Eléments de géodésie mathématiques Marianne Greff 2007-2008
Transcription
Eléments de géodésie mathématiques Marianne Greff 2007-2008
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y = r sin θ sin λ; z = r cos θ !"$#"$%&' ()+*, ° wq{|nUbºm} \?½ OaQO¿¹xh/nmPlph`pqlC nUe`hPh/\ r, θ, λ ³ ½PO ¿ ½ Oa Æ ¿ avec 0 ≤ r ≤ ∞; 0 ≤ θ ≤ π; 0 ≤ λ ≤ 2π °° b`h/lCmp`}kf6mp}ka\¤½PO Æ ¿¹b lplpnM`hPjw nM\M}f6}mpq.eh>jb7f6a``h/lpx \UeMbd\f6h]h/\|mp`h (x, y, z) him (r, θ, λ) ° xlpxMahyMl}k\ qihim`\U}M{ lpnUhi,h/xl/JMhi njhm¹l6eWmp`wShq/lp{|xnUqbºf6m}}U q\Ul7lC ½P}¶O mD'xObd¿` (x,n ½Py,O z)¿]eMlChq6lDU\Mf6}l lC`h/e\ m\MnM\M\¤q/h/fclByUfibdba\U`C^amhiq/7lp}h/h/\M\ \MmhelBh7lC f6}¶mDaxU`cbae` a\U(r,\Mqiθ,hliλ) lpbh/lDxfi||lC }mp`}keMaa\?\MeW\Mwq/nMh/\ l @BA)AC@ ¾¬¬DJUO]¬©N]N]¡TF ¦¡TJ£C®¯F§¬DJ¥M¬©ªD¬©NLBVCF ° hl)lCnU`Cuvb f6h/l r = cste ½vfi{ nU}jkjh¿Á θ!=+- cste h/n 2½À 7 ° qi`h/}l¹ef6}h/ \nM¿` Jlphh/l©fieahPnMxfihi|\| m`eMfcayU\Mb \Mf6nMq/h/\Ml h>r,eθ, r.lpae\|h/m¹nbds\lCbdnMjka}kº^abdnU\|h/m¹l©ebahnl¹sfibºanMh/!` l¹Jeh/lDhPbafixM|x `h/eMjq/ah/\Ml \Mq/h/l 21½À+DxUeWba3`w !nMba\»j*jkzi`0jkhihx36¿©zih6`m h] λ`h/=fÁ3Nmcbdcste ° hl©lpnM`Cuvb f6hl©ehf6a`cea\U\MqihlDlCλh M \ ^ M n j d b k } p ` h x, y, z f6 nMxhi\|mhi\uÀa`sbd\|meh/l)ba\M^ajkh/l)e`a}ml³jkh.lplCmpz/7heMh.f6a`cea\U\Mqihl[lpxMyMq/`p}{|nMh/l[hlXm `CmyMa^ a\Ubaj @BA)A ¾FQ¥F¯JU§¯]N]£X¥¡ OLBNP§VXF a¥ ¢FOF QS¬¬DJUO]¬©N]N]¡TF§|¦¡TJU£p®¯F  xU·bdhi m`pmp}¶m)}k`)j,beh/mJlhibajk jlph.hh/f6{|f/mpnMbdhih7`pnUmp`q³ l[lC}k\Mhi \M`p\Ushbd nM`CtmyMbdan\Uta`lpnM`pq/uvhPbaf6ehh>l eh/h>f6mpf6hinUa``cleanM\U\U\M}¶mcqibdh}kli`phla\e x h/enm]h6em qiUe\M}k `)nM\Mh JUbalph 3* $+-%*%Q E p r= p x2 + y 2 θ = arctan ; z x2 + y 2 + z 2 ; λ = arctan y x x y z r Eθ Eλ Er = Eθ = Eλ = −−→ ∂ OM ∂r − −→ ∂ OM ∂θ −−→ ∂ OM ∂λ ¹h6mpmph UJ b lChhlXm)a`pmpyUa^a \Ubdjkh7ba}kl\Ua\\Ma`7qih7³P\\Mdmh ; hr = p E r .E r ; hθ = p E θ .E θ ; hλ = p ° hl){|nUbd\|mp}mpql h h h6m h liw bdxUxh/jjkhi\|m)jhl[uvbaf6mphinU`l)eh>xU`p x `Cm} \M\Ubdjk}mpq³ r θ E λ .E λ λ hr = 1; hθ = r; hλ = r sin θ P\xh/nm]eq6\M}`]nM\Mh UJ balphxMylC}{|nMha`pmpyM \Ma`7qih³ P\gbbaj `l³ sin θ cos λ er = sin θ sin λ cos θ e −−→ er = ∂ OM ∂r −−→ eθ = r1 ∂ OM ∂θ eλ = −−→ ∂ OM 1 r sin θ ∂λ x ,ey ,ez cos θ cos λ eθ = cos θ sin λ − sin θ e ½PO ¿ ½PO ¿ x ,ey ,ez − sin λ eλ = cos λ 0 e x ,ey ,ez @BA)A _VC¡ ¢FN¹¥«O ÀL<JUQ'FW¥ OF ©¬©Vp¯] ¢F · }¶m.nM\¤xa}k\|m M xM`fcyUh7eMn«x }\|m&M eh»f6 `e \M\Mq/h/l>fiba`Cmq/lp}h/\M\Mh−l −−[x→+ dx, y + dy, z + dz] h6mGlpxMyMq6Y `}k{|nMhl [r + dr, θ + dθ, λ + dλ] MÂ]j `l[{|nMhjhl©q/jq/h/\|mleh]jka\U^anMh/nM`l[eh M M xM`d±Xhimpql¹lpnM` e , e , e lphi`a\|m [dx, dy, dz] UfihinxM`pa±Xh6mq/l)eMba\Uljhl)e}k`phfÁmp}ka\l e , e , e lphi`a\|m dr, rdθ, r sin θdλ 0 0 r θ λ x y z 8##"s#"%)+* ez .M . M , λ λ+d λ θ O ey λ λ ex P\xh/nmPfibajkfinMjkhi`jw bJSf6}llCh>f6nU`p}kj}k^a\Mh −−−→ −−−→ ds2 = M M 0 · M M 0 ds2 = h2r dr2 + h2θ dθ2 + h2λ dλ2 ³ ds2 = dr2 + r2 dθ2 + r2 sin2 θdλ2 ⇒ ° w qijkqi7h/\ m]eMh ajknM7hx nM`)jh.lplXmzi7h>eh.f6 `e \M\Mq/h/llCxUyMqi`}k{|nMhl[h/lCm]ea\M\Uqxbd`>³ dV = hr hθ hλ drdθdλ dV = r 2 sin θdrdθdλ ⇒ ½PO ¿ ½ Oa ¿ [bdjf6nUjh/`)xUbd`}k\ mqi^ `bdmp}ka\Tjb7lCnM`puvbafihh6m)jh> ajknM7h>eWw nU\Mh.lCxUyMzi`h>eh`b a\ R @BA)A ¤LBNªDF IFN©¥KOF QS¬¬DJUO]¬©N]N]¡TF e \MP\Mqi\thl)xhilCnMxUmyMqihi`}k{|xMnM`p}kh7l hi[A`jkh/,lAf6,aA`ce] \Mhim\Mqi|h}l©f6hif/Ybd `phimp`cq/lplpbt}khi³\M\Mhl [A , A , A ] eWw nM\ahfÁmph/nM` A hi\uÀa\fÁmp}ka\eh>lChlf6a`pY ; x r θ A = A x ex + A y ey + A z ez ⇒ Pnhi\Ufia`hPh/\}k\|mp`enM}lpba\ m)nM\Uh>7bdmp`}kfiheMh>xUballpba^ah³ sin θ cos λ Ax Ay = sin θ sin λ cos θ Az Pn y z λ A = A r er + A θ eθ + A λ eλ Ar cos θ cos λ − sin λ cos θ sin λ cos λ Aθ Aλ − sin θ 0 Ar sin θ cos λ sin θ sin λ Aθ = cos θ cos λ cos θ sin λ Aλ − sin λ cos λ cos θ Ax − sin θ Ay 0 Az ½PO |¿ ½PO |¿ @BA)A.7 !"