Eléments de géodésie mathématiques Marianne Greff 2007-2008
Transcription
Eléments de géodésie mathématiques Marianne Greff 2007-2008
"!$#
%'&)("*+&),,-/.("-10
2)3)35476283939:
! #" $%'& $%'")(+* ., -/, 0213)465713)8:9<;>==?3)468
@BAC@DAC@ EGFIHKJMLBNPORQSFTJUQWVXF
Y[e}kZ]}k\_lph^ahi`cbd\_\UeeghinMfihi`cf6yMjkq/h.h/}lClpm]xMnMyM\ozi`h/fil[hi`cq/f6^
jkh>bdnMmpS`cba fiq.rsjkbtlCnM`puvbafih.eWw nM\MhGlpxMyMz/`phG{|nM}~bsjkh.7i7hGe}bd7z6m`ph{
nw h/jjkh.h6m{
nU}jb
Y[YDaPngngh/hi\\UUfi\TaM`hajw}\|nMmp\gh/`^alp`ch/bdf6\mp}keta\f6h/e`h.fijhf6hihmClXmpmh.nMlpxM\_yMfiz/hi`p`ch>f6jkbh
lph/nMf]`)mpajb7nMmlCxMxMyUjkbazi\`hxbba|lbdlp\|bam\
m)jhxU7ba`)i7lC
h.\f6fih/hi\|\|mpmp``hha {|nTw hijkjkh
O
x
Z\g^
`ba\Uetfihi`cf6jkhhlXmjkhxMjknUl)^a`cbd\efihi`cf6jkh{|nMh>jw
\xUnM}kllphPm`b
f6h/`rjb7lCnU`Cuvb
f6h>eWw nM\Mh.lpxMyMzi`ha
@BAC@DA gL<JMLBVCVCVXF ¢¡TJU£COP£XFNP¤FW¥§¦>¨©VCF«ªD¡¬DªDJML<¦PP£C®¯F
° w nM\_ehl)^a`cbd\UeUl)f6hi`cf6jkh/l~±X
nMh>nM\_`²ajkhxUbd`pmp}f6nMjk}khi`.³M}kjxbd`pmbd^
hPjbs´hi``phhi\_jw yMqi7}lCxMyUzi`h.µa`ceh6m]jw yUqi7}¶Y
lp´xMayMnzim`h.fihi·`cnUf6jke?h7³Mmp`c
ba\fiq»jw balCxMnMx`Ghijkjkbojh>lpjxMw ¸¹yM{
z/`pnhsbºmphih/mnM`{|nM}¹lphsm`p
nMahseMba\Ul.nM\«xUjkba\«xUbd`cbdjkjz/jhsr_fihijknM}¹{|nM}[f6a\|m}h/\
mjw q/{|nUbdmphinU`
l/Zw ba\UxMh[xlph/xMjyMjkhz/`pnMh[\gxxU|bdlplp`cz/bdeMjkjkh¹zijkehah/ n.x²
jhl~^aqi
^a`cbdxMyU}k{|nMhliilp}mpnMql~r ehlC
\Gq{
nbºmph/nM`eh[xUba`Cm~h6mDeWw bdnm`ph[eh[fihijknM}Y¼fi}³
}k° jTbtteb`p
nM}¶\gmhGx{|²
nMjh.}~xUµ]baal`clCe¾hG½vxUxbd
`}\|jkh/m]lPµx¿[²ajkh6h/mlnMh\glXmxjk²ab»jkh>xh/·`pnUxe¤hi\U½ÀxeM}kafi}knM\
jkmPba·U}`¿Á90hG ban_xMjkba\oq/{|nUbdmp
`p}bdj{|nM}~xUb
lplph>xUbd`Pjhf6h/\
m`pheh.jb
´° hhil)``pehhi³M7a}\Y¼fijhiw `cbaf6xMjkxh/l©h/jmpjk`ch.baÂ)fiq/lh.lCenUh`[ljb7x²
lCxUjhyMlizi `h{|nM}W
\|mxanU`[himp`qi7}¶mq/ljkh/l)eh/ntx²ajkh/llpa\|meh/l[7qi`}keM}h/\Ul³Mf6h>lpa\|m
