HAL Id: inria-00072535
https://hal.inria.fr/inria-00072535
Submitted on 24 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,
A New Formulation for Non-Linear Camera Calibration Using Virtual Visual Servoing
Eric Marchand, François Chaumette
To cite this version:
Eric Marchand, François Chaumette. A New Formulation for Non-Linear Camera Calibration Using Virtual Visual Servoing. [Research Report] RR-4096, INRIA. 2001. �inria-00072535�
ISRN INRIA/RR--4096--FR+ENG
a p p o r t
d e r e c h e r c h e
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
A new formulation for non-linear camera calibration using virtual visual servoing
Éric Marchand and François Chaumette
N˚4096
Janvier 2001
THÈME 3
'&(#)* +&,)*'+-'+ )(
./103254768/12968:;=<68:;?>@/168:ACBD0FEHGI9=6KJLNMPO1O1MRQ
STVUXWZYH[\^]`_baYXcdReXagfihR_,TVhRWjWZY1klW5dReTmfn_VYpo
fnW5drqYsto+umhR_m_VvYstoVehR_m_+drfissdr_VeYs
wchrxyYXas"z{fisad
|{dr}m}hRcgaumY~cYeTVYXceTVY_bpp-\ bdr_b8fiYXcIppD{\p}+drqYs
!tCmp STmfis*}+dr}YXc*}mcYsYX_basHd_VYX hRcgWHmdragfihR_hRc{agTVY_VhR_Dklnfn_VYtdrcetdrnfnmcdragfihR_
}mchRmiYXW-{Y}mchR}hpsYdjWZYXagTVh8u,+dRsYuNhR_NagTVYH YXn¢¡8_Vht_£Kfis+dr!sYXcghRfn_mq¤dr}m}mchdReT@
Y¥ehR_VsgfiumYXcj}hpsY¥ehRWj}mmadragfihR_¦dr_Vu7iYX_VsZetdrnfnmcdragfihR_¦dRsagTVY§uD+dr}mchRmiYXW¨hr{8fisg+dr
sYXcghRfn_mqV©]ªa¤drnihts¤ah?adr¡YdRuDRdr_KadrqYhrdrn*agTVY£cYsYtdrceT«agT+dra¥T+dY£YYX_«etdrcgcgfiYu
hRma*fn_£agTmfis*umhRW5drfn_fn_NagTVY}+dRsgat~STVY}mchR}hpsYuNWZYXagTVh8uN¬YtdragmcYsdReeXmcdReXo=Y1®5eXfiYX_VeXo
setdrdrmfnnfnaª-cgat{ahZagTVYetdrWZYXcdZWZh8umYXdRs" YXndRs{cgatahagTVYH¬YtdragmcYsIY1¯8agcdReXaYuN¬chRW
agTVY*fnW5drqYpodr_Vu-sgfnWj}mnfieXfna`
°-±²³´µ C¶!p¸· drnfnmcdragfihR_@o&_VhR_Dklnfn_VYtdrcIdr}m}mchdReT@o+Kfncgag+dr¹Kfisg+dr@sYXcghRfn_mq
º¼»¹½`¾l¿tÀ½ÂÁ¹Ãn¾lÄÅÆ
ÇÈVÉlÊˬÌ`Í!΢ÏÑЬͪÒtÓRÔmÉlÏÖÕ1×Ø*ÒÉlÕªÙRÒÚRÛRÜ!ÏÝÉlÏÖÐlÒP×ÞßÉ
à
ÈVÉlÊˬÌ`Í!΢ÏÑЬͪÒtÓRáPɪÒâÚrÕãÊ1ÏÖÐy×äÙRÒâåCæÌãÍÍlÌÜ!ÏÝÉlÏÖÐlÒP×ÞçÉ
Unité de recherche INRIA Rennes
· Ycdr}m}hRcga~}mcvsYX_KaYjm_VY¬hRcgWmdragfihR_ hRcgfnqpfn_+driYj}hRmc*iY}mchRmiUXWZYZumYd
etdrnfnmcdragfihR_ umYsIetdrWZvXcdRstIhRVs}mchR}hpshR_Vs"m_VYdr}m}mch8eTVYcYX}hpsdr_KaIsmcmagfnnfisdragfihR_ umY
ÖdRssYXcgKfissYXWZYX_ba8fisgVYX¢fiumvYHehR_VsgfisgaYZehR_VsgfiumvXcYXc"djetdrnfnmcdragfihR_¬eÝYsgauDfncYHiY
etdrieXm=umYs}+drcdrWZUXagcYsfn_Kagcgfn_VsUKVYsum_VY*etdrWZvXcdYXa iY*etdrieXm=umYI}hpsYIYX_KagcY*detdrWZvXcd
YXa5m_VY-WjfncY5ehRWjWZY-iY-}mchRmiUXWZY£uD+dr{umY,ÖdRssYXcg8fissYXWZYX_Ka58fisgVYXl · YXagaY-WZvXagTVh8umY
ehR_VsfiumUXcYfnW5drqY"umY"ÝhRDxyYXa ehRWjWZYiYWZhRagf¼¹umvsfncvdragaYXfn_VuDcYpo8iYWZhRagf¼¹ehRmcdr_KaÂvXadr_ba
hRmaYX_KN}+drc{djcYX}mchrxgYeXagfihR_£uDNWZhKumUXiY[©umYÝhRDxgYXa{sgmc{iY}mdr_NfnW5drqYHum_VYetdrWZvXcd
8fncgagVYXniYp · YXagaYWZvXagTVhKumYH}mcvsYX_baYumY_VhRWmcYXD¯ dpdr_badrqYs Â}mcveXfisgfihR_£fiumYX_bagf!KVYdrD¯
WZvXagTVh8umYs umY"Wjfn_mfnWjfisdragfihR_5_VhR_Dklnfn_VvtdrfncYIeXdRssgf!KVYstoKsgfnWj}mnfieXfnavpY1®5etdReXfnavpoKYXaÂ}hpssgfnmfnnfnav
umY~ehR_VsfiumvXcYXc{uDf#"=vXcYX_baYfn_DhRcgW5dragfihR_Vs8fisgVYXniYs
$ µ
t
³
&%
·
drnfnmcdrqYpoVWZvXagTVh8umY~_VhR_Dklnfn_VvtdrfncYpodRssYXcg8fissYXWZYX_KaKfisVYX=KfncgagVYX
0 1
{+ 32¹&
· drnfnmcdragfihR_«fis¤dr_ fnWj}hRcgadr_ba¤fissgVY£fn_«ehRWj}mmaYXc¤8fisgfihR_ dRs¥sh8hR_ dRs¤dReeXmcdraY WZYXagcgfie
fn_DhRcgW5dragfihR_ T+dY?ah Y Y1¯DagcdReXaYuFchRW d sYXaNhr fnW5drqYsP¸STmfis£fis£d cYsYtdrceT
drcYtd§agT+dra~cYeYXfnYu WVeT?dragaYX_KagfihR_?sgfn_VeYagTVYjYtdrcgn54R Ýs76Vcsga*fn_agTVY}mTVhRahRqpcdrWjWZYXagcg
