Trigonométrie sphérique et calcul vectoriel

Trigonométrie sphérique

µ

h`B;QMQKûi`B2 bT?û`B[m2

2i +H+mH p2+iQ`B2H

Sb+H .mTQMi

JQib +Hûb , h`B;QMQKûi`B2 bT?û`B[m2- p2+i2m`X

_ûbmKûX .Mb +2 MmKû`Q /2 GQbM;2b- CX JBörBb  ûi#HB H2b T`BM+BTH2b `2HiBQMb /2 H i`B;QMQKûi`B2 bT?û`B[m2 2M /QTiMi mM TQBMi /2 pm2 [mB 2ȿi Tm āi`2 +2HmB /2 JöMöHȽb Qm /2b Ki?ûKiB+B2Mb `#2b [mB HmB QMi 2K#QBiû H2 Tb (j)X

SQm` BMiû`2bbMi2b [mǶ2HH2b bQB2Mi- +2b i2+?MB[m2b bQMi iQmi /2 KāK2 7Q`i HQm`/2b T` `TTQ`i ¨ mM2 TT`Q+?2 Ŀ KQ/2`M2 ŀ /2 H [m2biBQM- `2TQbMi MQiKK2Mi bm` H2 +H+mH p2+iQ`B2HX LQmb pQMb /QM+ T2Mbû [m2 H2b H2+i2m`b /2 +2ii2 `2pm2 b2`B2Mi ?2m`2mt /2 +QKTHûi2` T` +2 b2+QM/ TQBMi /2 pm2 H2m` ûim/2 /2 +2 #2m UKBb bm`MMûV bmD2iX

LQmb MǶpQMb- 2MbmBi2- Tb Tm `ûbBbi2` ¨ HǶ2MpB2 /2 KQMi`2` +QKK2Mi H2b 7Q`KmH2b /2 H i`B;QMQKûi`B2 `2+iBHB;M2 b2 ++?2Mi /Mb +2HH2b /2 b ;`M/2 UR<sub>V bƾm` bT?û`B[m2X</sub>

LQmb i`pBHH2`QMb bm` mM2 bT?ĕ`2 Σ /2 `vQM 1-/QMi H2 +2Mi`2 b2` MQiû OX GQ`b[m2 ABC 2bi mM i`BM;H2 bm` Σ- MQmb MQi2`QMb A- B  C H2b M;H2b-2i a- b  c H2b `+bBC-V CAV ABV `2bT2+iBp2K2MiX

S`QTQbBiBQM R U_2HiBQMb /2b +QbBMmbV .Mb H2 i`BM;H2

ABC-cos a= ABC-cos b ABC-cos c + sin b sin c ABC-cos A, cos b= cos c cos a + sin c sin a cos B, cos c= cos a cos b + sin a sin b cos C.

.öKQMbi`iBQMX S` bvKûi`B2- BH bm{i /2 T`Qm@ p2` H T`2KBĕ`2 û;HBiûX *QMbB/û`QMb H2 ;`M/ +2`+H2 A - BMi2`b2+iBQM /2 H bT?ĕ`2 2i /m THM T2`T2M@ /B+mHB`2 ¨ OA TbbMi T` OX S`QHQM;2QMb HǶ`+

V

AB U`2bTX ACV Dmb[mǶ¨ bQM BMi2`b2+iBQM BV ′ U`2bTX C′V p2+ A X G K2bm`2 /2 HǶ`+ B′CV ′ 2bi AX LQmb pQMb −−→OB = cos c ·−→OA + sin c ·−−→OB′ 2i −−→OC = 4 cos b·−→OA + sin b·−−→OC′X SmBb[m2 +2b /2mt p2+i2m`b bQMi MQ`Kûb- BMbB [m2 −−→OB′ 2i −−→OC′

-cos a = =!−−→OB""_"−−→OC#

=!cos c·−→OA+ sin c·−−→OB′_{"cos b ·}"" −→OA+ sin b·−−→OC′# = cos b cos c!−→OA""_"−→OA#+ sin b cos c!−→OA""_"−−→OC′#+