$#"$%&' ()+*, )FTJUQW£pQSF hi\_f/·bd `p}¶mpmq/lpnM}k\ghi\Mh/\Mjjkh}klilpxxUaba}e`)h.jw eq/h>{|nU`bdqimpa}k aj\?nMmp³ }ka\TMeh.eMhi7}¶Y¼^a`cbd\ebdh a himPeh.eh/7}Txhimp}mCY bºh c ·MbslCnM`puvbafihh/lCm]eq/fi`p}mphs x2 y2 z2 + + =1 a2 a2 c2 ¸©eh fi`p}k`p h)fih6PmCm\h\MlCnUa`Cmpuvh b f6hhi\nmp}kjk}kjlw bdbd\|xMmDjbºjkmh/}klBllpf6hi7 `hie\| m\Me\Mhq/h/jw l<hijklCjkxU}xUyMlpqia`}}ke{|h nM hli P\h6xM`}kh/`bPjkh)`b a\ r f6 7hnM\MhuÀ \UfÁm} \ θ λ a−c a =α ©¸b·a fih}¶`pmf]}k`pnMjkhh/\Uljhh``hiadjhbºmcfÁbºmmp}mh/ }nM \U`\l>|½Pbd}O mp }h|bdlp¿¹jklphahh6mGve½Ph>O =|}Ω¿Ámphlp∧lph>rbdhi\M\¾^ nMfijkbaa}``cheaΩ\M\U=qih/[0,l]fi0,bd`pΩ]mpqlC }khi\U\Mh/l]h6mhi\¾fia`cea\M\Uqih/l]lCxMyUqi`}k{|nMhli]q/`p}Uhi` ., - ?9 I0?0E4? F9CI63%8T=HGDF9CI65713 @BA )AC@ H#MJ LBN]OQSFTJUQWVXF ·nM`)nM\Mh.lpxMyMz/`ph>eh`cba \ R nM\g^a`cbd\Uef6h/`fijhh/lCmnM\_f6h/`fijheh>`cba \ R e \ m)jkh.f6h/\ m`phhlXmf6h/jnU}ehjkb lCxMyUzi`h½vh6¾³Ujhlq/`p}e}khi\Ul)lpa\|mehl)^a`cbd\UeUl[fihi`cf6jkh/lc¿ ·nM`jkblCxMyUzi`haUjb an "!$# % ½ÀjkhxMjknUlf6 nM`CmfcyMhi7}k\ eWw nM\ xa}k\|mrtnM\ banmp`h¿h/lCm nM\^ `ba\Uef6hi`cf6jkh @BA )A EGF ¥J£XLBNª©VCF'|¦P]¡TJU£p®[¯]F Z\m`p}bd\M^ jhlpxMyMqi`}{ nUhhlXmjk}k}mpq>xbd`ehl)bd`cfileh>^a`cbd\eMlf6hi`cf6jkh/l/ A b c C 90 90 B a jhl)f6²dmq/l>½ a, b, c¿[eWw nM\mp`}bd\M^ jh>lpxMyMqi`}{ nUhl/w h6xM`}7hi\|m)hi\nM\M}mpqeWw ba\M^ajkha jhlTbd\M^ jhlTe}kz/e`h/l©½ A, B, C ¿eh/lTxMjkba\Ul©½{|nM} eqimph/`p7}k\Mhi\|mjkh/lW^ `ba\UeMlTfihi`cf6jkh/lTeMhB`b a\nM\M}mpq¿liw h6xM`}kh/\|m hi\nM\M}mpq.eWw bd\M^ jh '& %( ) ³Mjkb7lpa77h>eh/l)ba\M^ajkh/l)eWw nM\mp`}bd\M^ jh>lpxMyMqi`}{ nUhe}+*Szi`h.eh 180 o D k5#"#"$ & %)+*, x3 X3 B c A a X2 b C x2 O x1 X1 ;<>=?<5@:ACBDE=F@GDHIEJ,K1LNMHOM2P D"V1LY<\YU?H ;<^<B[@CH c P LYE@:BNEA DH]UV L[JH !"#!$&%')(!*,+.-/01 HQ@ (O, x1 , x2 , x3 ) (O, x1 X2 , X3 ) R ;<>S1LNM:M:HTDH2UV E1<HXW x 1R U_La`aL[@:AC=?bQH7DH]S1LYMCM:LY\YHcDH 235476 (O, x1 X2 , X3 ) → (O, x1 , x2 , x3 ) (81:9- UV LYE@:ACH2SZL[AGE<HTA:BY@CL[@:=?BY< P 1 0 0 P = 0 cos c − sin c 0 sin c cos c ;<dbQL[U_beE1UFHcU?HIM bQBY`aSZB5M:LY<5@:HIMDEdfNHIbg@CHQEA x3 −− → OC DL[<ZM (O, x1 , x2 , x3 ) S1E=?MDL[<1M (O, X1 X2 , X3 ) B c A a b C O x1 180−B x2 sin a sin B −−→ OC (O,x1 ,x2 ,x3 ) = − sin a cos B cos a P !"