ehRWjWm_mfnaª ¬YpqVno98n:~agTVYX_7fn_ agTVY§ehRWj}mmaYXcKfisfihR_ dr_Vu?chRhRagfiesehRWjWHm_mfnagfiYs ¬YpqVno
8Ö[DoVVo=ppo=tmoV.:ªoVYXaep1
;Nhpsga~hr agTVYZdr}m}mchdReTVYsHehR_VsfiumYXcagTVYetdrnfnmcdragfihR_ fissgVYjdRsd§cYXqpfisgagcdragfihR_}mchRmiYXW
agT+draehR_Vsgfisgas~fn_ umYXaYXcgWjfn_mfn_mq£agTVYjcYXdragfihR_VsgTmfn}?YXa`ÂYYX_ [ehKhRcuDfn_+draYshr }hRfn_bas¬hRc
hRagTVYXc ¬YtdragmcYs !nfn_VYstobYXnnfn}VsYsto+nndr_Vu5agTVYXfnc }mchrxyYeXagfihR_Vs fn_ZagTVYfnW5drqY"}mdr_VYp!STVYsY
[ YtdragmcYs drcYVsg+drnnH}+drcga hrdIetdrnfnmcdragfihR_ZcgfnqHdr_VuagTVY}hpsgfnagfihR_jhr&agTVYsYÂYtdragmcYsfn_5d
hRcgiu¬cdrWZYÂT+dtYah*Y¡8_Vht_jfnagT5dIYXcgqh8hKujdReeXmcdReXw!YXcy¬hRcgWjfn_mqagTVYetdrnfnmcdragfihR_
iYtdRums¢ah{agTVYÂYsgagfnW5dragfihR_jhrVagTVYÂfn_bagcgfn_Vsgfie etdrWZYXcd{}+drcdrWZYXaYXcsFfnW5drqYÂeYX_KaYXctoRhKetdrKiYX_mqpagT@o
uDfisahRcgagfihR_ dr_Vu¤Y1¯8agcgfn_Vsgfie*etdrWZYXcd}+drcdrWZYXaYXcs FflÝYpnoDagTVY*}hpsY!}hpsgfnagfihR_,dr_Vu¥hRcgfiYX_badragfihR_
hragTVY~etdrWZYXcd¬cdrWZY*cgatagTVY~cgfnqZFcdrWZY1
{f#"=YXcYX_baetdrnfnmcdragfihR_?aYeTm_mf!bVYsY1¯8fisgaHdr_VuW5d YjeXdRssgf<6+Yufn_agTmcYYjW5drfn_?etdraY1k
qhRcgfiYs
=
>?(@! A(B 1STVYsYaYeTm_mf!KVYs ¬YpqVnoC8Ö[.:VsY¥d-iYtdRsaykªs b+drcYsWZYXagTVh8u7ah
YsgagfnW5draY*hRagT§agTVY*fn_Kagcgfn_Vsgfie*}+drcdrWZYXaYXcs{dr_Vu¥agTVYI}hpsYpSTVYXetdr_m_VhRaT+dr_VuDiYiYX_Vs
uDfisgahRcgagfihR_Vst
=ED
$? (FGHI? "KJ L NMO P?@/ A-BK XSTVYsYaYeTm_mf!KVYsHT+dtYZYYX_fn_Dk
agchKuDVeYubjagTVY{}mTVhRahRqpcdrWjWZYXagcgehRWjWm_mfnaªE8n(:ª]ªa ehR_Vsgfisgas fn_§Wjfn_mfnWjfNQfn_mqagTVY
YXcgchRcYXa`ÂYYX_jagTVYhRVsYXcgpdragfihR_Zdr_VuagTVY +dRe¡Kkl}mchrxyYeXagfihR_5hragTVY WZhKumYXlR;Nfn_mfnWjfNQPdCk
agfihR_,fis{T+dr_VuDiYu,Vsfn_mq5_bmWZYXcgfietdrfnaYXcdragfnYjdrnqhRcgfnagTmWZs*sgVeT dRsIYXahR_Dkª|{dr}mTVshR_
hRc @YXYX_KYXcgqRk?; drc K+drcgau=,STVYZW5drfn_ dRuDRdr_KadrqY¥hragTVYsYdr}m}mchdReTVYsjdrcY5agTVYXfnc
dReeXmcdReX dr_Vu agTVYXfncjYXcsdragfnnfnaª ¥agTVY¤etdrWZYXcd WZh8umYXetdr_ Y¥YXcg7qYX_VYXcdrl STVY
W5drfn_¤uDcd+dRe¡§fis agT+draagTVYXZW5dYIsgmDxgYeXa ahih8etdrWjfn_mfnW5ddr_Vu=o8ÂhRcsYpomuDfnYXcyk
qYX_VeYp STVYXcY1hRcY~agTVYX§Vsg+drnn¥cYbmfncYHdqh8h8u¥qpVYsshragTVYshRnmagfihR_-ahZYX_VsmcYHd
ehRcgcYeXa"ehR_bYXcgqYX_VeYp
=
STU (??JI 5?@!I A(K X STVYsY dr}m}mchdReTVYs ¬YpqVno8Ömo*ppo*tV:ehR_VsgfiumYXc,d?nfn_VYtdrc
YsgagfnW5dragfihR_¥hr=shRWZY}+drcdrWZYXaYXcs TmfniYagTVY{hRagTVYXcs drcY{YsgagfnW5draYu5fnaYXcdragfnYXn · hR_Dk
sgagcdrfn_bas ¬sgVeTdRs"agTVYS¹sdrfÝsIcdRuDfdrdrnfnqp_mWZYX_baehR_Vsgagcdrfn_Ka IdrcYehR_VsgfiumYXcYu£ah5nfn_Dk
YtdrcgfNQtY£agTVY£YsagfnW5dragfihR_ hrshRWZY£}+drcdrWZYXaYXcstSTVYsY drnqhRcgfnagTmWZs,drniht¸dldRsgaYXc
ehR_bYXcgqYX_VeYhr!agTVYdrnqhRcgfnagTmW-
¹YXa~VsW6V_+drnnN_VhRaYagT+drahRagTVYXc~aYeTm_mf!bVYsto¢etdrniYuX -
D
UYZ £dr}m}mchdReTVYsto!umh§_VhRa
VsYdr_K£etdrnfnmcdragfihR_ cgfnq[8ÖDo]\:ªHSTVYjetdrnfnmcdragfihR_fis*}YXcy¬hRcgWZYu Vsfn_mq,d5WZhPKfn_mqNetdrWZYXcd
agT+drahRVsYXcgYsdsgadragfie{seYX_VYp · hR_Vsgagcdrfn_basdrcY{}mcht8fiumYu§bZagTVYIseYX_VY{cgfnqpfiuDfnaª |{YXnfdrmiY
cYsmnas"drcYuDf¼®5eXmnaahZhRmadrfn_,Vsgfn_mqZagTVYsYdr}m}mchdReTVYst
!fnqpmcY?! #"$%'&($)"*+&($)*-,./"(01325476(8196$,;:=<?>(,@2A*BCDE,.3!",.
F
HGIJK #"*+&L*-,./"(02!>(,M6(
F
,NE,.%*POQ6;.*-,P$GD13'83*N9#6(
F
,NE,.%*!RSJTP"$%'&($
93
F
,.K1UVG$,.EV,N2*QE,.I!>(,WG,N*NV,+GJX,3!&$V,N*Y,!Z('9NE,+G[JEV
F
'>,\
F
2,[OE]V,+G0R^G_'>,
9.&$`V,.3^,W3Q;1,+GB! _((9.a'b-6(-cd,+9N!_OE?2I,+,.R':^<?>;*Le2I&V,\1&3*Q!K3E,N*)'>,[V,N2*Q!!