+ cos b sin c!−−→OB′""_"−→OA#+ sin b sin c!−−→OB′""_"−−→OC′# = cos b cos c + sin b sin c sin A.

c a b A A B B′ C C′ 6B;X R

RX LǶQm#HBQMb Tb [m2 H i`B;QMQKûi`B2 bT?û`B[m2- Mû2 /2b Mû+2bbBiûb /2 HǶbi`QMQKB2- 2bi Miû`B2m`2 ¨ H i`B;QMQKûi`B2 THM2X

Trigonométrie sphérique

S`QTQbBiBQM k Uh`BM;H2 `2+iM;H2V aB H2 i`B@ M;H2 ABC 2bi `2+iM;H2 2M A- HQ`b cos a = 4 cos b cos cX c a b A B B′ C C′ 6B;X k .öKQMbi`iBQMX AH bm{i /2 7B`2 A = π/2 /Mb H T`2KBĕ`2 /2b `2HiBQMb /2b +QbBMmbX S`QTQbBiBQM j U_2HiBQMb /2b bBMmbV .Mb H2 i`BM;H2 ABC-sin a sin A = sin b sin B = sin c sin C. S`2KBĐ`2 /öKQMbi`iBQMX S` bvKûi`B2- BH bm7@ }i /2 T`Qmp2` H T`2KBĕ`2 û;HBiûX aQBi D H2 TQBMi bBimû /m KāK2 +Ƭiû /m THM OAB [m2 C 2i i2H [m2 $−→_OA,−−→

OB′,−−→OD% bQBi mM2 #b2 Q`i?QMQ`Kû2X aB H2 ;`M/ +2`+H2 B 2bi H i`+2 bm` H bT?ĕ`2 /m THM Q`i?Q;QMH ¨ OB TbbMi T` O- D T2mi 2M+Q`2 āi`2 /û}MB +QKK2 +2HmB /2b /2mt TQBMib /ǶBMi2`b2+iBQM /2 A 2i B [mB 2bi /m KāK2 +Ƭiû [m2 C /m THM OABX _2K`[mQMb [m2 A 2i B bQMi BMi2`+?M;2#H2b /Mb H /û}MBiBQM /2 +2 TQBMiX

*QMbB/û`QMb H2 i`BM;H2 CC′_{D- `2+iM;H2 2M C}′_X

.ǶT`ĕb H T`QTQbBiBQM T`û+û/2Mi2-cosCD = cosV CCV′cosC′D =V

= cos$π₂ − b%cos$π₂ − A%= sin b sin A. JBb MQmb pQMb Q#b2`pû [m2 H /û}MBiBQM /2 D 2bi bvKûi`B[m2 T` `TTQ`i mt /2mt bQKK2ib A 2i B /m i`BM;H2X LQmb pQMb /QM+ û;H2K2Mi

cosCD = sin a sin B.V

c a b A B A B B′ C C′ D 6B;X j G +QKT`BbQM /2b /2mt /2`MBĕ`2b û;HBiûb MQmb T2`K2i /2 +QM+Hm`2 [m2 sin a sin A = sin b sin B. pMi /2 /QMM2` mM2 b2+QM/2 /ûKQMbi`iBQM /2 +2ii2 T`QTQbBiBQM- MQmb pQMb ¨ ûi#HB` mM H2KK2X

G2KK2 R aB ⃗u- ⃗v 2i ⃗w bQMi i`QBb p2+i2m`b MQ`@ Kûb- HQ`b

∥(⃗u ∧ ⃗v) ∧ (⃗u ∧ ⃗w)_{∥ = ∥(⃗v ∧ ⃗}w)_{∧ (⃗v ∧ ⃗u)∥ =} =∥( ⃗w∧ ⃗u) ∧ ( ⃗w∧ ⃗v)∥ . .öKQMbi`iBQMX 1M miBHBbMi H 7Q`KmH2 /m /Qm#H2 T`Q/mBi

p2+iQ`B2H-(⃗u_{∧ ⃗v) ∧ (⃗u ∧ ⃗}w) =_{⟨⃗u|⃗u ∧ ⃗}w_{⟩ ⃗v − ⟨⃗v|⃗u ∧ ⃗}w_{⟩ ⃗u =} =_{⟨⃗v| ⃗}w_{∧ ⃗u⟩ ⃗u,} TmBb[m2 ⃗u ⊥ ⃗u ∧ ⃗w c 2i- +QKK2 ∥⃗u∥ =