$#"$%&' ()+*, O X3 B c A a X2 b C sin b sin A −− → OC (O,X1 X2 ,X3 ) = sin b cos A cos b A O P\gb7jkb`hijbºmp}ka\¾³ P\6 Jm}h/\ m]baj `l[jkh.lClCmpz/h7³ X1 −−→ −− → OC (O,x1 ,x2 ,x3 ) = P . OC (O,X1 X2 ,X3 ) sin b sin A − sin a cos B cos a w NKjkh/luÀ `pnMjkh/l³ ( 7³ ½POaQO |¿ = sin a sin B = − cos b sin c + sin b cos c cos A = cos b cos c + sin b sin c cos A ½POaQO6O¿ ½PO ' O ¿ sin a sin b sin c = = sin A sin B sin C ( s³ % cos a = cos b cos c + sin b sin c cos A % % ! Æ ! 7³ ¸D\_f6a,JM}k\Ubd\|m)jhlq/{|nUbdmp}ka\Ul>½ O¿¹h6mG½ ¿en_lp mzi7h aO QO Ua\6Jm}h/\ m>³ @BA )A )F ¢¦PVCF ½PO 'O Æ ¿ sin A cotgB = cotgb sin c − cos c cos A P\eq/j}lphjb7´h/`p`hxUbd`)nU\Mh.lCxUyMzi`heMh>`b a\ R = 6371 g Pole o 90 60o 60o Gran d ce rcle L M Parallele bkG S Q bkG hQ PVU SQ QhVP U FL\gG G Zg _F _ H L\b P UG ODPQMS b b O S PY UQMP PU MQ P ∆ QMSsQ K δ QTSvO Y YI KT`ZG FG L\bF K H \L b F H F F]sKT`ZG NL H Ps P LNH UXPQMSsH O Y n S P U x ODU PeP XU P SU{SQMQhyVQhPP UWP M P L M P H L n D k5#"#"$ & %)+*, ONO [bdjf6nUjh/`[jb7e}lXmcbd\Ufih>hi\|mp`h L ab `p}l[h6m _M |lpfian ; latitude longitude Paris : ϕo = 48o 500 11.2” λo = 2o 200 13.8” latitude longitude Moscou : ϕ1 = 55o 450 39.4” λ1 = 37o 390 53.1” @BA )A.7 _®¯L~¥M£X¬¹N O v¯]N¦>FT¥£X¥§QSFTJUQWVCF ¯JKVXLI ¦TJUF nM\Mh.·lp xM}¶myMz/nM`p\¾hnMx\Uh6m}¶m}¶m>q.f6lih/w`qfif6j`h}`cf6bth/\ ³ m`pqhi\ P ½veh7f6 jkbdmp}mpnUeMh θ h6m>ehjka\U^a}mpnUeh λ ]¿ h6m>eWw ba\M^ajkh α W Z\Mh7f6 nM`!Jh7lCnU` o o f (θ, λ) = 0 N λo−λ θo θ M x P Px α θo λo Âf6 xU`bae`C m}\M`)\Meq/h>h/ljbeWw uÀnMa\g`GxnM jk}h>\|m ehGÃbabdxMnUxUllba`Cxmahi\UnMba`)\ jkm)hbdmp`n}kbax\Mh6^ampjk}h>mPlCfixMhi`cyUf6qijk`ha}kM{|eWnMwhsbd\M½ N^ jPh MU¿6f6h/a\ \m`pqmp`h/a\ nM hj³ w q/{|nUbdmp}ka\aqi`}uÀqihxUbd`jkh/l M α P ½POaQO ¿ cos α = cosθ cos θo + sin θ sin θo cos(λ − λo ) @BA )A _®¯L~¥M£X¬¹N O v¯]Nª©JMLBN]ORQSFTJUQWVCF ¯]J VCL |¦PJMF L anM`m`p nMah/`¹jwMq{ nbºmp}ka\geWw nM\g^a`cbd\Uef6h/`fijhf6h/\|mp`qh/\ P (θ , λ ) M \uvba}¶m α = 90 eMbd\Ul>½ OaQO ¿Á w NK?