9+3"(,.%2,.9+,SGf!>(,
F
(g1+hi&(,.9+,TJj!>,B;kl33!H:
]ª_NagTmfis{}+dr}YXcIÂY}mchR}hpsYd5_VYX hRcgWHmdragfihR_hr agTVYetdrWZYXcd§etdrnfnmcdragfihR_£fn_KhRnKfn_mq
dZ¬mnsetdriYH_VhR_Dklnfn_VYtdrchR}magfnWjfNQPdragfihR_@HYehR_VsfiumYXcIagTVY}hpsYehRWj}mmadragfihR_}mchRmiYXW¸dRs
agTVY§uD+drhr{ 8fisg+drÂsYXcghRfn_mq 8nmV:dr_Vu Y§Y1¯DaYX_Vu?agTmfisHfissVYZah£etdrWZYXcdNetdrnfnmcdragfihR_@
· hR_VsfiumYXcgfn_mqNetdrnfnmcdragfihR_ dRsHdr_ fnW5drqY1kl+dRsYu Kfis+dr sYXcghRfn_mq,}mchRmiYXW 8Vo¹.: adr¡YsdRu8k
pdr_badrqYHhrdrn@agTVY*+dRe¡KqpchRm_Vu£dr_Vu¤agTVY~cYsgmnasfn_¤agTmfis96+YXiu= ]`a"drnihts"Vsah}mchR}hpsYHd
YXcg§sgfnWj}miYdr_Vu¥YXcsdragfniY*hRcgWHmdragfihR_Nhr!etdrWZYXcdjetdrnfnmcdragfihR_@ z drcgfihRVsetdrWZYXcdWZh8umYXis
etdr_ YH}YXcyhRcgWZYu dr_VuetdrnfnmcdragfihR_ ¬chRW uDf#"YXcYX_Ka*¬YtdragmcYsIW5dtNYehR_VsfiumYXcYu fnagTmfn_
agTVYHsdrWZY*}mchKeYsstgmmcgagTVYXcgWZhRcYÂYetdr_,YtdRsfnn§ehR_VsgfiumYXc{uVdradFchRW WHmnagfn}miYfnW5drqYs"ah
fnWj}mchPYagTVY K+drnfnaª¤hragTVY~fn_bagcgfn_VsgfieH}+drcdrWZYXaYXcs"YsgagfnW5dragfihR_@
]ª_«agTVY cYXW5drfn_VumYXc£hragTmfis¤}+dr}YXctoIÂY}mcYsYX_ba,fn_po8YeXagfihR_ 7agTVY}mcgfn_VeXfn}miYhragTVY
dr}m}mchdReT@]ª_qo8YeXagfihR_ DÖ[- Y§sTVht TVhthRmcdr}m}mchdReT fisHWZh8uDf<6+Yu7ah,T+dr_VuDiYWHmnagf¼k
fnW5drqYsetdrnfnmcdragfihR_@ ]ª_Po8YeXagfihR_,[Dob YI}mcYsYX_baagTVY*eTVhpsYX_¥etdrWZYXcdWZh8umYX@dr_Vu Y*umYXcgfnY
agTVY5Kfis+drÂsYXcghRfn_mq Yb+dragfihR_7cYXdraYu7ah-agTmfisWZhKumYXlN]ª_qo8YeXagfihR_?Vo! Y5}mcYsYX_KajshRWZY
cYsmnashR_-pdrcgfihRVsfnW5drqYsYXas"dr_Vu-ÂYehRWj}+drcYhRmc"WZYXagTVhKu¤fnagTNhRagTVYXc{dr}m}mchdReTVYst
g +-)*'+-'+ )( %'+ 2
¢{# #!g% &
s"drncYtdRuD-sgadraYu=o+agTVY~+dRsgfieHfiumYtdZhrhRmcIdr}m}mchdReT£fisah5umY6V_VYagTVY}hpsYehRWj}mmadragfihR_
dr_VuagTVY etdrnfnmcdragfihR_}mchRmiYXWdRs!agTVY uD+drK}mchRmiYXW©hr 8fisg+drDsYXcghRfn_mqL8Ömot:ª¢]ª_8fisg+dr
sYXcghRfn_mqVoagTVY5qhdr fisHahNWZhtYagTVY§etdrWZYXcd,fn_7hRcumYXcah£hRVsYXcgY¥dr_7hRDxgYeXajdrad,qpfnYX_
}hpsgfnagfihR_fn_agTVYfnW5drqYp5STmfis~fisHdReTmfiYXYu b Wjfn_mfnWjfNQfn_mq,agTVYjYXcgchRcHYXa`ÂYYX_7d¥umYsfncYu
sadraY hragTVYfnW5drqYYtdragmcYsjdr_VuagTVYeXmcgcYX_ba sadraYÂ]ª+agTVYYeXahRc hr&8fisg+dr8¬YtdragmcYs
fisI YXn eTVhpsYX_@o@agTVYXcYfis~hR_mn£hR_VY6V_+dr}hpsfnagfihR_ hr agTVYetdrWZYXcd§agT+draHdrnihts~ah¤dReTmfiYXY
agTmfisWjfn_mfnWjfNQPdragfihR_@ Y*_VhPY1¯D}mdrfn_¤Tb¥agTVYetdrnfnmcdragfihR_,}mchRmiYXW fisYXcg¥sgfnWjfndrct
@YXa*Vs{umY6V_VYjd5Kfncgag+dretdrWZYXcdfnagTNfn_Kagcgfn_Vsgfie}+drcdrWZYXaYXcs5ih8etdraYu dra*d5}hpsfnagfihR_
sVeT-agT+draagTVYHhRDxyYeXa"FcdrWZY~fiscYXdraYu-ahjagTVYetdrWZYXcdFcdrWZY~K¥agTVY~TVhRWZhRqYX_VYhRVsI
W5dragcgf¼¯! #"R$ #"umY6V_VYs agTVY}hpsYTVhpsY}+drcdrWZYXaYXcsdrcYetdrniYu5Y1¯Dagcgfn_Vsgfie}+drcdrWZYXaYXcst
STVY}hpsgfnagfihR_-hragTVYhRDxgYeXa}hRfn_ba%'&fn_¤agTVYHetdrWZYXcdFcdrWZYfisumY6V_VYu¤b
&#(
)"
" &
dr_Vu-fnas}mchrxyYeXagfihR_,fn_-agTVY~uDfnqpfnagfNQtYu-fnW5drqY~K
*(,+.-0/21
&435(6+.-0/71
#"
"
&43 y
TVYXcY8+.-0/91;:<3~fisagTVYZ}mchrxgYeXagfihR_ WZhKumYXdReehRcuDfn_mq ah-agTVYfn_bagcgfn_Vsgfie}+drcdrWZYXaYXcs=D,STVY
qhdrÂhretdrnfnmcdragfihR_?fis*ah¤Wjfn_mfnWjfNQtYZagTVYjYXcgchRcYXaª YYX_ agTVYZhRVsYXcgYuuVdrad¤umYX_VhRaYu>?
FVs+drnn5agTVYI}hpsfnagfihR_§hrdHsYXahr@YtdragmcYshR_¤detdrnfnmcdragfihR_¤cgfnq dr_Vu§agTVYI}hpsfnagfihR_¥hr¢agTVY
sâdrWZY¬YtdragmcYsehRWj}mmaYu?b+dRe¡Kkl}mchrxgYeXagfihR_ dReehRcuDfn_mq£ah-agTVYZeXmcgcYX_KaY1¯Dagcgfn_Vsgfie§dr_Vu
fn_Kagcgfn_Vsgfie~}+drcdrWZYXaYXcs@ dRsumY6V_VYu¤fn_BAb+dragfihR_ 1]`_-hRcumYXcahjYX_VsgmcY*agTmfisWjfn_mfnWjfNQPdCk
agfihR_£ÂYWZhtYagTVYKfncgag+dretdrWZYXcd Ffn_mfnagfdrnnNfn_ DC " dr_Vu-WZh8uDf¼¬¤agTVYHfn_Kagcgfn_VsgfieetdrWZYXcd
}+drcdrWZYXaYXcsFfn_mfnagfdrnn
C Vsgfn_mqjd8fisg+drsYXcghRfn_mqjehR_bagchR=d¦TVYX_§agTVY{Wjfn_mfnWjfNQPdragfihR_
fisdReTmfiYXYu=oDagTVY{}+drcdrWZYXaYXcshr@agTVY{Kfncgag+dretdrWZYXcdfnn&Y E #"RoKagT+draÂfisÂagTVY"cYtdr}hpsYpo
dr_VuB
E
STmfis"}mch8eYssfis{fnnnVsgagcdraYuNfn_ !fnqpmcY5pYfnn¢sTVht fn_NagTVYH_VY1¯Da"}+drcdrqpcdr}mTVsITVhP
ah£}YXcy¬hRcgW agTmfisWjfn_mfnWjfNQPdragfihR_ Vsgfn_mq Kfisg+drsYXcghRfn_mq dr_ agTVY¥fn_KaYXcYsgasZhr{ehR_VsgfiumYXcgfn_mq
agTmfis{dr}m}mchdReT@
F GHJI9K5LM@I7NPORQRTSULMVWXLYJZPL\[2QR
STVYqhdr fisahNWjfn_mfnWjfNQtY§agTVYYXcgchRc^]_6`a?R]PNY§agTVYXcY1¬hRcY§umY6V_VY§agTVYYXcgchRcfn_7agTVY
fnW5drqY8b5K§agTVY~sfnWj}miY~cYXdragfihR_
b(c^`H?