1-∥(⃗u ∧ ⃗v) ∧ (⃗u ∧ ⃗w)∥ = | ⟨⃗v| ⃗w∧ ⃗u⟩ |. .2

KāK2-∥(⃗v ∧ ⃗w)_{∧ (⃗v ∧ ⃗u)∥ = | ⟨ ⃗}w_{|⃗u ∧ ⃗v⟩ |} 2i

∥( ⃗w∧ ⃗u) ∧ ( ⃗w∧ ⃗v)∥ = | ⟨⃗u|⃗v ∧ ⃗w⟩ |.

P`- H2 T`Q/mBi KBti2 DQmBi /2 H T`QT`Bûiû /ǶBMp@ `BM+2 T` T2`KmiiBQM +B`+mHB`2 ,

⟨⃗v| ⃗w_{∧ ⃗u⟩ = ⟨ ⃗}w_{|⃗u ∧ ⃗v⟩ = ⟨⃗u|⃗v ∧ ⃗}w_{⟩ .} G T`2mp2 2bi /QM+ i2`KBMû2X

Trigonométrie sphérique

a2+QM/2 /öKQMbi`iBQM /2 H T`QTQbBiBQM jX *H+mHQMb H MQ`K2 /2

&−→

OA∧−−→OB'∧&−→OA∧−−→OC',

2M MQmb bQmp2MMi [m2- T` /û}MBiBQM- H MQ`K2 /m T`Q/mBi p2+iQ`B2H 2bi H2 T`Q/mBi /2b MQ`K2b /2b /2mt 7+i2m`b 2i /m bBMmb /2 H2m` M;H2X A+B-

/Ƕ#Q`/-( (

(−→OA∧−−→OB((_{( =}((₍OA−→((₍((₍−−→OB((_{( sin !}AOB = sin c 2i /2 KāK2

( (

(−→OA_∧−−→OC((_{( = sin b.}

1MbmBi2 U`2pQB` H };X jV- H2 p2+i2m` −→OA ∧ −−→OB 2bi T`HHĕH2 ¨ −−→OD 2i- /2 KāK2- −→OA _∧ −−→OB 2bi T`HHĕH2 m p2+i2m` −−→OE UMQM `2T`ûb2MiûV i2H [m2 $−→_OA,−−→

OC′_,−−→_OE% bQBi mM2 #b2 Q`i?QMQ`Kû2 /B`2+i2X

BMbB-!&−→OA∧−−→OB,−→OA∧OC−−→'=!&−−→OD,−−→OE'= =!&−−→OB′,−−→OC′'= A.

6BMH2K2Mi-( (

(&−→OA∧−−→OB'∧&−→OA∧−−→OC'((_{( = sin c sin b sin A c} /2

KāK2-( (

(&−−→OB_∧−−→OC'_∧&−−→OB_∧−→OA'((_{( = sin a sin c sin B} 2i

( (

(&−−→OC∧−→OA'∧&−−→OC∧−−→OB'((_{( = sin b sin a sin C.} *QKK2- b2HQM H2 H2KK2- +2b i`QBb MQ`K2b bQMi

û;H2b-sin c û;H2b-sin b û;H2b-sin A = û;H2b-sin a û;H2b-sin c û;H2b-sin B = û;H2b-sin b û;H2b-sin a û;H2b-sin C, 2i MQmb Q#i2MQMb H i?ĕb2 2M /BpBbMi H2b i`QBb K2K#`2b T` sin a sin b sin cX

