3 ä st r3 s r s 3 s r ÿ s s R i s P t s t r t ür s r ä r s s t R 1 = ( 0,31, 1,364,1), R 3 = (0,948,0,516,1), R = (1,00, 0,35,1), R 4 = ( 0,784,0,91,1), R 5 = ( 1,015, 0,074,1). t R i r s r z = 1 s s r t t r r ss s r s 3 t s r s s r äÿ P t s t s s r t s r r 3 r r t R i R 1 = (0,653, 0,899,1), R = (1,057,0,343,1), R 3 = (0,1,11,1), R 4 = ( 1,057,0,344,1), R 5 = ( 0,653, 0,899,1). t R i r s r z = 1

58 r 3 rü 3 t s r t s r 3 r r t 3 r 3 s r t R i s P t s r rts t r t ri := (x i,y i,1) T,i = 1,...,5. s x i y i tür 3 ä s t p i s P t s rt s p i = P i+ P i+3 = Pi+ MP i+3, i = 1,...,5. r r t r r Pr t t R i s P t s tr r rs r M t P i s s är s P t s s s t tür 3 s t 3 s rs r s ü r i, r i+1 = R i MR i+1 = P i MP i+1 = p i+3, 3 p i = r i+, r i+3. t t ss P t r r ä π P i P i+ = Pi MP i+ = π, i = 1,...,5. tt ts r ü r ä t ü r st t r s ür s r t R i π = P imp i+ π = R imr i+ π = r i 1,r i+1 0 = x i 1 x i+1 +y i 1 y i s s t3 r rt ür x i y i s r r t 1 = 0 = x i 1 x i+1 +y i 1 y i+1 +1 G G cosφ i 1 G G cosφ i+1 + G G = cosφ i 1cosφ i+1 + G G sinφ i 1sinφ i+1. G G G sinφ i 1 G sinφ i+1 ür t ss r s rst s st r r ä st t t 3 3 rü 3 ÿ s t r t r s t 3 s 13 tr s r t s P t s st

59 s s t rt t r t3 ür 13 tr s φ i,i = 1,...,5 r t R i,i = 1,...,5 s r s s P t s t sin( φ i +φ i+ ) cos( φ i φ i+ ) sin( φ i 1 +φ i+1 ) cos( φ i 1 φ i+1 ) cos( φ i 1 +φ i+1 ) cos( φ i 1 φ i+1 ) = G G sinφ i, = G(G 1) G (G 1) sinφ i = cos( φ i +φ i+ ) cos( φ = G i φ i+ G ) cosφ i, G(G 1) G (G 1) sinφ i, = G(G 1) G (G 1) cosφ G(G 1) i = G (G 1) cosφ i. s r ts r ä t t ÿ r s r t s r ss r rt s r s r r r ö t ä st 3 r s s s s st r r st r r s r φ i r φ i 1 φ i+1 rs t3 r t r x i x i +y i y i +1 = 0, x i x i+ +y i y i+ +1 = 0. ös r rst s r x i r t r x i = y i y i +1 x i. s s t3 r 3 t r t ös y i r y i = x i x i+ x i+ y i x i y i+. s t3 s s r ss s rst r t s ÿ s ös x i r t x i = y i+ y i x i+ y i x i y i+. s s t3 r ür x i s r r s s r x i = y i+ y i x i+ y i x i y i+

60 G G cosφ i = G G cosφ i = G G cosφ i+ G G G sinφ i+ G sinφ i G G G G sinφ i G cosφ i G sinφ i+ sinφ i+ sinφ i cosφ i+ sinφ i cosφ i sinφ i+. tr tr s s t3 t ür 3 x y s t3 sinx siny = cos( x+y cosx cosy = sin( x+y )sin( x y ), )sin( x y ). rst s r t3 r ä r 3 r r r t st r r ür s s r G G cosφ i = G G cosφ i = G G cosφ i = G G cosφ i = cos( φ i+ +φ i )sin( φ i+ φ i ) sin(φ i+ φ i ) cos( φ i+ +φ i cos( φ i+ +φ i sin( φ i+ φ i cos( φ i+ +φ i )sin( φ i+ φ i ) sin( φ i+ φ i ) ). cos( φ i+ φ i ) )sin( φ i+ φ i ) )cos( φ i+ φ i ) s ts r t r 3 t s rst s r t r r s t3 r r t3 r 3 s ä st t x i x i+1 G 1 + y iy i+1 G G 1 = 0. r t s r st ür s r 3 r r s s ss r s s st r ü rs t s s rt s r s ür x i 3 y i r t ö r s t x i = G 1 G 1 y i = G 1 G 1 ür s s y i+1 y i 1 x i 1 y i+1 x i+1 y i 1 x i+1 x i 1 x i 1 y i+1 x i+1 y i 1

61 r s r t r r tr tr s s 3 s rst t s t 3 3 ss t G(G 1) G (G 1) = G(G 1) G (G 1), G(G 1) G (G 1) = G(G 1) G (G 1). r r rst s r 3 3 t s ö s ässt st s r rst ö r q r rt s G (G 1) G (G 1) = G(G 1) G (G 1). s st s t äq t 3 G (G 1) G(G 1) = G (G 1) G (G 1) s 3 G(G 1) G 1 = G (G 1) G. 1 s ss t s G s G r t ös r 0 = 4t 3 4t +(1 ω) t+ω s s ö r r ÿ r 0 = 4t 3 4t +(1 ω) t+ω ω = t(t 1). t 1 G G s s ös s r r s t ss r ü t s ss t rst s G ös st ässt s 3 t s 3 s ÿ r r s s t t r r r s r rö ÿ α,β,γ,δ ǫ s r s t3 α i := tan p i+3, i = 1,...5. t t s α 1 = δ, α = ǫ, α 3 = α, α 4 = β, α 5 = γ, α i = α i+5.