STVY"WZhRagfihR_§hr@agTVY¬YtdragmcYsÂfn_5agTVY{fnW5drqYfisÂcYXdraYuahagTVY"etdrWZYXcdHYXih8eXfnaª{dr_Vu5agTVY
agfnWZYRdrcgfdragfihR_,hragTVY~fn_Kagcgfn_Vsgfie~}+drcdrWZYXaYXcs{b
(
[
agT+dra{etdr_¤Y*cYXcgfnagaYX_ dRs
*( fnagT (
F
; dragcgf¼¯ fiseXdRssgfietdrnnetdrniYu7fn_baYXcdReXagfihR_ W5dragcgf¼¯?hRcfnW5drqY¥ bdRehRmfdr_ fn_?agTVY8fisg+dr
sYXcghRfn_mq5ehRWjWHm_mfna`Â]`afisqpfnYX_
(
"!
/$#
\
]l!ÂYsg}YeXf¼¬¥dr_-Y1¯D}hR_VYX_Kagfdr¢umYehRm}miYuNumYeXcYtdRsYhragTVYYXcgchRc%bagT+drafis
b (#`&%.b
TVYXcY'%¤fis"d}mchR}hRcgagfihR_+dr¢eh8Y1®5eXfiYX_ba{agT+draagm_VYsagTVY~umYetdt¥cdraYpomÂY76V_+drnn¥qYXa
(#`&%(*)?b U4
TVYXcY+ ) fisagTVY}VsYXVumh~fn_KYXcsYhrW5dragcgf¼¯, - ) (W1-/.0 31324.-f¼3fis d{¬mnVcdr_m¡
W5dragcgf¼¯ 1
5 µ ±76 tIm µ ¹{98 ± :8 µ<; ±jµ(= ?> _b¡Kfn_VujhrVYtdragmcYetdr_jYÂehR_VsgfiumYXcYuZfnagTmfn_
agTmfisehR_KagchR@d dRssh8hR_-dRsÂYdrcY~drmiY*ahehRWj}mmaY~agTVYIfnW5drqY~ bdRehRmfdr_@]`_58Ö.:ªoVd
qYX_VYXcdrbFcdrWZYX hRcg¡~ahehRWj}mmaY
BA
fis@}mchR}hpsYu=DC{_agTVY hRagTVYXc¹sgfiumY
/
fis@sYXiumhRW uDf¼®5eXmna
ah~ehRWj}mmaYIdRsfnnVYsgTVht_Zfn_jagTVY_VY1¯8a sYeXagfihR_@STmfis fis hR_VYhragTVYdRuDpdr_badrqYs hragTmfis
dr}m}mchdReTZfnagTjcYsg}YeXaah~hRagTVYXc_VhR_Dklnfn_VYtdrc etdrnfnmcdragfihR_dr}m}mchdReTVYsP]ª_VumYYu YdrcYdrmiY
ah}YXcy¬hRcgWetdrnfnmcdragfihR_¥¬chRW dHdrcgqYIpdrcgfiYXaª¥hr¹}mcgfnWjfnagfnYs F}hRfn_bastoDnfn_VYsPo8eXfnceXiYsPomYXaepnn
fnagTmfn_¦agTVY£sdrWZY¤¬cdrWZYXÂhRcg¡pVmcgagTVYXcgWZhRcYpo"ehR_VsgfiumYXcgfn_mq pdrcgfihRVs5¡8fn_Vu hr}mcgfnWjfnagfnYs
fnagTmfn_,agTVYsdrWZY~etdrnfnmcdragfihR_NsaYX}-fis"drish}hpssgfnmiYp
FE G KTD[PIH KJ LYS.I XLMZTPL\[PQ
STVY fn_Kagcgfn_Vsgfie}+drcdrWZYXaYXcs hRmadrfn_VYujVsfn_mq*hR_VY fnW5drqYW5dtYpoRfn_j}mcdReXagfieYpopYXcguDf#"=YXcYX_ba
¬chRW«agTVY}+drcdrWZYXaYXcs!hRmadrfn_VYufnagTdr_VhRagTVYXc¢fnW5drqYadr¡YX_~¬chRWdr_VhRagTVYXc@8fiYX}hRfn_KatoRYXYX_
f¼@agTVY*sdrWZY{iYX_VstoKagTVY{sâdrWZYIetdrWZYXcdDo8agTVYIsdrWZY"FcdrWZYIqpcdrmYXcdr_Vu§agTVYIsdrWZY{etdrnfnmcdragfihR_
cgfnq"drcY VsYu ¬sYYagTVYcYsmnas=}mcYsYX_baYuHfn_HS¢dr 1¢]`a=fis=agTVYXcY1hRcYfnWj}hRcgadr_Ka¹ahehR_VsgfiumYXcd
Wmnagf¼klfnW5drqY"etdrnfnmcdragfihR_Z}mch8eYssagT+drafn_KaYXqpcdraYs fnagTmfn_5dIm_mf!KVYWjfn_mfnWjfNQPdragfihR_Z}mchKeYss
uVdradFchRWpdrcgfihRVsfnW5drqYst
STVYZm_VumYXcgn8fn_mqNfiumYtd,fisahNehRWj}mmaY§d-m_mf!bVYsYXahrfn_Kagcgfn_Vsgfie5}+drcdrWZYXaYXcsagT+drafis
ehRcgcYeXa{hRcIdrn!agTVYfnW5drqYs FflÝYpnoVhRc~drn¹agTVYetdrWZYXcdZ}hpsgfnagfihR_Vs 1w mqYXa~dr_Vu oK¡hRcuVdRsL84.:
ehR_VsfiumYXcYu agTmfisfiumYtdD STVYX ehRWj}mmaYu¦agTVY 6V_+dr*fn_bagcgfn_VsfieN}+drcdrWZYXaYXcs-dRsagTVYNWZYtdr_
pdrnVY§hr"agTVY5}+drcdrWZYXaYXcsjehRWj}mmaYu?hRcYtdReT fnW5drqYstoagTVY§etdrWZYXcd,}hpsgfnagfihR_VsjdrcYagTVYX_
cYehRWj}mmaYu cgat*ahagTVYsY_VYX fn_bagcgfn_Vsgfie}+drcdrWZYXaYXcst'C{mc~dr}m}mchdReTfis*uDf#"YXcYX_Kat*Y
ehR_VsfiumYXc dIKfisg+drmsYXcghRfn_mqseTVYXWZYagT+dra ehRWj}mmaYsagTVYWZhRagfihR_hr¥Kfncgag+drVetdrWZYXcdRsÂdr_Vu
agTVYpdrcgfdragfihR_Zhr&agTVY&fn_bagcgfn_Vsfie}+drcdrWZYXaYXcsagT+draT+dYah*Y agTVYsdrWZYÂhRc drn8agTVYfnW5drqYsP
+hRc fnW5drqYsto¢ÂY5agTVYXcY1¬hRcYT+dY
m_m¡K_VhP_?pdrcgfdrmiYsdr_Vu
C 2
C
Yb+dragfihR_Vs
FTVYXcY
C
fisagTVY_bmWHYXc{hr!YtdragmcYshRVsYXcgYu¤fn_-agTVYfnW5drqY1
]l5C!fis"agTVYHsYXaIhr¬YtdragmcYsIY1¯8agcdReXaYu-FchRW fnW5drqYpo&agTVYfn_KaYXcdReXagfihR_ W5dragcgf¼¯-VsYu
fn_¤agTVYHetdrnfnmcdragfihR_,}mch8eYssfisagTVYX_-qpfnYX_,b§agTVY~cYXdragfihR_
2
(
2
8m
fnagT
(
"! : : :
/
"!
: : :
/
: : :
9!
/
;Nfn_mfnWjfNQPdragfihR_,fisT+dr_VuDiYu¤Vsgfn_mqZagTVY~sdrWZYWZYXagTVhKumhRihRqp
2
(#`&%( ) 2 `H 2
`H
`H
yt
g% &X " {&'
E L J NL$J Q VT.