G i`B;QMQKûi`B2 `2+iBHB;M2

+QKK2 HBKBi2 /2 H i`B;QMQKûi`B2 bT?û`B[m2

G2 THM- +2 MǶ2bi `B2M /Ƕmi`2 [mǶmM2 bT?ĕ`2 /2 i`ĕb ;`M/ `vQM- TQm` BMbB /B`2X LQmb H2 bpQMb ûpB@ /2KK2Mi #B2M- MQmb H2b ?mKBMb- [mB pBpQMb bm` mM2 bT?ĕ`2 2M vMi HǶBKT`2bbBQM /2 pBp`2 bm` mM THMX _TT2HQMb [mǶBH  7HHm ii2M/`2 H2b i`pmt ;ûQ/ûbB[m2b /2 :mbb TQm` [mǶBH bQBi KBb 2M ûpB@ /2M+2 [m2 H2b U;`M/bV i`BM;H2b /û}MBb ¨ H bm`7+2 /2 MQi`2 ;HQ#2 b2 +QKTQ`iB2Mi +QKK2 /2b i`BM;H2b bT?û`B[m2b 2i MQM +QKK2 /2b i`BM;H2b `2+iBHB;M2bX LQmb /2pQMb /QM+ MQmb ii2M/`2 ¨ +2 [m2 H2b 7Q`@ KmH2b /2 H i`B;QMQKûi`B2 bT?û`B[m2 b2 `û/mBb2Mi ¨ +2HH2b /2 H i`B;QMQKûi`B2 `2+iBHB;M2 HQ`b[m2 H2 `vQM /2 H bT?ĕ`2 i2M/ p2`b HǶBM}MBX PmB- KBbĘ BH v  mM T2iBi T`Q#HĕK2 , MQmb pQMb bmTTQbû- iQmi H2 HQM; /2 +2 [mB T`û+ĕ/2- [m2 +2 `vQM ûiBi }tû ¨ 1X 1M 7Bi- bB MQmb pQMb Tm MQmb H2 T2`K2ii`2- +Ƕ2bi iQmi bBKTH2K2Mi [m2 +2 `vQM MǶBMi2`pB2Mi MmHH2 T`i c 2M 2z2i- H2b +Ƭiûb /ǶmM i`BM;H2 bT?û`B[m2 bQMi /2b `+b 2i MQM /2b HQM;m2m`b c 2i- /ǶBHH2m`b- BHb MǶBM@ i2`pB2MM2Mi /Mb H2b 7Q`KmH2b [m2 pB H2b 7QM+iBQMb i`B;QMQKûi`B[m2bX HQ`b \

.B`2 [m2 H bT?ĕ`2 2bi i`ĕb ;`M/2- +Ƕ2bi /B`2 [mǶ2HH2 2bi i`ĕb ;`M/2 T` `TTQ`i m i`BM;H2 ûim/Bû c Qm 2M+Q`2- [m2 +2HmB@+B 2bi i`ĕb T2iBi T` `TTQ`i ¨ H bT?ĕ`2X SHmb T`û+BbûK2Mi- [m2 H2m`b +Ƭiûb a- b  c bQMi i`ĕb T2iBib ě KBb H2m`b M;H2b A- B  C MǶQMi m+mM2 `BbQM /2 HǶāi`2X .Mb H2b 7Q`KmH2b +B@/2bbmb- MQmb HHQMb `2KTH@ +2` H2b 7QM+iBQMb /2b i`QBb p`B#H2b a- b  c T` H2m`b TQHvMQK2b /2 J+Hm`BM /ǶQ`/`2 bm{bMi Um KQBMb 1VX *QKK2MÏQMb T` H2b `2HiBQMb /2b bBMmbX *QKK2 H2 TQHvMQK2 /2 J+Hm`BM /m bBMmb 2bi T1sin = X-H2b i`QBb K2K#`2b /2 sin a sin A = sin b sin B = sin c sin C /2pB2MM2Mi a sin A = b sin B = c sin C. Zm2HH2 bm`T`Bb2 5 *2+B  mM T2iBi B` /2 /ûD¨ pm 5 SQm`bmBpQMb p2+ H2b `2HiBQMb /2b +QbBMmbX LQmb b@ pQMb [m2 T1cos = 1 2i T2cos = 1−1<sub>2</sub>X2X .Mb H T`2KBĕ`2 /2 +2b