elphih`plBm)7eTaw
}mp`p}k}kq/^alB}k\Meh7h³M^
fd`w bah/\UlCmeMl<jkbf6h/7`fiqij`h}lie }h/° \w nUÅ/\7qi`eTgw h/³Mn}WjTafif6ah/nMjnMx}h>{
jnUw q}M{
xUnbabºlmplph/h[nMx`)bdhi`©\ÃnM`\ghihix\aÄ)}k\
}fcmPyývxM `z/lDeh ° a\UeM`phl<hi\sÂ\M^
jhimph/`p`h¿6
Æ
o
!"$#"$%&'
()+*,
N
ri
Me
di e
n
Parallele
G
Equateur
-./-0.21
354(687&689$:;<;=6?>A@BC7A6D9$:;
E,FGDH5IJFKMLNH
E
H FZG8[ G G\L^]+_ K K La`NG8KbFG8I
_e`NG H G L^G8L^IJFKMLZH
L\g
U P L8FH
n E f,Q2c S MQ P
H I8I M_ ODFPRL\gQTS KhHWG PVUWUXPRGiY/P IJFKMU LNH PRY bkFZU ]+I8HW_ IJU F O KlP HXKh[ ] LNH[ PRQdc HeS nJo$P L 5
PVUXI Y"_Qdc mYLZH LNH hQ P ] ] PVUXS]sλ_ K K LF KMgKML n
(GOE)
MH j
p
FZL8gPeKhHWY G S PFQ LNH0Q bkFZ]+I ODK P5O LNH
b P GDY q q%P H P]
PeYQ
ODPVYY
U Y/PeYP UXP 0o P 360o j PeYODP P UXr PeY,UXP U Y/P S QhP r P
U O P U
P
N
M
θ
ϕ
O
λ
G
H
E
S
p L\g
H I8I _
HWKhHWG
GuIJFKMLNH
HvL8FHW_
nw%K
H L
_]sK I
F
c H
S bFQM]s
P ID(EOM
HW_ I|F ) KhHWPeKMY [ ] S LNPVH Q Pt
HFQMS L^I ODPO
HXKlHXG M P F
KMP L8FZϕL FL^MI PVY O\S YsQdcHXx KlHXG Y wDx8G yVUWPH z U{O=jPQMQhP
PVbkFZY ]sIDHW_ L8_P g HXKh[ Y ] P LNHVnZP p P HWKhHWG SUWQMPR
PRIJQTS
ODF P Lv
OI P PQMQhLNPRH0YWI PU{S
[ ODPR
K LNQTS H LNH ODP}z U{oO~GvY IJ j w%G SH UXQhPVUXStODG+
P
S
P
P
V
P
0
Y
M
Q
S
D
O
e
P
Y
S
U
P
P
X
U
P
−90 S
M
Q
P
s
O
P
+90 S
M
Q
P
z
{
U
O
P
SYXYXS
SU
LHXFGDHIJFKMLNH
_V`NG H G n
0o P
D
O
P
2
Q
c
S
P
U
p L\g
LNH
K bkHWKMFL GI|
F
H
_L\KlH
K bHWKMFL
[ HXKMb
b L LNH
G
z bkFZU{]+
O
I P _V] QMPv LNH K xu2O UWP HWKhHWG ODPvQTS FL PVU L8SFQMH PsSZY P O\S P$O n
IJFc S KMLZH QMP (N OM
H I8I) P _bUWFPsQTS
HXKlHXO G UXP K O H$_g QhPRG
o
QTS GiODP0I|~ Q,PVY F SQS
p bF MHWKhHWPVG Y S [ PK Q
GiQ IJ P w%S G UXPODLtP5I QTSvQTS LNHI ODP}f o QMP _V`NG PU{S H θG =n 90 − ϕ
o
o
S QTS
ODP SU PODP 0 j\S
QhP5z U{O=j8 180 j8S
QMP O=j\P SYXYXS
SU 90 Qdc S P U
-./-0.