³ P½ OaQ O ¿ cos(λ − λ ) = −cotgθ cotgθ o o o o o @BA )A EGL¢EG¬ )¬)OJU¬© £XF lhipa\?nUxMlPZ`nM\Ud\¾h±XhjkfÁbamp\M}k^aajk\¾eMhG`pefi ha\UMo}klChtmhiba½`c\ efimn¾bºma^ Z]``p\MhxUhf.bd`pjk` TnnM½Àmp\U|hh¿ Y[jjk}hi^ mP\MehsY e`e`aa`7a7}k}{|mp}khhnMhGfi7ah/banMlC}k`cmllC`hG\Mhih6xMJM``q/jkhi}klpxM{|hi`nM\|q/hmlp¿qihihG\|hmplXlphm>nM`xUnMba\UnMlh\Mh7jfibafinMeMbd`!`p}klCJmpmhGhba7\U{|f6nMbah}D`p}kjkf6\Mb hnMxMaxjknUn¾hl>jkbah/f6qilP `nMa`p\Umpq/bah`pn}ehimp}\|}k{|himpnM\U`hhl e° h/bn`anxmah}k\|jml/¸De\g`ah 7*S}kh6{|m]nMjhbh/`plC m)nnMmp\Mhh>jb`axUnjnUmh l]f6af/nUbd`CxgmhPf6h lX\UmlXmcjbbd\|`pm/ nmpha`pmpyUeM`p }{|nMha !"$#"$%&' ()+*, O Loxodromie : route constante. Trajectoire en spirale s'enroulant autour du pôle. 50 70 90 30 10 )FTJUQW£CQSF − dθ V ( % θ λ − λo = tan V log tan 2 ds sinθ dλ ( ! Bmp¸ }ka\gxM`x}7anMhi`` tan V h/\7uÀa\Uf6mp}ka\seh dθ dλ him θ \|mpq/^a`hi`Dfih6mCmhq/{|nUbdY m`p nMah/`[jw q/{|nUbºm} \eMhjb7jke`a7}kha • % ( % ¸B7axM`cf6`}h7bdnh/`>eMjh>w qijjkqi7hie\|`mGa7ehs}h jk a\U^anMh/nM` ds h/\?uÀa\Uf6mp}ka\¤eh dθ dλ h6m θ \|mpq/^a`hi`xanM`m`p nMah/`jbj \M^anUhinM`.eWw nM\ ( ! fibajkfinMjh/`>jkbge}lCmbd\f6hhi\|mp`h ¹`h/lCm>h6m>jkh ]qi\UqiÅinUhijblpnM`jb`anMmph7jMe`a7}{|nMhGxMnU}kl.lpnM`jb • `anmh `CmyMe`a7}k{|nMh • colatitude longitude Brest : Venezuela : colatitude longitude θo = 42o λo = −4o θ1 = 75o λ1 = −60o @BA )A ¾ JM¬|FQ¥M£C¬©N FTJUQSL~¥¬DJ L `d±XhfÁmp}ka\lCnU`)nM\_f6j}k\Ue`hmba\M^ah/\ mr7jw q{|nUbºmhinM`)r7jblpxMyMz/`ph7³ YDYD7xUbaqi``ba}keMjjk}zih/jk\Uh/ll>³M³jkjk}}^ ^ \M\Mh/h/l)l)eMe`p` a}¶}mmph/hllxUxUbabd``cbabdjjkjkjkzizijkjkh/h/l[l[\Mq/ {|\nM}eq{ }lXnUmc}kbde\|}mplChmbdl \|mh y = k log[tan( Π ϕ + )] 4 2 6K y h lXm)7h/lpnM`qhi\_f6 ehixMnU}kljw q{|nUbºmhinM`)him ϕ h/lCmjkbjbºm}¶mnUeha 8 ( &#A(ks*,'$'(( O ,.