62 ÿ r ö r t r t r α i 1+α = γδ, 1+β = δǫ, 1+γ = ǫα, 1+δ= αβ, 1+ǫ = βγ s r ÿ r r 1+α i = α i α i+. s ss ässt s t ss r r s t3 r 3 r s r t r r r r t 3 tr s t t r s t3 r r tr r t rr ö r s r α i t r rts t r r i r t s s P t s s rü α i = tan p i+3 = r i r i+1 < r i r i+1 >. t r t3t s t3 r ür r ä st t β i := sin p i+3. st rstä ö r β i ä t r t r r i β i = sin p i+3 = r i r i+1 r i r i+1. r P r 3 tät r p i s s t3 β i tür s s s t β i = β i+5. tr tr s s t3 t ür x r s β i s t sin x = tan x cos x = tan x sec x = tan x tan x+1. β i = α i α i +1. r st α i ö r t r β i sα i r r t s r rt β i = α i α i +1 α i = β i 1 β i.

63 s röÿ α i β i s s t s r t 3 tr s

64 t s t s P t r r r ÿ ss t3 P t P t s ÿ ss t3 t r P t r t3 r r tr t r s t t tt 3 ä st r r 3ös s r r s r s t r r3 r s s t r ü r t t tt r r r s ss 3 t 3 rst ü r r r s s t t r s ÿ tt s r s 3 rü r r ä r s r t s r s r s r r r ÿ s t st r s r té s r r étés r t s s r s s r r r t tr t ä t rö t t r t tr rs r rts r är ss s s r s P t s r rü r s r r r r t tr 3 ö s r t t P t s Pr ss r ür r rs tät t3 t r s t r r 3 3 r ss rrä r s t 3 r ss r r t ss 3ü s s s Pr 3 s r 3 r t r ät r r r t r r tärs r r s st rs s st rt tt P t st r P r s r s r r3 s t t 3 P ts ÿ ss t3 r s r ä t r r té s r r étés r t s s r s t t st s t3 t str t 3 r K 1 K 3 r s t tt t M 1 3 M R 1 3 R s R 1 röÿ r s s R K r K 1 rü r s 3 r st M 1 M r tt t t a s t s t r r M 1 M röÿ r r r s K 1 3 P t P P s ss s t M 1 P = M 1 P = R 1, M P = R 1 +a, M P = R 1 a. s ä st s ä r P t A 0 röÿ r r s K 1 t t 0 s r r s s K r s P t A 0 s t K 1 t r P t A 1 3 r 3 t t t 1 K r A 1 t r P t A A 0 K 1 r ä t s str t r r r r s ss r tt t A n r t t K 1 3 t t n 1 t n K r s r Pr 3 ss s s ÿ s ss ür s r A n t A n = A 0, n N, n >, r ss s t ÿt s str t n s ü r r s K 1 s r s K s r s s t3t P t

65 A 1 t 0 t1 M A 0 A 3 t M 1 K A K 1 s r str t s t rs s r str t 3 t r ss t s t t t3 P ts ÿ ss t3 r 3 3 rs t t s s r s s r r r s r st s t s t s r s t ÿ ss t3 P t t r rt ÿt s s r t str t n r tt s r s s r t P t A 0 r s s s t r str t ä s t t3 3 rt P t s t r r té s r r étés r t s s r s r r ts r t tür t s r r r t s tt t s t3 3 r3 t r s r t à s t q q q t s ér ts s ts à 1 t s rt t à tr s s t s q s t ê s t t s s r èr s t q rs ê s t q r q s s t s t q r à s s rs s t q t t s s s t s q s s t s q rs t êtr q q r èr rr q s t r r rès s t s s t r t t s s ts rs s ôtés é r r t é t tr s s t s q s t ê s t t s s s r sé s

r r t r r t t r t P s r t r P s r s r r rs tr t r r t s ss r P s s t r t t tr r r t t r t r r t t s r t rr t Ü rs t 3 r r r 3 rträ 3 röÿ r t

r r t r r t t r t P s r t r P s r s r r rs tr t r r t s ss r P s s t r t t tr r r t t r t r r t t s r t rr t Ü rs t 3 r r r 3 rträ 3 röÿ r t r t t r t ts r3 s r r t r r t t r t P s r t r P s r s r P s r 1 s r rs tr t r r t s ss r P s s t r t t tr r 2s s r t t r t r r t t s r t rr t Ü rs t 3 r t r 3 s3 Ü rs t 3 r r r 3 rträ 3 röÿ r t r r r rs

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