]`_ZagTmfis sYeXagfihR_@oKÂYVsY{dsfnWj}mnf<6+etdragfihR_§hr=agTVYWZh8umYX+VsYuZfn_5}mTVhRahRqpcdrWjWZYXagcgE8n:ª!Y
fnnhR_mn ehR_VsfiumYXcjfn_?agTmfisWZh8umYXagTVY5cdRuDfdruDfisgahRcgagfihR_@{hPÂYXYXcto ehR_VsfiumYXcgfn_mqhRagTVYXc
¡8fn_Vu7hrqYhRWZYXagcgfietdruDfisgahRcgagfihR_?fis}hpssgfnmiY ¬sYY }m}YX_VuDf¼¯ ~mmaY1¯D}YXcgfnWZYX_basjsgTVhP
agT+dra{sgVeT-WZhRcYHehRWj}miY1¯¥WZh8umYX@umh8Ys_VhRacYtdrnn§fnWj}mchtYagTVY b+drnfna`¥hragTVY~cYsgmnasP
@YXaÂVsumY6V_VYIb (W1 3 . agTVYIeh8hRcuDfn_+draYshr¢d}hRfn_bafn_§agTVY{etdrWZYXcdH¬cdrWZYp
STVYeh8hRcuDfn_+draYshr+agTVY }YXcs}YeXagfnY }mchrxgYeXagfihR_Zhr+agTmfis!}hRfn_Kafn_agTVY fnW5drqY}mdr_VY fis!qpfnYX_
K (W13
.
fnagT
(
(
yp
]ª& Y"umYX_VhRaY81X3agTVY}hpsfnagfihR_ZhragTVY"ehRcgcYsg}hR_VuDfn_mqH}mf¼¯DYXVfn_5agTVYuDfnqpfnagfNQtYuZfnW5drqYpoKagTmfis
}hpsgfnagfihR_¤fiscYXdraYu,ahagTVYeh8hRcuDfn_+draYs41N3 K
(!
+#"$
%'&
13
(
+#()
%'*
1X3
yt
TVYXcY
% &
dr_Vu
%'*
drcY~qYhRWZYXagcgfietdruDfisgahRcgagfihR_Vs"fn_bagch8uDVeYu-fn_-agTVY~etdrWZYXcdZWZhKumYXl STVYsY
uDfisahRagfihR_VsdrcYuDVYah§fnWj}YXcy¬YeXagfihR_Vs*fn_ agTVYiYX_VsYs~umYsgfnqp_dr_VudRssYXWmn,agTVYXcYVsg+drnn
drcY¤shRWZY¥}hpsgfnagfihR_+drYXcgchRcsZagT+draZT+dY¥ah Y¥adr¡YX_ fn_bah dReehRm_Kat
% &
dr_Vu
%'*
etdr_ Y
WZh8umYXiYu,dRshRniht
% &
1X3(+-,- /.
%'*
1X3(+-, - 0.
yt[
fnagT . ( `1 (W1=`1. 3dr_Vu -( .
. p
STVY76VY*}+drcdrWZYXaYXcs"ahZY~YsgagfnW5draYuNdrcY~agTbVs4(32!+ " +#(4 +5,76b
EF 8>NDO5J5S [49T6JR[2NLY [PQR J L\[2;:
Y{T+dY{ahehRWj}mmaY{agTVY{fn_baYXcdReXagfihR_¥W5dragcgf¼¯@ZagT+dra nfn_m¡8sÂagTVY{WZhRagfihR_=<>(W1 < 3 hr¢d
}hRfn_ba@ (W1X3fn_¤agTVY~fnW5drqY~ah>
$?. YVhRc"hR_VY*}hRfn_bato+agTmfis bdRehRmfdr_,fisqpfnYX_-b
(
"!
/ #
yX
TVYXcY4
"!
fisd= W5dragcgf¼¯¤dr_Vu
/
fisd=HW5dragcgf¼¯= · hR_VsgfiumYXcgfn_mq§detdrnfnmcdragfihR_,fnagT
}hRfn_bastoVagTVY*Fmn=fnW5drqYH bdRehRmfdr_,fisqpfnYX_,b¥agTVY>~W5dragcgf¼¯
(W1-@ @ : : : 3 .
y'\
C{_VYZhragTVYZfn_baYXcYsahragTmfisdr}m}mchdReT?fis~agT+drafnafis}hpssgfnmiYahNehR_VsgfiumYXcagTVY5+dRe¡Kk
qpchRm_Vufn_58fisg+drsYXcghRfn_mqV STVYfnW5drqYI bdRehRmfdr_
"!
agT+dra cYXdraYs agTVY{WZhRagfihR_§hr¢d}hRfn_Ka
fn_¤agTVYfnW5drqY~ahjagTVY~etdrWZYXcdjWZhRagfihR_,fisbmfnaY~eXdRssgfietdrT8:?8ÖV:¢dr_Vu¤fisqpfnYX_-b
(
( + "
+#(
yt
fnagT
( ` 2
"
! `=1
3
` 2
(
` ! `
yO4
mmcgagTVYXcgWZhRcYpoV¬chRW yt 1oVuDf#"=YXcYX_bagfdragfn_mq £dr_Vu 5¬hRciYtdRums"YXcg§YtdRsgfnn§ah
( `+5, -
. ` +5, . . . -
` +-, . . `+5,81-
. 3 .=-
y3m
YdrcY_Vht drmiYah*cgfnaYagTVYcYXdragfihR_ZagT+dra nfn_m¡agTVYWZhRagfihR_5hr=dI}hRfn_bafn_5agTVYfnW5drqY
ahagTVYHetdrWZYXcdWZhRagfihR_£dr_Vu¥agTVY~pdrcgfdragfihR_,hragTVY~fn_bagcgfn_VsfieHetdrWZYXcd}+drcdrWZYXaYXcst
¹YXa Vs 6V_+drnn_VhRaYagT+dratob]ª&ÂYumh*_VhRa dr_ba ah~ehR_VsgfiumYXc uDfisgahRcgagfihR_ZfnagTmfn_jagTVY"etdrWZYXcd
WZh8umYXlomagTmfisYK+dragfihR_,etdr_-Y~sgfnWj}mnf<6+Yu,dr_Vu-ÂY~cYX}mdReY
/
b
(
yt
fnagT ( 1<+#" + ( ! 7 3 .