`2HiBQMb-cos a = `2HiBQMb-cos b `2HiBQMb-cos c + sin b sin c `2HiBQMb-cos A,

TQm` Q#i2MB` H2 TQHvMQK2 /2 J+Hm`BM /ǶQ`/`2 n /m K2K#`2 /2 /`QBi2- BH 7mi MQM b2mH2K2Mi miBHBb2` H2b TQHvMQK2b /2 J+Hm`BM /ǶQ`/`2 n /m bBMmb 2i /m +QbBMmb- KBb 2MbmBi2 Ŀ `#Qi2` ŀ- /Mb HǶ2tT`2b@ bBQM Q#i2Mm2- iQmb H2b i2`K2b /QMi H2 /2;`û UiQiH T` `TTQ`i ¨ a- b  cV 2bi bi`B+i2K2Mi bmTû`B2m`

Trigonométrie sphérique

¨ n U+7X (k)VX § HǶQ`/`2 1- MQmb pQMb /QM+ Up2+ /2b #mb /Ƕû+`Bim`2V ,

T1(cos a) = T1(cos b cos c + sin b sin c cos A)⇔

⇔ 1 = T1(1× 1 + bc cos A)

⇔ 1 = 1.

oQB+B [mB M2 MQmb TT`2M/ `B2M 5 _û2bbvQMb 2M Tb@ bMi ¨ HǶQ`/`2 2X

T2(cos a) = T2(cos b cos c + sin b sin c cos A)⇔

⇔ 1 − 12a2= T2 $$ 1₋1₂b2% $1₋1₂c2%+ bc cos A% ⇔ 1 − 12a2= T2 $ 1₋1₂b2₋1₂c2+1₄b2c2+ bc cos A% ⇔ 1 − 12a2= 1−21b2−12c2+ bc cos A ⇔ a2= b2+ c2_{− 2bc cos A,} 2i MQmb `2i`QmpQMb HǶmM2 /2b 7Q`KmH2b /ǶH@Eb?BX 1M}M- bB MQmb T`Q+û/QMb /2 H KāK2 KMBĕ`2 p2+ H 7Q`KmH2 /2 H T`QTQbBiBQM k- [mB +QM+2`M2 mM i`B@ M;H2 `2+iM;H2 2M A ,

cos a = cos b cos c,

BH pB2Mi

T2(cos a) = T2(cos b cos c)⇔

⇔ 1 −1 2a 2 _{= T} 2 $$ 1−1 2b 2% $₁₋1 2c 2%% ⇔ 1 −12a2 = T2 $ 1₋1₂b2₋1₂c2+1₄b2c2% ⇔ 1 −12a 2 _{= 1} −12b 2 −12c 2 ⇔ a2 = b2+ c2,

[mB MǶ2bi `B2M /Ƕmi`2 [m2 H +ûHĕ#`2 û;HBiû /2 Sv@ i?;Q`2X *2HH2@+B 2bi mM +b T`iB+mHB2` /2 +2HH2 /ǶH@Eb?B- iQmi +QKK2 H2 `ûbmHii /2 H T`QTQbB@ iBQM k 2bi mM +b T`iB+mHB2` /2 +2mt /2 H T`QTQbB@ iBQM RX SQm` 2M bpQB` THmb (R) ?iiTb,ff7`XrBFBT2/BXQ`;frBFBf h`B;QMQK&ö'i`B2nbT?&ö'`B[m2X *QMbmHiû H2 y8fRkfRNX (k) .mTQMi SX 2i obi LX- G2b TQHvMƬK2b /2 hvHQ` , /û+Qmp`QMb +QKK2Mi H2b +QMbi`mB`2 2i H2b miBHBb2` 2{++2K2MiX _2Tĕ`2b- jj- TTX RR8ĜRke- RNN3X (j) JBörBb CX- Ji?ûKiB[m2b /2 H bT?ĕ`2X GQ@ bM;2b- 9d- TTX k3Ĝ99- kyRNX G2 iQK#2m /2 :HBHû2X

48