u7W96
p Kh]GDH H L8g
F K FLNH
b HW K
L
I LH L8g LNH
I
GIJFKMLNH
LNH
YQMgP _F QTS _ K S b H/P H _kt_ QMSL\Y b x8yVUWPR
K bHXc SKhFZ L G8LPVY FZDQd c S H H"QMPsG8xL U K bHXSKhQ
FZL c PeY _ _ O UXL\Psb O\n S Li
H S L\F g_VM
Fg P I KMUW`NPsG QTS n
Op UWP Kh]GDH O=c H] G P _ P I8G8P
ODL PU g _UXP L P m
U{S FLNx H P
K O UWP L8F
L O _kY H PFj g P P$U L UWP P$PVY KhgZG8QMP K U{O G8L ]s
c S
PeY PVY U ODP YQMP U{OtP ODP U O8S YQhPY/P YU U UXSZODPsYWP Y$ODPVYS
QhQMPVYOc P
UXPef
8##"s#"%)+*
bd}k\Ulp}Wjw ¸©lCmh/lCm]bdn 90 Mjkh.lCnUebdn
o
180o
h6m)jw PnUh/lCm]bdn
Mx
270o
N
a
!#"$&%'&)(*,+-.0/1"
2*,3+42+
365
@[email protected] 8£C
¥L<NPQSF:9W¡N]£ ¥M]L<VCF
° b>eM}klCmba\Uf6h)Å/qi\M}mpyUbajheTw nM\Uh]e}k`phfÁmp}ka\7h/\snM\sx
}\|m¹eMa\M\Mq)hlXm¹jw bd\M^
jh)ehPf6h6mpmph]eM}`h/f6mp}ka\sbahf¹jbah/`Cm}kf/bdjkh
jkf/bdjkha< ;Pw h/lCmbd}k\UlC}{
nUhjb7f6
jkbdmp}mpnUeh>hlXm)jb7e}klCmba\Uf6h>Å/qi\M}mpybdjkheMnx²ajkh.µ
`eT
,.- = >@?A?>C9 BD?0E4E4F)3)8:8TH= GDFC9 I65713)8
`b
a¿Á\ reW
w bºxM`h>hi\Ueah/\UllBfinUd\smh/`plh/jxbsz/`peh`p
eW}¶wmah `}k^a}k\Mh
O6K eTÂ)w bdh/\»lCh>meejkhh.h`lf6hihixb\|Jqim`lp`phifih.`©}kleelCh>hh]lj7bsjkb7balp\MxMe}kyM`ziazi`}`hmphah fi bdOG`pmpqjkh.lC}keWxhi\Uw
bd\M}\|h)hm]jheMelBh/n_l)x7a
}k`q/\
e`pmc}
l©e\M}ke\Mhih]\gq/h/jkÅib.lq/jlp`pbxMsyMeM{|z/`pnM`p
h}}¶mlpeMh»hh½ mpOH
ON
`
a
M
n
>
h
7
r
j
w
/
q
|
{
U
n
d
b
p
m
i
h
U
n
`
6
h
m
O
G
H jkh.x
}\|m]eh
jw q{
bnxbºamph/nUnM``f6eh>a`cjkeaa\M\U^
\M}¶mqinUhel h 90 < ;¹h/l]bºh/llpa\|m)a`}h/\
mq/l)ehmhijkjh.lpa`pmph>{|nMh7³
G
(r, 0, 0)
H b7x
nM`f6a`ce
\M\Mqihl (0, r, 0)
N b7x
nM`f6a`ce
\M\Mqihl (0, 0, r)
L anM`~6JMmphi\U}`<jkh/l~f6a`cea\U\MqihleWw nM\x
}\|m M rjbPlpnM`puvbafih¹eWw nM\MhlCxUyMzi`h¹eh[`cbaa\ r df6
\M\Uba}kllpba\|mlpbPf6
jkbdmp}mpnUeh
jjkzijkhi76 JMhimp\|him.\bdr \|ON
³Tx
a\?}k\|mN Jmp}kehi\|h»mGf6jkhsa`cxea
}k\
\Mm\MqiLh
mph6h/mjDlp{|bnMjkh a\MOL^
}¶mnU=eharsina\«h6θmGxM~`jkPh»d±X\?xhimCaxMmp}kh7\
`dm jk±Xh»h6mpxmplpahnM}k\
hi`.&m\UjlpwM nMbº}mph lphsnMjk`.h7jkhsx
6xM}J\|jbdmm \ hi°\UGOH
lbaC\|nMmG`>jhtjw bºxUxba
h `}ba\|OG
m
U
n
?
\
I
m
(r sin θ cos λ, 0, 0)
L
OH
J ehtf6
`e
\M\Mq/h/l (0, r sin θ sin λ) ¸D\
xM`d±XhimCmba\|m~jkh x
}\|m M lpnM`<jw bºh ON xUba`bajjkzijkhi7hi\|mBbdnxMjkba\ (GOH) d
\GN Jmp}khi\|mBnM\x
}\|m K eMhfia`cea\M\Uqih/l
(0, 0, r cos θ)
o
N
K
M
O
J
I
L
G
S
H
° w bJlpfi}kllCheh I jw a`cea\M\Uqih»eh J ih mjb fidmpheh K Cl
\
m`phlCxh/f6mp}kahi7h/\
mjw bJUlf6}llCh
jw
`e
\M\Mq/h»h6mjbofidmph
engxa}k\
m M P\gb7ea\f
P½ OaQ O¿
OM = [x, y, z] avec x = r sin θ cos λ; 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