STVYsYYb+dragfihR_VsdrcY agTVYX_cYtdrnnsgfnWj}miY fnagTcYsg}YeXa¢ah{agTVYYb+dragfihR_Vsg+drnnumYXcgfnY
hRceXdRssgfietdr _VhR_Dklnfn_VYtdrcWjfn_mfnWjfNQPdragfihR_?aYeTm_mf!KVYs¬sYY
}m}YX_VuDf¼¯ hRcd¥ehRWj}+drcgfishR_
fissgVY1
#{& '+y&'
STVY5}mchR}hpsYu7drnqhRcgfnagTmW T+dRsHYYX_ aYsaYu?hR_?RdrcgfihRVsfnW5drqYst¤Y§ehR_VsgfiumYXcYu?fnW5drqY
dReKmfncYujhR_ZhRmc etdrnfnmcdragfihR_Zcgfnq~fnagT5RdrcgfihRVsiYX_Vs ytD\rWjW ^!fnqpmcY"DÖd~dr_VuZrWjW j!fnqRk
mcY*D dr_Vu§fnW5drqY"¬chRW agTVY · drnfnmcdraYu¤]`W5drqpfn_mq ¹drhRcdrahRcg§dra · ; X!fnqpmcY~DÝe1
d e
!fnqpmcY5
F
2,SJCV
F
!>(,C,N*Q[G$E$(*-,@O RS
F
2,BJW& 93$n1(!@82?'>:
FTF
OQ.R
FTF O`9RU
F
2,\JC
F
!>(,SBU
T G QR5Q KJ LYS6XLYJZPL\[2QR)P.I9KT[2I
S¢h¥fnnnVsgagcdraYjagTVYYXT+d8fihRcHhr hRmcdrnqhRcgfnagTmW ÂYjT+dtY6Vcsga~}YXcy¬hRcgWZYuagTVYjetdrnfnmcdragfihR_
hrdrWjW#iYX_Vs*agT+dra{¬YtdragmcYsdr_ fnWj}hRcgadr_ba~uDfisgahRcgagfihR_Vsgfn_mq¥hRmc~etdrnfnmcdragfihR_cgfnqV~STVY
fnW5drqY"sgfNQtYfisa4R[pYT\DtDSTmfiscgfnqfisW5dRumY"hrtp*}hRfn_bas hR_jagTmcYYhRcgagTVhRqhR_+dr+}mdr_VYs¬sYY
!fnqpmcYD 1&C{_mnO4R5}hRfn_KasIdrcY8fisgfnmiYdr_VuNagTVYX_£ehR_VsfiumYXcYu,fnagTmfn_£agTVYetdrnfnmcdragfihR_
}mch8eYsst¤STVYfnW5drqY§ehKhRcuDfn_+draYshr"YtdReT7}hRfn_bafisehRWj}mmaYu7fnagT7d-}mcYeXfisgfihR_ hr"Dn
}mf¼¯mYXl
% µ ;98 ± 8¢! ;Fµ > !fnqpmcY £sTVhts¥agTVY£+dRe¡Kkl}mchrxyYeXagfihR_ hragTVY"ªWZhKumYX$#Nhrjd
smDkl}+drcgaIhr agTVYHcgfnq5¬chRW sgf¼¯-uDf#"=YXcYX_ba*8fncgag+dr¹}hpsgfnagfihR_£hr agTVYetdrWZYXcd5uDmcgfn_mq¥agTVYehR_Dk
YXcgqYX_VeYpg!fnqpmcYI[sgTVhPs agTVYWZYtdr_§YXcgchRcFfn_}mf¼¯DYX ehRWj}mmaYu§dra YtdReTfnaYXcdragfihR_§hr=agTVY
Wjfn_mfnWjfNQPdragfihR_ drnqhRcgfnagTmW-
s{cYbmfncYu£b£Yb+dragfihR_ 1oagTVYYXcgchRc*umYeXcYtdRsYsIY1¯D}hR_VYX_Dk
agfdrnn§]`_agTmfisHY1¯D}YXcgfnWZYX_Kato!ÂYZT+dtY 6VcsgadRssgmWZYu d¤uDfisahRcgagfihR_ ¬cYYZetdrWZYXcd¤fn_?hRcumYXc
ah,qYXad6VcsgaHqh8h8uqpVYsshragTVY5fn_Kagcgfn_Vsgfie§dr_Vu7Y1¯Dagcgfn_VsgfieZ}+drcdrWZYXaYXcst£STVY5ehRWj}mmaYu
shRnmagfihR_£fis{agTVYX_ cY6V_VYu ehR_VsgfiumYXcgfn_mq¤agTVYFmn!WZhKumYXlIYetdr_ sYYHagTVYfnWj}mchPYXWZYX_baHhr
agTVY*Wjfn_mfnWjfNQPdragfihR_£TVYX_-agTVYuDfisahRcgagfihR_-fisehR_VsgfiumYXcYu dC¬aYXcfnaYXcdragfihR_73mphR_ !fnqpmcYH[D
%'&
ÙrÌ ÏÑæÒ)(1ÌyÐ+*ÌãÉlÌ ÍªÒ-,1ÌyÚ.*@ÏÝÍlÙjÒ"ÐÕyÏÖÌyÚÍlÏ/RÕÕgÒæÌãɪÒ{ÏÖÚÒâÚÏÖÚrÛCÊÊâÉ ÐÌ`ÍÍlÏÑÚ0(C×
&
ÙCÌyÐÌÛrÒͪÒâÐÌãÍlÐ ÕyÊâÚͪÒâÏÖÚ
æ å21ÝÍlÏ4321ÖÌÏÖæÒ)(1ÌyÐ@ÊâÞDЬͪÒÍlÏÖÕÐÕyÌyÚCÌyÐ5*@ÏÝÍlÙ~ÒÕyÕyåPɪÒÍlÌÏÖÚCÞçÊâÉlæÒÍlÏÖÊâÚ~Ò-6pÊâåC͹Ê76tˬÌyÕ`Í81ÖÊÕgÒÍlÏÖÊ1ÚrТÏÖÚ:9-;Â×
0 50 100 150 200 250 300
0 50 100 150 200 250 300 350
residual
d
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
50 100 150 200 250
residual
!fnqpmcY[ .S,3B,.` 6;Z.,.iOg,9+3*NG$,.L'>,GI$GI*QE%!B+JXE,. C,.! R]O .R
C,.K3! E OQR C,.! E
480 500 520 540 560 580 600 620
0 50 100 150 200 250 300
px py
d
200 250 300 350 400 450 500
0 50 100 150 200 250 300
xc yc
-5e-07 -4.5e-07 -4e-07 -3.5e-07 -3e-07 -2.5e-07 -2e-07 -1.5e-07 -1e-07 -5e-08 0
0 50 100 150 200 250 300
K
e
!fnqpmcY~H"(&TJj'>,S13!`3*N9T6(K
F
,NC,.%*TO RR+ " $G +!( OQR G- O`9R +-,
¢fnqpmcY¥sTVhtsjagTVY¤YXhRnmagfihR_ hrIagTVY¥fn_bagcgfn_Vsgfie-}+drcdrWZYXaYXcs5htYXc5agfnWZY¤uDmcgfn_mqagTVY
Wjfn_mfnWjfNQPdragfihR_5}mchKeYssP¢STVYWjfn_mfnWjfNQPdragfihR_¥drnqhRcgfnagTmWfis!dRsadr_Vujumh8Ys_VhRacYKmfncYW5dr_b
fnaYXcdragfihR_Vs drchRm_Vu tpfn_,agTmfisY1¯8}YXcgfnWZYX_ba{TVYXcY'% ( :Hdr_Vu-TVYXcYagTVY*fn_mfnagfdr¹qpVYss
hRc!agTVYÂshRnmagfihR_fis!YXcgHuDf#"=YXcYX_ba¬chRWagTVY ehRcgcYeXa shRnmagfihR_ 1!S¢drmiY~ sgTVhtsagTVYÂ_bmWYXc
hrDfnaYXcdragfihR_Vs@cYKmfncYu~ahWjfn_mfnWjfNQtY agTVYYXcgchRc@fnagTcYsg}YeXa@ahagTVY}mchR}hRcgagfihR_+drKeh8Y1®5eXfiYX_ba
% ¬sYYYK+dragfihR_ U4 agT+draehR_bagchRisagTVY§umYetd cdraYpNSTVY¤drnqhRcgfnagTmW YX_VumYu?TVYX_7agTVY
uDf#"=YXcYX_VeYYXa`ÂYYX_agTVYehRWj}mmaYuYXcgchRc bNdra{fnaYXcdragfihR_ dr_VudraIfnaYXcdragfihR_
fis"iYss
agT+dr_
1¹STVYT6Vcsga!ehRnmWj_ "ªzIzo # @sgTVhts!agTVY _bmWHYXchrmcYbmfncYufnaYXcdragfihR_Vs¢FchRW dr_