- ¤ 399<3%3)464I6=8 ? B;.4i3 I .19~3 B 39CFF9<3 E[3 Æ @BA]AC@ _VC¡ ¢FN¹¥«O]FªD¡T¬)O¡£CF ¢L~¥M¡ ¢L~¥M£C®¯F ° hlBxM`hi7}kzi`h/lD}k\Ue}fibdmp}ka\Ul©{ nUh]jb.´h/`p`heMba\Ul©lpa\sh/\UlCh/&JMjkhhlXm©jkqi^ zi`hi7hi\|mDh/jjk}xlCeMbajh alpa\|m¹r>jbuÀa}l©eMh \UxUbdbamp`nM;[`h)balhilC7}k\MxM}W}k`ph6}m{|mnM`h)bºh6bdm[nMehe\UhbdmpnMnM`h^ mpyMyMq/hia\U`l}k{|h6nMmhµhih/m¹ÄeMmpbºam\_hi\|lpm¹nM`)enHjbO xh/lCl}kbdz/\|fijmh.hinM½v6`Á¿ÁJU lphi`ºbºmp}ka\7ehjw bdxMjbºm}kllCh/7hi\|m©eh nMxM}mphi` P½À\U\7dmpxq h/nm[ba¿Áj `l<`hixU`pqlCh/\ mhi`<\Mamp`h)xMjkba\Mz6mhxUba`DnM\shijkjk}xUlp¼eh)eh`qi ajknmp}ka\xUba`D`baxMxa`pm©r>lC \7bdheh)`pambdmp}ka\ Oz z xM B , xM x O F A ¯ FVp®¯F O¡N]£ ¥M£C¬©N] ° w hijkjk}xUlp¼eh>lpheMq/enM}m]eWw nM\_f6h/`fijh Me}¶m]fihi`cf6jkhxM`p}k\Ufi}xUbajxbd`)nM\Mh.bs\M}¶mqxUbd`cbdjkjz/jh>r Oz ³ M 0P a = MP b b·a h}¶m f OAjh=uÀº ahijk`)h.eeh>hi7jw h/}jYjk^ }x`balC\Ueteh bdPh\him \UOB dmph/`=b cb jkjb7h>ee}h/lX7mcbd}¶\UY¼xfihih mpc}mP=bºOFh am=hijkj6378136 h.{|nMh c = √h6am b−=b6356751 g F P\gbdxMxhijkjh>jw h6Mf6h/\ m`p}f6}mpq e = mph/jjkh.{|nMh e = P\gbdxMxhijkjh.baxMjbºmp}llCh/h/\|m^ qi qimp`}k{|nMhjkh`cbdxUx `Cm α = 2 c a 2 a2 −b2 a2 2 a−b a `´hihi\U`fi`p¸Dah¹\«\|}kempfc`qyUh)bdb jkxUf6h©bdnMhi`[\¤m<bdje}kw jbºh/jkhilnUh¹x`el©ah[}kj\|w`bºmqil.ah] eMjehnMhPmpj}kb_`paq/\GlpanM r]`Cjuvnjb w mbºf6} h7h¹\TeeM hs·n}j7w xUh/jabajk\`©}xeh6lCha mhiae\U\Gh7lpx} \¾\TnMax`p `chib\»nm>xbabdlm`lC`}kjb 7h/f6`h/}ejk`>h]h[nMfijk\Mh6bdmChsmpm}mph]\MnU \MeM`p hs`pseTbdw jknMbahj\h]7xr.bd mpjk}b>yM\|mBq/ shie`ph[bºmpmp}jkfi}b{|balCnMjnUh]hg`CeTuv³Tb w nMhif6jk\Uh]jkhh ³ fxU6hba`<lphifi`ch6bmpmpjh)w bd\M\U ^a`pjksheMbdh)jkh[jkbh6mB\Mj w bº`psh)baejhnhi7\safi\Uh)exh }° \|bm©b\Maahmpf[}kajk\b>eTew}kba`h/Åi}kfÁGm}n \mBh6eMm©nseMxMhjkbaeM\7}klCmq/ba{|\UnUf6bºhmaÅi`q/}k\Mba}jmp yU° bah)jh)xMlpjkh[ba\7^ qi\Mq/q/`p`}baej}}klphih©\sqilp^|hibd`cjkbhi7eMq6hiU\|\Mm} !"$#"$%&' ()+*, O xM , n P\geq6\M}¶m]eM} *Sqijbº`himp}\|mpmpnhel)h>jbº^ampq/}mpneMeq/lph/}kl{|nM³ h jbºjkbdmmp}¶m}mpnUnUeeh>h>^ `qiq/eMf6nMh/}¶\ mmh `p}{|nMh • (ex , n) = Ψ • (ex , OM ) = ϕ • (ex , OM 0 ) = u x M u ψ ϕ e O x B@ A]A )FTJUQW£pQSF OY©¸©fi`p}k`phjkh/l`hijbºm} \Ul[hi\|mp`h tan u tan Ψ tan ϕ a him b YP\xh/nm7a\|mp`hi`{|nMh MlC} tan y = p tan x Ma\gbg³ x−y = ∞ X n=1 −1 n+1 q n sin 2nx n avec q = 1−p <1 1+p `¸D½}¶Pm\oyM\ e7uÀqah.e`nMe7}kh>`phGhGf6jh6jbtmpw h6mp{|h.nUxMebd`h/h/\|`plm\UlC}¶m}k}az/q \g`ptan(x heh6h.xMjb7`−h/jlbºlpy)mp}} mp\na¿Áe\ h.h/^a\q/ef6qhie\|mnM`p}}m {|enMh him]`xMq/nUe}knMlP}mpah \ uvbdhi}\mnMuÀa\¾\fÁempqi}k ahi\ojkaeMxMh.xjkhib77jhibº\|m}¶mmnUj}ke7h.}¶m^ qqieMen qlCjka}{|^ nMbdh Y ϕ u Ψ MbdngxM`hi7}khi`a`ce`hhi\ α L ÂxUxMj}fibdmp}ka\g\nM7qi`}k{|nMh7³Mr bd`}li Ψ = 48 50 11, 2” Æ Y M_ \ m`ph/`){ nUh cos u = cos Ψ him sin u = (1 − e ) sin Ψ 1 − e sin Ψ 1 − e sin Ψ ¸D\_eqenM}k`ph>jw q{|nUbºm} \ehjw h/jjk}kxUlC eMh³ 2i(x−y) o 2 2 2 0 2 2 2 2 r=a q 2 2 1 + (e4 − 2e2 ) sin2 Ψ p 1 − e2 sin2 Ψ ÂxUxMj}fibdmp}ka\g\nM7qi`}k{|nMh7³Mf/bdjf6nMjkhi`)jb7e}klCmba\Uf6h>eh L ba`p}l)bdngf6h/\ m`ph>eh>jb7´h/`p`ha Y©¸©fi`p}k`phjw q{|nUbºm} \ehjw h/jjk}kxUlC eMhh/\_f6a`cea\U\Mqihl©xajbd}k`phl>½À} h Mjbºmp}mpneh>^aq/fihi\|mp`}{ nUh¿Á O 8 ( &#A(ks*,'$'(( Y _ \ m`ph/`){ nUhjw qijkqi7hi\|m]eh>jka\M^ nMhinU` ds = √dx + dz lCnU`)nM\gq/`p}e}khi\gliw h/fi`p}m>³ M 2 ds = a 2 1 − e2 dΨ (1 − e2 sin2 Ψ)3/2 ]¦ ¦VC£pQSL¥M£C¬©N N¯] ¢¡TJ£C®¯F Â)·fi}h/\ \Uf6ehhtl)mpehih `7L }bd\U`hi}k`l jkbo`p^|{|bdnM\Mh/}lp`pbh/jjkehhinMlpnM`7jw}kbdllpxU}jk bd\Ump}l)lplpbºhiU7\ghie\|h>m7eMhth/lpjnMb `hi´`hinM``p\_hte{|h/nM^a}`qenMeM`ch>b eMq/h`p}Oe}khidÆ\?³Mr jw nMO \UÆ h.bdTngjw Â)µfiaba`ceMe¤qi7½veM}bah\Uleh/jkhl ^a° bj¶uÀxMh>`ehi7h }k[zia`hmp\Mh6}khx¿6qjew bd}mpn}kam`p\gh>6 baJn m}L\|m)q/`pnM \_nT eh/^a`qeMh7q/`p}e}khi\_eh Æ Gmp }klph/l)rjbjkbdmp}mpnUeh>7º hi\M\Mheh ° b7lph/fia\UehxM`enM}meh/l)`q/lpnMjmbdml)ballphiÅe}klpxhi`clCqliMeh>7º hi\M\Mh mp }klph/l)rjb7jkbdmp}mpnUeMh.eh −1 6630 20 ;[bdjf6nUjh/`)jw bdxMjbºm}kllphi7hi\|m[mphi``h/lCmp`hNJmph/\nrjw q/x{|nMha o o "6" % A 1"3 , %Q "$!3436 2 N54 0 0