drcgmfnagcdrcgfn_mfnagfdrnfNQPdragfihR_ FagT+drafisHagTVYjfn_bagcgfn_Vsgfie5}+drcdrWZYXaYXcsHqpfnYX_?b agTVYZW5dr_KDdReXagmcYXc
dr_Vu d£}hpsgfnagfihR_ ehRcgcYsg}hR_VuDfn_mqah agTVY6VcsafnW5drqY¤hR_D!fnqpmcY p STVY¤sYehR_Vu ehRnmWj_
"`S¢hpsetdr_mf {zIzo # ehR_VsgfiumYXcsagTVYZsâdrWZY5drnqhRcgfnagTmW¸mmaagTVY5etdrnfnmcdragfihR_ }+drcdrWZYXaYXcsdrcY
fn_mfnagfdrnfNQtYu Vsfn_mq§agTVYS¢hpsetdr_mf¼k +drmqYXcdRsdrnqhRcgfnagTmW 8Ö[.:ªSfnWZYsdrcYfn_£WjfnnnfisYehR_Vumsdr_Vu
drcY~hRmadrfn_VYuNhR_Ndw!YX_bagfnmW#]] bp; Q
qbdrfn_ zIzo S¹hpsetdr_mf¢agTVYX_-zIzo
% fnaYXcdragfihR_Vs agfnWZY FWZs fnaYXcdragfihR_Vs
Dn tD mpp pO4
DÖ\ p[ \Dt[ $m
D\ \ p;m p[
D4 p '\p '\
t[ p[ 4
S!drmiY5A +"(,.%2,.9+,fC, ?!>fV,N*+6;,+9N E %MO8*-,+,],5&(3!ORQR':
5 µ<6 ± 0 ±76 ±,;t ± > ]`_mfnagfdrnfNQPdragfihR_ZfishrFaYX_ZehR_VsfiumYXcYu5dRs dI}mchRmiYXW©hRcagTmfis¡Kfn_Vujhr
dr}m}mchdReT@ ]`_VumYYu=o_VhR_Dklnfn_VYtdrchR}magfnWjfNQPdragfihR_Vs WZYXagTVh8u5sgVeTdRsdrVssyk IYXahR_@o&IYXahR_Dk
|Idr}mTVshR_5hRc @YXYX_bYXcgqRk?;£drc K+drcgau¤drcY"sgm}m}hpsYu5ahcYKmfncY{d*qh8hKuZqpVYssÂhr=agTVY"fn_mfnagfdr
}+drcdrWZYXaYXcsPS¹sdrf}hRfn_KashRma agT+dra agTmfis 8fihRdraYs agTVY}mcgfn_VeXfn}miY*hr¢drmahRW5dragfihR_@ IhtÂYXYXcto
hRmc*aYsgas~sTVhtohRcIagTmfis*}+drcgagfieXmdrcHdr}m}mnfietdragfihR_@o¹agT+dra~fnWj}hRcgadr_KaYXcgchRcs*W5d£Yfn_bagchrk
uDVeYu¥fn_¥agTVYIfn_mfnagfdr=YeXahRchr¹}+drcdrWZYXaYXcst STmfisfiscYX}hRcgaYu¥fn_¤S¢drmiYdr_Vu¤[D|"Ysmnas
sTVht agT+dra~YXYX_fnagT£fnWj}hRcgadr_baYXcgchRcs*fn_ agTVY}hpsYHfn_mfnagfdrnfNQPdragfihR_@o¹agTVYjdrnqhRcgfnagTmW ehR_Dk
YXcgqYsHahtdrcuagTVYZsdrWZY5shRnmagfihR_@-Iht YXYXcto¹hRc*YX©fn_mfnagfdr }hpsgfnagfihR_VshRcHshRWZYZfn_mfnagfdr
fn_Kagcgfn_VsgfieI}+drcdrWZYXaYXcsto8fna etdr_m_VhRaehR_KYXcgqY Fnfn_VYIfnagT · dCFaYXc agTVY{fn_mfnagfdrWZYtdr_¥YXcgchRcfn_
agTVYHadrmiYs 1 !fnqpmcY \ZuDfis}mdDs"shRWZYhragTVYHKfncgag+dr!etdrWZYXcdZagcdCxgYeXahRcgfiYs{YXa`ÂYYX_£agTVYXfnc
fn_mfnagfdr¹}hpsgfnagfihR_,aht drcu,agTVYXfnc96V_+dr@}hpsgfnagfihR_@
]ªacYXW5drfn_VsagT+dratoVfn_¤}mcdReXagfieYpomfn_mfnagfdrnfNQPdragfihR_,fisagTVYX_-_VhRacYtdrnn¤d¬m_VuVdrWZYX_badr¹fissgVYp
· hR_VsfiumYXcgfn_mqagTVY¥W5dr_bDldReXagmcYXc§uVdradNhRcjagTVY¥fn_bagcgfn_Vsfie¤}+drcdrWZYXaYXcs5}mchPKfiumYsPo f¼I_VhRad
qh8h8u?etdrnfnmcdragfihR_@odNqhKh8u7qpVYssjhr"agTVY¤shRnmagfihR_@7mmcgagTVYXcgWZhRcYpo agTVY§}mchRmiYXW etdr_ Y
¬mnnshRnYuf¼Vfn_mfnagfdrnfNQPdragfihR_fis¹}YXcy¬hRcgWZYuagTmchRmqpTZdsgfnWj}miYÂnfn_VYtdrcetdrnfnmcdragfihR_jdrnqhRcgfnagTmW
sVeTNdRsagTVY~S¢hpsetdr_mf¼k +drmqYXcdRs~drnqhRcgfnagTmW-
5 µ !m ; µ<6 ´ ;98 µ 98 ± b%; !CD ;Fµ<6 % µ ;98 > VhRc¤ehRWj}+drcgfishR_«fissVY, Y
ehR_VsfiumYXcYu?pdrcgfihRVsjdr}m}mchdReTVYs 5agTVY¥S¹hpsetdr_mf¼k +drmqYXcdRsdrnqhRcgfnagTmW 8Ö[:ªoagTVYS¢sdrfÝsetdr¼k
fnmcdragfihR_7WZYXagTVh8u 8Ö.:ªod¥¬mn setdriY5_VhR_Dklnfn_VYtdrcetdrnfnmcdragfihR_ F_VhR_Dklnfn_VYtdrchR}magfnWjfNQPdragfihR_ fis
}YXcyhRcgWZYuHVsgfn_mqIagTVY @YXYX_bYXcgqRk?;£drc K+drcgauZdrnqhRcgfnagTmW ¬chRW agTVY nfnmcdrcg1ophRmc¢8fncyk
ag+dr&Kfis+dr+sYXcghRfn_mqFzIz]o drnqhRcgfnagTmW Vsfn_mqHhR_mnjagTVYcdRuDfdruDfisgahRcgagfihR_ FflÝYpnoKagTVYWZh8umYX
}mcYsYX_baYufn_o8YeXagfihR_?[Dn1o@hRmc~zIz]oNdrnqhRcgfnagTmW¸Vsgfn_mq¤agTVYFmnWZh8umYX dRs*}mcYsYX_baYub
YX_mq 8np: ¬sYY*agTVY }m}YX_VuDf¼¯ 1
ÌårÐÌ ÍlÙrÌ Ê-3CÍlÏÖæÏyÌgÛXÌãÉlÐÏÖÊ1Ú ÊâÞ?ÍlÙrÌÏÑæ301ÖÌyæÌyÚͪÒÍlÏÖÊ1Ú 6
^ Ì ( Ï141ÖÐÊ1Ú$Ò1ÒÏ1ÑÒ)621ÖÌÒÍ
! #"$% #"&' (*)'+-,./0/$1234)(5 67&98
fn_mfnagfdr¹}hpsgfnagfihR_ fn_mfnagfdr@WZYtdr_
S=¯ S S*Q |¯ |" |CQ YXcgchRc
D\ D\ p tp tDtpD\
D\ p\ p tp \r!\D\D
k p\ p tp \\D\Z4
k p\ p k`p mp 4RZ48np
k VppÖ[Z4
k D\ \p k\p p [p;m;mDÖp
k D\ \p k\p [p VtZ4848
[ [ [ ppDÖZ4 ·
fn_mfnagfdr¹}hpsgfnagfihR_ 6V_+dr@WZYtdr_
DÖp[;m DÖp[p pÖ;m [Db DpnmZ4 p3mDÖZ4 DÖp[p;m
S!drmiY +",.%2,.9' @C,N*QUJX$Y"`&3*B1;!$6;.*N! JW'>,"1%'&(i93
F
,.H4!>*Q,
*+>( '>,T;k6;.*N![J)'>,[93
F
,.T$GB'>,SV,N*+&132
F
,3],.`:g<U>,_;kH3'813*N9
93
F
,.K 6(
F
,NE,.%* V, !>(.*-, 6&$n*+>,+G !>, 93
F
,.
F
$&+J`9N'&V,. : .*Q ,J '>,
Q$,S*A'>,\e ($H9+
F
6$&3C,+G 6;.*-,;:WOY
F
,3$3*W'>( '>,]$2`'>
F
GGP. 9++"(,.%2,R
−2
−1 0 1
0 −0.5 1 0.5
2 1.5
−2
−1.5
−1
−0.5 0 0.5
x y
z
!fnqpmcY\ W1%'&(93
F
,.KT'CcC,+9NC$/ TE YVGf'>,]9+
F
6&3E,+G@6;.*-,
fn_mfnagfdr@fn_bagcgfn_Vsfie}+drcdrWZYXaYXcs fn_mfnagfdr¢WZYtdr_
+ " +#( +5, YXcgchRc
[p\ \p pp pp pZ48\p[
[p\ \p pp pp kY1k` ppDnmr
pp pp \prVÖp[
pp pp kY1k` prVnm\
pp pp kY1k\ tpDpÖ ·
tpp tpp ppDÖ[p
\ppp \ppp \pp \pp Z4\Z48Ö[
ehRWj}mmaYu,fn_bagcgfn_VsgfieH}+drcdrWZYXaYXcs 6V_+dr@WZYtdr_
[Z4RDnt r$mDnmr \\Z48Ö[;m \\pDÖp[ klVÖpRY1k`Z4 DÖp[p;m
S!drmiY[ +",.%2,.9' _E,N*Q^J8 "8&3*\;kH!813*N9]6(
F
,NE,.%*?G !>,_V,N*+&132
F
,3$
,. V : <?>, ;kL6;.*N!P*T9+3*QQBO ( 1;`
: : 9:P3_$G ( 1
3QR':
;*Qn(, J?!>(, Q,@*]!>(,Be (9+
F
6&3E,+G73'8*N9 6(
F
,NE,.%*3: O F
,3$3*_!>('>,
$2$8'>
F
GIG 9++",.%2,R
rWjW iYX_Vs +#" + ( ! WZYtdr_,YXcgchRc
S¹hpsetdr_mf ppDb ppDnt [;mDpÖ;m p[\Dp DÖ;mp
S¹sdrf \\Z48n3m \\pD4\ [Z4RDnt r$mDÖ\ DÖp[Dt
; \\Z48Ö[;m \\pDÖp[ [Z4RDnt r$mDnmr DÖp[p;m
z{z]o \\Z48Ö[;m \\pDÖp[ [Z4RDnt r$mDnmr DÖp[p;m
z{z]o 3Fmn \\;mDÖD \\Z484R \\;mDÖD \\Z484R DÖppD
S!drmiYY F
68*-?J 933!?$2$8'>
F
*LJ8
F_F
,.3*3: #, 9+3*NG,.I'>,W< .*-93;b
$&32,.*28!>
F
'>,@<i*;L$2$8'>
F
JE& *-93$,#+b-,3 933!MO .,'"(,.$.,.V2.b
5&(%EG]63!
F l33!R
& $2$8'>
F
$GT& $2$8'>
F
?!>W'>, JC&$393
F
,.
F
HG$,.
S¢drmiYsgTVhtsÂetdrnfnmcdragfihR_5cYsgmnas Vsgfn_mq~agTVYsYpdrcgfihRVsÂdrnqhRcgfnagTmWZstY"umh~_VhRa sgTVhP
agTVY~uDfisahRcgagfihR_¤}+drcdrWZYXaYXcs"sgfn_VeYIagTVY~ehR_VsgfiumYXcYu¤etdrWZYXcdZdrcY*_VhRadrndDsagTVY*sdrWZYp S!dCk
miY7\sgTVhtsagTVY 6V_+drWZYtdr_¤YXcgchRctoDagTVYIsadr_VuVdrcu§umYX8fdragfihR_-dr_Vu§agTVY{W5dC¯DfnWHmW YXcgchRc hRc
agTVYÂagTmcYYÂfnW5drqYs!hr!fnqpmcY{dr_VuagTmcYY uDf#"YXcYX_Ka!WZYXagTVh8umst C{_mn_VhR_Dklnfn_VYtdrc dr}m}mchdReTVYs
YtdragmcYjcYsgmnasHfnagT7d¤sD®5eXfiYX_badReeXmcdReX/C{mcWZYXagTVhKuetdr_?YldhRcdrmnehRWj}+drcYu
fnagT agTVY§S¹sdrfdr}m}mchdReT dr_Vu?qpfnYssdrWZYcYsgmnasagT+dr_ agTVYFmnsetdriY_VhR_Dklnfn_VYtdrcjWjfn_Dk
fnWjfNQPdragfihR_ Vsgfn_mq£agTVY @YXYX_bYXcgqRk?; drc b+drcgau drnqhRcgfnagTmW ¬sYY }m}YX_VuDf¼¯ HYXYX_ f¼agTVY
umYXcgfnpdragfihR_,hragTVY bdRehRmfdr_NfisYXcg§uDf#"=YXcYX_bat
WZYXagTVh8u eXcgfnaYXcgfihR_ trWjW rWjW e1k`pDtD
WZYtdr_ D44.m Dn3mp[p DbZ4R
S¢hpsetdr_mf sau¤umYX DÖZ44R DÖp\Z4 D\ppp
W5dC¯ [DÖ[rb[r DÖpp[p p48tD
WZYtdr_ Dnt\ DÖp[D Dnptp
S¹sdrf sau¤umYX DÖ\p Dntr DÖpp[p
W5dC¯ DÖpZ4 p4R;m DÖ[ppp
WZYtdr_ DÖpr DÖp[D Dnp'\p
z{z]o sau¤umYX DÖ\p[ Dntp DÖprb
W5dC¯ DÖ[p;m pnmDt DÖ[Dpt
S¢drmiY\ W F
6(8*-f.,Ng,+,.933!
F
,N!>(HG;*
TF G KTD[P KJ LYS,XLMJZTL [PQR
+hRc5YtdReT fnW5drqYpoÂagTVY¤}+drcdrWZYXaYXcs§drcY¤fn_mfnagfdrnfNQtYu fnagT agTVY¤}+drcdrWZYXaYXcsehRWj}mmaYu hRc
hR_VY¥hr"agTVY§fnW5drqYsVsgfn_mq agTVY§}mchKeYss}mcYsYX_baYu fn_ agTVY5}mcYX8fihRVsjsYeXagfihR_@STVYWHmnagf
fnW5drqYetdrnfnmcdragfihR_ }mch8eYssIfis{agTVYX_ ehR_VsgfiumYXcYu£ah¤dReTmfiYXYjd5qpihR+drhR}magfnWjfNQPdragfihR_@HSTmfis
dr}m}mchdReT drnihts~ahqpcYtdragn,fnWj}mchPYs~agTVY b+drnfna`Nhr agTVYetdrnfnmcdragfihR_ ¬sYYS!drmiYZ 1]l
agTVYqpihR+drWZYtdr_ YXcgchRcW5dt fn_VeXcYtdRsYpo agTVY5qpihR+drWZYtdr_ YXcgchRcfn_?WZhpsg?hr"agTVY5fnW5drqYs
umYeXcYtdRsYs~sgfnqp_mf<6+etdr_bagn dRsIetdr_ YsYYX_ hR_ !fnqpmcYjDmmcgagTVYXcgWZhRcYjagTVYehR_bYXcgqYX_VeYZhr
agTVYdrnqhRcgfnagTmW fisKmfnaY*dRsa iYssagT+dr_ tfnaYXcdragfihR_Vs ¬sYY]!fnqpmcYH 1
fnW5drqY + " +#( +-, cYsgfiuD+dr
[;mpDÖ[pZ4 p[rVntp pXbDb p'\pDÖp k`DÖ[p\RY1k`;m DÖpp[;m
[;mp[Dn'\Z4 p[rVÖZ4R p'\pDb p'\DpÖ[;m k`Dntp[RY1k`;m DÖppr
[ [;mp[DÖ[\p p[Z48V'\ p'\rVÖp p'\rV4R k\DÖ[ppRY1k`;m DntZ4R
[;mp[DÖ\p p[\D4Rp p'\pD4R p'\pDÖp k`DÖppRY1k`;m DÖ;m\p[
\ [;mp[D$m\ p[pD48t p'\pDÖp p'\p[DÖp k\D4RpRY1k`;m DÖpr$m
drn [;mp[D/4R p[rVnmpD pXbD\p p'\pDÖr k`DÖ;mDY1k`;m DÖp;m\
S!drmiY S (G
F
&
F
2, 93$n1(!B&3*N132BG ,.V,.3 ",7 *WJ)'>,[*
F
,\93
`25: #,B93$3*-,.%"(,B*-
F
,_"`$;X @f!>,B,N*Q!
F
3!BJ !>(, 3'8*N96(
F
,NE,.%*!>3
".G$C,N* '>,B,+9+,N*Q*NX TEB9+3*NG,.
F
&16,T
F
32,N*\ST93$n1(! 6V09+,N*Q*3: