HAL Id: inria-00072789
https://hal.inria.fr/inria-00072789
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- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Object Rotation Axis from Shading
Jürgen Stauder, Henri Nicolas
To cite this version:
Jürgen Stauder, Henri Nicolas. Object Rotation Axis from Shading. [Research Report] RR-3865, INRIA. 2000. �inria-00072789�
ISRN INRIA/RR--3865--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
Object Rotation Axis from Shading
J¨urgen Stauder, Henri Nicolas
No 3865
January 2000
TH `EME 3
)+*,.-0/2143576+89:/;,;<>=?/;1,A@CB(@ED2F0GH6JI
KMLONQP#R&STVUXW;YZRA[]\2^_Y]`ba2W?LOa2PcP#RQdEP\e^fLg`bWOR2h
`iP\kj2RAlAhnmOaeWOWOoARAlAhO^a2WOWg\k`pl]lZ\eWg^R.l
q>[]akrsRQYtKuRAPc`p^Ql
vt\kwOw+ae[ZYtmxR[]RA^fLORA[]^fLORyW0zSe{2|2}~TV;\kWng\e[Z?eee&T wg\ejeR.l
?:AQ7Oee UXWY]LO`pl[]RQw+ae[ZYAh\kWlZ`bP#wOiR\eWgmWORQR.lsY]`bP\7Y]ae[`pl?mxRAeRAba2wJR.mae[Y]LORYZ`ibY$\kWgjeiR
aeY]LOR[]akYf\7YZ`iaeW\7x`plak\[]`bj2`imaeOrsRA^_YY]Lg\7Y~`il`iibOP#`iWg\7Y]RAmn\wJa2`bW;Y&i`bj2L2YlZaeg[]^QRe(KMLgR#YZ`i Y
\eWOjeiRmxR¡gWgRAl~YZLORa2[Z`iRQW;Y]\kYZ`iaeW¢aeCY]LOR£k¤¥wO[]akrsRA^YZ`iaeW¦aktY]LORSe¤aeOrsRA^_Y#OWO`bY[ZaeY]\kYZ`iaeW¦\kx`ilc`iW
Y]LOR£`iP\kjeRwOi\eWOReyKMLOR§\kWOj2bR`pla2WOR£akY]LOR§|¨mxRAje[]RQR.lak[]RQRAmOaeP©ak&Se¤ª[Z`ije`pmPcaeYZ`iaeWyUHY
`pl#\kWg\eb;Y]`i^A\kibl«Lga7W¨YZLg\kY~Y]LOR$YZ`i Y\kWgjeiR$^A\kW¢+R(R.lsY]`bP\kYZRAm£rsgl«Y#nRxwOiae`bYZ`iWOj§YZLOR$RQ[][]ae[ae
P#aeYZ`iaeW£^a2P#wJRAWglZ\kYZ`iaeW+RYs¬RQRAWYsa?`bP\kj2RAltmOOR&Y]a?aeOrsRA^_Yl«Lg\2mx`iWOjgCvtRA®;O`i[ZR.m`bWOwgxYmg\7Y]\?\k[]R
\mxRAWglZRmx`ilZwOp\e^QRQP#RQW;YeR.^_YZa2[¡gRQpm§ae[P#akY]`ba2W¨^aeP#w+RQWgl]\7Y]`ba2W\eWgm\k¤O`iWg\e[Z£a2xrsRA^Y&P\elZ¯+
KMLgRR.lsY]`bP\7Y]ae[\el]l«gPcR.lu[]`ije`pm°hk±u\eP+RQ[ZYZ`p\kW&a2xrsRA^Y]l` Y]L~R.®2J\kibmx`plsY][Z`iOxY]RAmcl«g[«²\2^RWgae[]P#\eil:\el
¬RQi\el\#l]^RAWOR`bibgPc`iWg\kYZ`iaeWn?\#mx`il«Y]\eW;YtwJa2`bW;Yb`ijeL;YtlZaeO[f^R\kWJm(\ePO`iRQW;Yb`ijeL;Y.´³´xwJRA[Z`iP#RQW;Y]l
`bYZLLOR.\emndX\kWJmndHlZLOa2OimORQ[c;`pmxRAa¢l«R.®2gRQWg^QRAl#n`bRAim®;g\ki`bY]\7Y]`b2RQi¢jeanaxm¦[ZR.l«ObY]lcY]Lg\7Y?\e[ZRmxRA[Z`ieR.m
gbi¢\kxY]aeP\7Y]`i^A\kib2µwJRA[«a2[ZP\kWJ^R?\eWg\kixl«`pllZLOa7tlY]Lg\7Y~YZLOR$RAl«YZ`iP\7Y]ae[¬ae[]¯xl&+RAl«Y~`bCi`bj2L2Y
^QaeP#RAl[]aePª;`iRQ`iWOjmx`i[ZR.^_Y]`ba2W¶`iWg\ebieh:\kW¨\kwOwgb`p^Q\kYZ`iaeW£Y]an`imxRAa`bWgmORx`bWgjlZLOa7tlY]Lg\7Y&LOR.\emnd
\eWgmndXl«LgaeOpmxRQ[lZRA®;ORAWg^R.l^Q\eW·JR£Obi\kxY]aeP\7Y]`i^A\kibl«RAjeP#RQW;YZR.m`bW;YZa¢YZRQP#w+ae[f\kClZRQj2P#RQW;Y]lae
LgaeP#aej2RQWORAaeglaexrsR.^_YtP#akY]`ba2W´µymxRQP#a^A\kW$JRa2OWgm\7YL;Y«Y]w¸º¹2¹`b[]`plZ\g[¹.Y]RQP#`i^Alf¹k¤CRAP#a2lA
»¼;½0¾]¿À 7Á:e¦Â akY]`ba2W(R.lsY]`bP\7Y]`ba2WhOlZLg\emx`iWOjJhx`bibgPc`iWg\kYZ`iaeW
ÃpÄ°ÅZÆHÇAÈMÅÉ:ÊiÆHËEÌÍ
ÎAÏÐÐkÑ_ÒÓEÔsÕÖ×جÙJÏÒ²Ñ_ÐkÔZØfÚ&ÛØfҲܺÔ>Ý0ÏҲܺÔ>ÞuÔsßàÔZØfÒ²áHâãxÒàØfܺÚܺÚätå:ÒàØfÚÓ
Unit´e de recherche INRIA Rennes
Rt[f\kwOw+ae[ZY´wO[Zo.l«RAW;YZROWORWgaeO2RQibRP#oY]LOaxmxRClZ`bP#wOiRw+RQ[]PcRQY«Yf\kW;YR.lsY]`bP\7Y]`ba2WmxRi\
wg[ZaersRA^_Y]`ba2Wmg\kWgl°iR>wOp\kW`iP\kjeR>mOR´\7xR>mxR>[]akYf\7Y]`ba2WS2¤ mOWaeOrsRY[]`bj2`imxR´`iibOP#`iWOo>wg\e[OWOR>lZaeg[]^QR
w+aeWJ^_YZgRQibR2 RYZYZRwO[]akrsR.^_Y]`ba2Whmxo¡JWO`bRwJ\k[OWOR\eWOjeiRcYZ`i Y.h^Qae[][ZR.l«w+aeWgm$OW¢mxR.ll«`bmxRAje[]oAlmxR
i`iJRA[«Y]o¬mx~P#aeO2RQP#RQW;Y´Se¤[Z`ije`pmxR2AUXnRAl«YPca2W;YZ[]o`p^`x®;OR¬^QRY´\kWOj2bRwJRAxYY][ZRRAl«YZ`iPcoMRAWRxwOiae`bY]\eW;Y
RQ[][ZRAO[mxR^aeP#w+RQWgl]\7Y]`ba2W?mxR Pca2OeRAP#RQW;Y`plZlZOR mORAl>7\e[Z`p\7Y]`ba2WglmaeP~O[f\kjeRtmORCa2xrsRYM^aeWJl«`pmxoQ[]oe
RQY«Y]R$\ewOwO[]an^fLgR(xY]`bi`pl«R$OW^fLg\kP#w¦mORQWglZR$mOR(2RA^_Y]RQO[fl#mxR$;`bYZR.lZlZR$RYOW P\2lZ®;ORk¤maexrsRQY]l
wg[Zo.\kp\kOiRQP#RQW;Y^Q\ei^QObo.lQx¤ RwOiglQhxp\cP#oYZLganmORwO[Za2wJa;l«oAR lZOwOw+a2lZR®2gRiRAlMa2xrsRYfll«a2W;YM[]`ije`pmxRAl¬RY
±\kP~JRA[«Y]`bRAWgl\eRA^mxRAlWgae[]P#\ebR.lmORMlZO[Z²\e^QR¬mx`pl«YZ[]`bOgoQRAloAj2\ebRAPcRAW;YA7±`biiOP#`bWg\kYZ`iaeWcRAl«YlZOwOw+a2lZoQR
lZR#^aeP#w+a2lZRQ[mOWOR#iOP#`bNA[ZR#\kP~O`i\eW;YZRcRY&mOWORl«a2O[f^RiOP#`iWORQglZR#wJa2Wg^_Y]ORQiiR(`iWx¡gWO`E RQY«YZR
\ewOwO[]ax^fLOR¬\oQYZo7\ki`imOoQRl«g[>mxRAl>lZoA®;ORAWg^R.l´mxR¬YsnwJRn`pl«`iax^aeWOoQ[]RQWg^QRw+aeO[´iRAl]®;ORQiiRAl´mxR.l´[Zo.l«g Yf\7Y]l
l]\7Y]`il«²\k`pl]\kW;Y]la2W2Y>oQYZota2xYZRAWnglQtWORC\eWg\kixl«RmR ^A\e^`bYZoP#a2W2Y][ZRt®;ORRAl«YZ`iP#\kYZRAO[>P\k[f^fLORP#`bRAx
lZ`´p\$mO`b[]RA^YZ`iaeWmxRi\lZaeO[f^R#iOP#`bWgRQglZRl«RlZ` Y]OR#wO[]an^fLgR#mOR\7xRmxR#n`ilZomxR#i\^A\kP#oQ[f\O RQY«YZR
P#oQYZLOaxmxR\oQYZo\kwOwgb`p®;OoQR\kwg[Za2ObNAP#RmxR`iWgmxRQx\kYZ`iaeWn`pmxoQaJhx^R®2g`0\wJRA[ZP#`plMmxR P#a2W2Y][ZRA[¬®;OR
mORAlulZoA®;ORAWg^R.lmxR´Ysnw+R>n`ilZ`ian^QaeWxoA[ZRAWg^R´w+RQgeRQW;Y!QYZ[]RlZRQjeP#RAW2Y]oQR.l°RQW;YZ`iNQ[]RQP#RAW2Y:\kxY]aeP\7Y]`i®;ORAPcRAW;Y
RAW&l«RAjeP#RQW;Yfl°YZRAP#wJa2[ZRAilmxR>P#aeO2RQP#RQW;YuLgaeP#aej2NQWOR>\e&l«RAWglumxR\kWgjeiRmxR´Y]`bbYA¤C`#"0oQ[]RQW;Y]l[Zo.l«g Yf\7Y]l
RQxwJoA[Z`iP#RQW;Y]\ex(lZaeW;Ytmx`ilZw+aeWO`iOiRAl$c\emO[ZR.lZlZRe¸nL;Y«Y]w¸º¹2¹`b[]`plZ\g[¹.Y]RQP#`i^Alf¹k¤CRAP#a2lA
% À . ¾ & ³lsY]`bP\7Y]`ba2WmxRP#aeO2RQP#RQW;YAhxa2PO[f\kj2Reh;`biiOP#`bWg\kYZ`iaeW
* +
~ u%-,( &
KMLORR.lsY]`bP\kYZ`iaeWcakJ[]`ije`pma2xrsRA^Y>PcaeYZ`iaeWc[ZRAi\kYZ`ieRAbY]a\^A\kwxY]O[Z`iWOjn`imORQa^Q\ePcRA[]\ `pl´\^Qi\2lZlZ`p^Q\k
wg[Za2ObRAP0KMLgRcP#aeYZ`iaeW`plYZLgRQ[]Rae[]R~mORAl]^[]`b+RAm§n£Se¤ []akY]\kYZ`iaeW\eWgmSe¤ YZ[f\kWJl«p\7Y]`ba2WhJ`ERe°n£l«`b
mORQje[]RQR.l¬ak:[ZRARAmxa2P$
KMLORSe¤ aeOrsRA^_Y$P#akYZ`iaeW \kWgm Se¤ aexrsR.^_Y$lZLg\ewJR^Q\kW JRRAl«YZ`iP\7YZR.m·[]aeP k¤ w+a2lZ` Y]`ba2Wgl?ae
^Qae[][ZR.l«w+aeWJmx`bWgj `iP\kjeRMw+ae`iW;Y]l´`iWl«g^A^RARAmx`iWOj`bP\ejeRAlae0\n`pmxRQa`iP\kjeRlZRA®;ORAWg^RM`iW^Q\elZRMak0¯nWOa7W
\eWgm#OWO¯nWOa7W`iW;YZ[]`bWJl«`p^C^A\kP#RQ[f\wg\k[f\kP#RQYZRQ[flAhel«RAR/.º}100\eWgm.º20+ae[[ZRAn`bRAtlQhk[]RAlZw+RA^_Y]`b2RQie2UXWYZLOR.l«R
\ewOwO[]a2\2^fLORAlAh7wOO[]RtjeRAaeP#RY][Z`p^`iWxae[]P\7Y]`ba2W`ilRxwOiae`bYZRAm;KMLORCaexrsR.^_YP#aeYZ`iaeWRAl«YZ`iP\7Y]`ba2WwO[]aeOiRQP
^A\kWJRlZ`iPcwgb`b¡gRAmhu`bY]LORSe¤a2xrsRA^Yl«LJ\kw+Rc`pl&mxR.lZ^Q[Z`i+RAm§;\$wg\k[f\kP#RY][Z`p^#l«LJ\kw+RcP#axmxRA.|30ae[
`bY]LORSe¤ a2xrsRA^YlZLg\kw+R#`ilRAeRQW§¯nWOa7W`iW\2mx7\kWg^QR4. 105.iAS60H°UXWY]LORAlZR#\kwOwO[]a2\2^fLORAlAh0\emOmx`bYZ`iaeWJ\kib
Y]awOg[ZR$jeRAaeP#RY][Z`p^`iWxae[]P\7Y]`ba2WhuY]LOR$`bP\kj2R(lZ`bj2Wg\k¬`bY]lZRQbC`ilcRQnwgba2` Y]RAm¢mx`i[]RA^_Y]b¨n¨\kwgwObn`iWOj§\
j2[]\2mx`iRQW;Y¬P#RQYZLOaxm7.b.S60H
UXW·Y]LOR\ewOwO[]a2\e^fLgRAl#P#RQW;YZ`iaeWgRAmOW;YZ`iWOa7h>wOLOaeYZa2PcRQYZ[]`i^£R "0RA^_Yfl`iW·YZLgRn`imORQa¨`iP#\ejeR.lb`i¯eR
lZLg\2mx`bWgjghu^A\el«Y~l«LJ\emxa7tl&\eWgm¨lZwJR.^Op\k[~[ZR8gRA^YZ`iaeWglLJ\eR+RQRQW RQ`bYZLORA[~\ea2`imORAm:9nglZ`bWOj[Za2Ogl«Y
w+ae`iW;YM^Qae[][ZR.l«w+aeWJmxRQWg^QRAl;ae[¬WORAjeiRA^_Y]RAm<9àmxO[]`bWgj~RQnwgba2` Yf\7YZ`iaeW?ak°YZLgR`iP\kj2R lZ`bj2Wg\k';_;UHY¬`plM\el]l«gPcR.m
Y]Lg\7YC\cP#a7n`bWgj~a2xrsRA^YtwO[ZR.l«RA[Z2RAl` YflO[]`bj2L;YZWOR.lZlMmxO[]`iWOj#PcaeYZ`iaeWhJl«RAR=.i.60ae[t\#mx`plZ^Qgl]l«`iaeW
UXWwO[]RAlZRQWJ^Rak>\mOaeP#`bWJ\7YZ`iWOjJhOwJa2`bW;YZdEi`b¯2Rb`ijeL;Y lZaeg[]^QRehglZg^fLwOLOaeYZaeP#RQYZ[]`i^R "+R.^_YflC\k[]R&lZLOa7W
`iW>.b}10°YZa+R&l«YZ[]aeWOj#\kWgm(Y]a#^QaeW;Y]\e`bW$`iWxae[]P\7YZ`iaeW$aeW$P#akY]`ba2W$\eWgml«Lg\ewJRaeuYZLgRaexrsR.^_Y]lA
KaCxY]`bi`@?QRwgLOakY]aeP#RY][Z`p^´R "0RA^Y]lAhQY]LORS2¤ a2xrsRA^YlZLg\kw+RP\JRRAl«YZ`iP#\kYZR.m[Za2P lZLg\2mx`bWgj/.A30.ºk|0Eh
[]aeP lZw+RA^gi\e[[]R8gR.^_Y]`ba2WglB.2}10xae[:[Za2P ^Q\2lsY´l«LJ\emxa7tlC.b.|605.{605.230H7UXW&YZLgRAlZRM\ewOwO[]a2\2^fLORAlAhY]LORa2xrsRA^Y]l
\e[ZRRQ`bYZLORA[:l«Y]\kYZ`p^.A605.e|605.{30.230xae[uYZLORMP#akY]`ba2W&`pl¯nWOa7W`bWc\emx7\kWJ^R>[]aePy\CYZO[]W;Y]\eObRD.2}10H¶gO[ZYZLORA[Ah
Y]LOR#aexrsR.^_Yfl\k[]R~aeYZRQW¨\2lZlZOP#RAm£YZaJRgWO`i^Qaeiae[]RAm°h0`ER2°Lg\n`bWgj\$lZwg\7Y]`i\ebi£OWO`bae[]P []R8gR.^_Yf\kWg^QR
.A605.e|605.{30E
KaxY]`bi`E?AR&wOLOaeYZa2PcRQYZ[]`i^R "+R.^_Yflta2[CSe¤ aexrsR.^_YCP#akY]`ba2WRAl«YZ`iP#\kYZ`iaeWhgYZLORglZ\ejeRae:Y]RQP#w+ae[f\kib
^fLJ\kWOj2`bWOjaeOrsRA^_Y?lZLg\2mx`bWgjLJ\el#+RQRQW·wg[Za2wJa;l«R.m°>¶Oa2[RO\kP#wOiRehY]LORlZLg\emx`iWOjak\wJRA[]lZaeWJlc²\e^R
^fLJ\kWOj2RAlAhx` YZLgRwJRA[]lZaeW$YZg[ZWgltLO`ilt²\e^QR&\¬\#[]aePYZLOR&i`ijeL;YCl«a2O[f^RexqRAW2Y]i\eWgm7.b}10wg[ZR.l«RAW2Y]RAm$`iW
1A&AgM\eW\kwOwg[Za;\e^fLcae[¬l«Lg\ewJRCRAl«YZ`iP#\kYZ`iaeW#[]aeP WOakY]LO`iWOj&Y]Lg\kW#Y]RQP#wJa2[]\ebi#^fLg\kWOj2`bWgj&l«Lg\2mx`iWOjg;¶Oae[
S2¤ P#aeYZ`iaeW£R.lsY]`bP\7Y]`ba2Wh+`bWF.b7 G0.S60YZRAP#wJa2[]\ebi$^fLJ\kWOj2`bWOj(l«Lg\2mx`iWOj?Lg\2lt+RQRAW£Y]\e¯eRAW£`bW;Y]a(\e^A^a2OW;Y
mOO[Z`iWOjtRA7\kig\7Y]`ba2WaenYZLOR`bP\kj2RlZ`bj2Wg\kkn\je[f\emO`bRAW2YP#RQYZLOaxm°.UXWYZLOR.l«R¬\kwOwg[Za;\e^fLOR.lQh_wOLgakYZa2P#RYZ[]`p^
3H% j2RQa2PcRQYZ[]`i^R "0RA^_Yfl~\e[ZR?gl«R.m°¶OO[ZYZLORA[AhuYZLORSe¤aeOrsRA^_Ycl«Lg\ewJR(`ilc\el]lZOP#RAmYZa§JR(¯nWOa7W¢`iW
\2mx7\kWg^QReI¬RA^A\kglZR¬\ je[f\emO`bRAW2YP#RY]LOaxm&Y][]\2^f¯nlY]LORMPcaeYZ`iaeWcak0\Ciax^Q\kx`iP\kj2R[]RQj2`ba2W&`bWJmxRQw+RQWgmORQW;YZi
[]aePakY]LORQ[t`iP\kj2R[]RQj2`ba2WglAhnYZLOR&a2xrsRA^Y]lt\e[ZR\ebia7R.mY]aJRPObYZ`bdH^Qaeiae[]RAm°hx`EReOY]LORQ(P\?Lg\2R\
lZwg\kYZ`p\kib7\e[Zn`iWOj~[]R8JRA^_Yf\kWg^QRe
UXWYZLg`il[]RQw+ae[ZY=.ºe60.ºk60Hh0YZLgRSe¤ P#akY]`ba2W§ae¬\kW§aexrsR.^_Y`plRAl«YZ`iP#\kYZR.m£[]aeP WgakYZLg`bWOj(YZLJ\kWYZLOR
wgLOakY]aeP#RY][Z`p^&R "+R.^_Yak>Y]RQP#wJa2[]\ebi£^fLg\eWOje`iWOja2xrsRA^YlZLg\emO`bWOjJ Â ae[]R~lZwJR.^`b¡J^Q\ebieh0aeWgRakYZLOR#l«`b
mORQje[]RQR.lCae>[]RQRAmOaeP ak¬aexrsR.^_YP#akYZ`iaeW§`ilR.lsY]`bP\7Y]RAm°KMLgR~R.lsY]`bP\7Y]RAm§RQW;YZ`bYs`plCY]LORcYZ`i Y&\eWOjeiR~ae
Y]LORcaexrsR.^_Y[ZaeY]\7Y]`ba2W£\7x`plCY]Lg\7YmxRQ¡gWORAl Y]LOR~a2[Z`iRQW;Yf\7YZ`iaeWae>YZLgR#e¤ wg[ZaersRA^_Y]`ba2W£akYZLOR#Se¤ aeOrsRA^_Y
[]akYf\7Y]`ba2W\kx`il`bW¨Y]LOR?`iP\kj2RwOp\kWOR2:KMLOR?wO[ZaersRA^YZRAma2xrsRA^Y~[]akYf\7YZ`iaeW¨\kn`pl&`plc\wJa7¬RQ[ZO¯eRQ§YZa
`iP\kj2RgWgmxRQ[fl«Y]\kWJmx`bWgjg
KMLOR\ewOwO[]a2\e^fL§\kWgm§` Yfl[ZRAi\kYZ`iaeWYZa$YZLORi`bYZRQ[f\7Y]O[]Rc\e[ZR#\el aeiia7tlQKJ·LORA[ZR.\el`bWL.º}605.320.i 10.S0
P\e`bWOiCj2RQa2PcRQYZ[]`i^R "+R.^_Yfl\k[]R´RQ7\kig\kYZRAm a2[P#akYZ`iaeWR.lsY]`bP\kYZ`iaeWh_Y]LO`plu\kwgwO[Za;\e^fL RQ7\kig\7Y]RAl°WgakYZLg`bWOj
Y]Lg\kW§YZLgRwOLOakY]aeP#RY][Z`p^cR "0RA^Y&akY]RQP#w+ae[f\kib^fLJ\kWOj2`bWOj$aeOrsRA^_Y&lZLg\emO`bWOjJMJ·LORQ[]RA\2l`bW:.º}10.º230Se¤
"!$#&%
'
$()!#
'
#+*,#.-"/102
$()!#
'
#+-3#.*4/102
$) 5!$#
'
#.*.#.-67/80
5 !#&%
*
$) 5!$#.*9#.-:,#
'
7/80
!#
'
#.-;<#+*"/10
!#+*=#+-3#
'
/10
>$)!#.%
- ?@
9s1;
P#aeYZ`iaeW \eWgm¨Se¤¥l«LJ\kw+R(\k[]RR.lsY]`bP\7Y]RAmrsa2`bW;Y]b2h`iW¢Y]LO`il~[]RQw+ae[ZYaeWOiP#akYZ`iaeW `il~RAl«YZ`iP\7YZR.m¨n\
iRAl]l¬^aeP#wOiRP#RY]LOaxm°xUXW?a2wOw+a2lZ` Y]RtYZa-. G0.iAS0.i 60.S60Hh2Y]LORSe¤ aexrsR.^_Yl«LJ\kw+RC`plWgakY\el]l«OP#R.m#YZacJR
¯nWOa7Wc`iW#\2mx7\kWg^QRe7UXWgl«YZR.\em°hY]LORt\el]lZOP#wxYZ`iaeWcak0RA®;g\ebimx`plsY][Z`iOxY]RAm#lZO[«²\2^RMWOa2[ZP\kpl:`il\kwOwOi`iRAm°h
¯nWOa7W[]aeP `ibiOP#`bWJ\7YZ`iaeWRAl«YZ`iP#\kYZ`iaeW\ewOwO[]a2\2^fLORAl.i205.A05.e|60EKaCa7eRA[]^QaeP#R:YZLOR¬l«wJ\7YZ`p\kii 7\k[]n`bWgj
\eWgm~OWO¯nWOa7Wa2xrsRA^Y´[ZR8gRA^Y]\kWJ^Reh.Ys¬a`bP\kj2RAl:`ibx+RRQ7\kig\kYZRAm.UXWJl«wO`i[]RAm~;q:RAW;YZp\kWgm4.b}10Hh7YZLOR
`iP\kj2RAl`bi°JRP#akY]`ba2W$^QaeP#w+RQWgl]\7Y]RAm 9²glZ`bWOj\~je`ieRAW(mx`pl«wgi\2^RQP#RAW2YMeR.^_Y]ae[¬¡gRQpm$;¬\kWJm?^a2PO`iWOR.m
n\WOa2WxdHb`iWORA\e[wg[Zax^R.lZlZ`iWOjCY]aRQi`bP#`iWg\7Y]RMYZLORMOWO¯nWOa7Wc[]R8gR.^_Y]\eWg^R2UXW~Y]LO`pl[ZRAwJa2[«Y.hk\el:WOaeWOdEi`bWgRA\k[
wg[Zax^R.lZlZ`iWOjtY]LOR
%BAC@&%C 3
9ठ¶v;´`il`iW;YZ[]anmOg^R.m°7UHY´`biO+RMlZLOa7WYZLORAae[]RYZ`p^Q\ebiehY]Lg\7Y
Y]LORQ[]R>`plu\tl«YZ[]aeWgjb`iWO¯JRQYsRARQWYZLgR¤¶:v¢`bWglZ`pmxR´YZLgR¬k¤ aexrsR.^_YlZ`ibLOa2ORYZYZR\eWgmY]LOR>aeOrsRA^_Y[ZaeY]\7Y]`ba2W
\kx`ilY]`bbYA;Kua~l«LOa7¦YZLJ\7Y>Y]LORtRAl«YZ`iP\7Y]RAmcYZ`ibY¬\kWgjeiR`il>\&gl«RQOg`iWxa2[ZP\7Y]`ba2Wh7Y]LORtRAl«YZ`iP\7Y]RAmcYZ`ibY`ib
+R&\kwgwOb`p^Q\kYZR.mY]acn`pmxRQa#`iWgmxRQn`iWOj.ºxQ0H
KMLOR[]RQw+ae[ZY`pla2[Zj;\kWO`@?QR.m\elaeiba7tlA2UXWEDnRA^YZ`iaeW(Oh;YZLORmx`pl«wOp\e^QRAm#[f\kP#R []\kYZ`ia49²¤¶v;`bi0JR
`iW;YZ[]axmxg^R.m°xUXWFDnRA^YZ`iaeWSOhJ\~P#RQYZLOaxm(ae[t[]akYf\7Y]`ba2W(\7x`pl¬YZ`i Y RAl«YZ`iP#\kYZ`iaeW$[]aeP¤¶v4aeglZRQ[]7\7YZ`iaeWJl
`ii+R~mxRAeRAba2wJR.m\kWgm`bY]l wJRA[«a2[ZP\eWg^R&`ii+R~mx`pl]^gl]l«R.m°gUXWDnRA^YZ`iaeW2gh+Rxw+RQ[]`bP#RAW2Yf\k[ZR.l«ObY]l
\eWgm$\kW\kwOwOi`p^Q\7Y]`ba2W?Y]an`imxRAa#`bWgmORx`bWgjc`iiJRwg[ZR.l«RAW2Y]RAm°GDxRA^_Y]`ba2W}~j2`b2RAl\#^a2Wg^iglZ`ba2W
H I
! % :~ ª
UXW¦YZLO`pllZRA^_Y]`ba2Wh:YZLORmx`plZwOi\2^R.m¢[f\kP#R$[]\kYZ`iaF9²¤¶v;#`plc`bW;YZ[]axmxg^QRAm°´KMLOR¤¶:v¥`ib+R$YZLOR
a2glZRQ[]\kYZ`iaeW?ae[¬YZLORR.lsY]`bP\kYZae[M`iW$YZLORWORQnYlZRA^YZ`iaeWOKa#aeglZRQ[]eR YZLgR¤¶vhg\cwO[]RQn`iaeglM`iP\kjeR\7Y
Y]`bP#R`iWgl«Y]\kW;Y3J K
`bYZL$Y]LORbgPc`iWg\eWg^R
LNMPOQ SRT!
VU
XWY!TZ[]\F!
9à&;
\eWgm·\ ^QO[][ZRAW2Y`iP\kj2R\7Y(YZ`iPcR§`bWJlsYf\kW;Y^J`bYZL·Y]LORiOP#`iWg\kWg^QR
L M XR4!
\k[]R£WORQR.mxRAmUXW³®+xh
R
S_`a5!$b
`il~\k¤`bP\ejeRw+a2lZ` Y]`ba2Wh
\
Y]LOR?WOa2[ZP\e>akY]LOR?a2xrsRA^Y~l«g[«²\2^Rwg\kY]^fL¢\kY&YZLOR$Se¤
w+a2lZ`bYZ`iaeW
W
Y]Lg\7Y`pln`ilZ`iObR£\kY
R
hcU ]WY!
YZLOR[]R8JRA^_Yf\kWg^QRakCY]LORlZO[Z²\e^R$wg\kY]^fL·\kWgm Z[]\F!
YZLOR
`i[][]\2mx`i\eWg^R2h`ER2¬YZLORwJRA[]^QRQ`ieR.m·b`ijeL;Y$wJa7¬RQ[(wJRA[$a2xrsRA^Y$lZO[Z²\e^R2¬KMLgR§`i[Z[f\emO`i\eWg^RY]RQ[]P
Z[X\F!
mORAl]^[]`b+RAl¬Y]LOR&l«LJ\emx`iWOjcaeW$YZLORa2xrsRA^Ytl«O[Z²\e^QRmxRQw+RQWgmO`bWOja2W?Y]LORWOa2[ZP\k
\
.b.{605.b1A60E
KamO`bn`pmxR§a2xYYZLOROWO¯nWOa7W \kWgm l«wJ\7YZ`p\kii7\e[Zn`iWOj a2xrsRA^Y$[]R8gR.^_Yf\kWg^QR£[]aeP YZLgR`iP\kj2R
iOP#`iWg\kWg^QRAlAhOY]LOR~^QO[Z[]RQW;YC`iP\kj2R`il P#akYZ`iaeW£^QaeP#w+RQWgl]\7Y]RAmn\je`ieRAWmO`ilZwOp\e^RAP#RQW;YCeR.^_Y]ae[t¡JRQpm
d M XR4!
\eWgm?[]RQp\7Y]RAm?YZa#Y]LORwO[]RQn`ba2gl¬`bP\ejeReOKMLg`ilM[]RAlZObY]lM`iW(Y]LOR&¤¶:v
egfCh SRT!
L M SRi
d M SRT!!
L MPOQ SRT!
Zj
\F!
Z[X\F!
`
9²S;
`bYZL) YZLOR#[]akYf\7YZ`iaeW§P\7YZ[]`bmxRAl]^[]`bg`bWOj(YZLgR[ZaeY]\7Y]`ba2Wg\k:wg\e[«YakYZLORa2xrsRA^YP#akY]`ba2W[]aePY]`bP#R
`iWgl«Y]\eW2YjJ
YZa,J0>KMLOR[]akYf\7YZ`iaeW¦P\7YZ[]`b ^Q\kW¦JR£mORAl]^[]`b+RAm¦n¢\§gWO` Y[]akYf\7YZ`iaeW\7x`pl~2RA^YZae[
X#
' ` #+* ` #+-N!
b
\kWgm(\&[]akYf\7YZ`iaeW\eWOjeiR
.b2Q0°\e^Q^Qae[fmx`iWOjYZa~³®+ \kYY]LORtY]aew(ak°YZLgR wO[]RQn`iaegl
wJ\kjeR2OKMLOReR.^_YZa2[£hxO[ZYZLgRQ[tmxRAWOakY]RAm\elM[]akYf\7YZ`iaeW$\7x`plQhO^A\kW$JR&mxRQ¡gWOR.m(n?\cY]`bbYC\kWgjeiR
\kWJm
\#lZi\eW;Y\eWOjeiR&U
\2^Q^Qae[fmx`bWgjY]a
# '
# *
# - ?@ 7/10
U
7/10
U 7/10
?@ 92 ;
9àl«RAR¶:`bj);_:e+KMLOR~¤¶v mxRAl]^[]`iJR.lYZLgRwOLOaeYZa2PcRQYZ[]`i^&R "0RA^_Yak´Y]RQP#wJa2[]\ebi(^fLJ\kWOj2`bWOj?aexrsR.^_Y lZLg\7d
mO`bWOjJ
x
y z
R
$' '"$ 3 53 3 G6 53 ' " % 1 %'H3 5 6Q
X_6`7aC`!
"$ n C)(9E
% HD " Q$ 7 D '"$:AH
' 5= " _ a!
A EK#"5
' (
A EK%$N& "$ '"$ E3$
U
" =)(9@ ' " )( G*"5
G' ( %3' +$4H%
¶gae[\£l]^RAWOR`ibiOP#`bWJ\7YZ`iaeW¨n\mx`il«Y]\eW;YwJa2`bW;Yi`ijeL;Yl«a2O[f^R\kWgm¨\ePO`iRQW;Ymx`#"0gl«R?b`ijeL;YAhuYZLOR
`i[][]\2mx`i\eWg^R
Z[]\F!
-,
7:.0/ \F!
9à}&;
\kY¬\eW?aeOrsRA^_YMl«g[«²\2^RCwJ\7Y]^fL?`il¬mxR¡JWORAmdgwYZac\^A\kP#RQ[f\mxRAwJRAWgmxRAW2Y^a2Wgl«Y]\kW;Y,Md´;cY]LORlZO[Z²\e^QR
Wgae[]P#\e
\
\kWgm£YZLOR#¬RQ`ijeL;YZR.m£`ibiOP#`bWJ\7YZ`iaeWmx`i[ZR.^_YZ`iaeW
/
KMLORmx`i[]RA^_Y]`ba2W£ak
/
`ilmxRQ¡gWORAm§nYZLOR
\eWOjeiRAl
1
\eWgm U 1
\2l lZLOa7Wa2[.ª`bW¶:`bjJ:232
/ 2 54#`pltY]LORc[]\kYZ`ia?+RYs¬RQRQWwJa2`bW;Yb`ijeL;YlZaeg[]^QR
\eWgm\ePO`iRQW;Yb`ijeL;Y`bW;YZRAWglZ` Y]`bR.l .iA{0.iGA60H0UXW³®+u}Oh
.
h0` YZLORcwJa2`bW;Yb`ijeL;Yl«a2O[f^R`iibOP#`iWg\7Y]RAl
Y]LORaeOrsRA^_YtlZO[Z²\e^R2hxRQpl«R
.
76g
8
> ¥ :&:9y f <; ° Q &
UXW§YZLO`pllZRA^_Y]`ba2Wh°\kW§RAl«YZ`iP#\kYZa2[ae>YZLgRc[]akYf\7Y]`ba2W§\7x`pl YZ`i Y&\kWgjeiR~`ib+RmxRQ[]`b2RAm§\kWg\eb;YZ`p^Q\ebie
KMLgRQ[]Rae[]ReheYZLORmxR.lZ^Q[Z`iwxYZ`iaeW?ak°YZLgR¤¶v \2^Q^a2[]mO`bWOjY]a~³®JxS`il¬l«`iP#wOi` ¡gR.mglZ`iWOjY]LOR\el]l«OP#wxY]`ba2W
Y]Lg\7Y
'
O 1
1 1 O 1
Q
1
O 1
O 1 O
1
1
1 O 1
1
Q
*
O 1
1 O 1 1
Q
1 O 1 O 1 O 1
1 1 O 1
1
Q
9à|;
'"$ B Q n
lZg^fLcYZLJ\7Y´YZLgR[ZaeY]\7Y]`ba2W~P\kYZ[]` ~[Za2P ³®+gt^A\kWcJRti`iWORA\e[Z`@?QR.m`bYZL[]RAlZwJR.^_YYZaY]LORt[]akYf\7YZ`iaeW#\kWgjeiR
n`iRQpmx`bWgj
#.-P
(#+*N
(#.-
# '
#+*N
(#
'
?@
9E 3;
I¬`iWglZRQ[ZYZ`iaeW$ak
`bW³®+gSghO\kpl«a~YZLOR&¤¶v
egfCh SRT!
Z[
\F!
Z[X\F!
`
9²{;
`pli`bWOR.\k[`bYZL[]RAlZwJR.^_YYZa$YZLgR[ZaeY]\7Y]`ba2W§\eWOjeiR
uµCl&\$WORQnYl«YZRAwh°Y]LORl«wJ\7YZ`p\kj2[]\2mx`bRAW;Y]lak¬YZLOR
¤¶v `bi+R&mxRQ[]`b2RAm° l«`iWOjcYZLgR&\el]l«OP#wxY]`ba2W?Y]Lg\7Y
"!
'"$& #&3 (E1 9n A AHG' 3 5%= / BA "$
QA$ "
Y]LORl«g[«²\2^RMWOae[]P\k
\ S_:$# _6`7a4$# a5!
`iWc\ian^A\kxWORA`bj2LnJa2[ZLOanaxm&ae+\^a2WglZ`imxRA[ZR.m&`iP\kjeR¬w+a2lZ` Y]`ba2W
R X_6`a5!
b
^Q\kW$+R[Z`bY«Y]RQW\el
\ h # _
# a
%
h %
&# _ %
&# a % ?@ `
9A;
`bYZL h YZLORM[f\emx`iglakJYZLORtlZwOLORA[Z`p^Q\enwg\7Yf^fLkKMLO`pl´\kwOwO[]ax`bP\kYZ`iaeW&Lg\2lJRARQW#glZRAm~nva2lZRQWORQpm`iWc\
lZ`iPc`ip\k[P\eWOWORQ[ae[YZLOR `bP\kj2RtiOP#`bWg\eWg^R .A0E'ta7h2YZLORl«g[«²\2^RtWOa2[ZP\eO`bW?³®+g{&`pl>[]RQwOp\e^QRAm#n
Y]LORlZwOLORA[Z`p^Q\e2\kwgwO[Zax`iP\7YZ`iaeW AO.KMLORQW³®J{`plumxRA[Z`ieR.m` Y]L[ZR.l«w+RA^YYZa
# _
\kWgm
# a
hA[ZR.l«w+RA^YZ`ieRQie
¶:`iWg\kiiehYZLgRMlZwOLORA[Z`p^Q\ex\kwOwO[]ax`bP\kYZ`iaeW/A`ilRAb`iP#`bWJ\7YZR.m`iW~YZLORmxRA[Z`i7\7YZ`ieR¬[]RAlZO Yje`in`bWgjCY]LORtl«wJ\7YZ`p\k
j2[]\2mx`iRQW;Y (
egfCh SRT!
'
SRT! ` * SRT!!
b
9«A;
`bYZL ( b
*)+,)C_Y`)+,)Ca5!
YZLORj2[]\2mx`iRQW;Y¬a2wJRA[]\kYZa2[M\eWgm
'
SRT! ` * SRT!
\ellZLOa7W$`bW$³®+g|O
4 U 1 7/10
U 1 1 U 4
/10
U
7%
U 1 4 /10
% U 1
/10
% 1
7/10
U
c
U 1 /10
U
/10
U 1 1
U 4
7/80 % U 1 7/10
1 1
7/10
U 7/10
4 $%
U 1 7/10
U 7/10
7/80
U 1 4
U 1 /10
1
U 4
7/10
% U 1
1 /10
1 /10
U
4 7/10
% U 1
7%
1 7/80
U 7/80
c
U 1 /10
U /10
/10
U 1 /10
1 U
9«e1;
KaO[ZYZLORA[(mxRAeRQiaew¦YZLgR^Q\ei^QOi\kYZ`iaeWh\kW`imORA\akqRQW;YZp\kWJm .bG20C`plglZRAm°tR£wO[]aew+a2lZRAm YZa
^A\kp^Op\7Y]R¬YZLORtlZwg\7Y]`i\exP#RA\kWJlak0bOP#`iWg\kWJ^Rje[f\emO`bRAW2YflYZa[]RA^a72RQ[YZLORw+ae`iW;Y>b`ijeL;YlZaeO[f^RMmx`i[ZR.^_Y]`ba2W
Y]`bbYA)JR \ewOwOi`i^A\7YZRtYZLO`pl¬`imxR.\YZaYZLOR¤ ¶v je[f\emx`iRQW;Y]l>YZa&[]RA^Qa7eRA[´YZLOR aexrsR.^_Y¬[ZaeY]\7Y]`ba2W\7x`pl´YZ`ibYA)J§R
\2RQ[f\kj2RMYZLOR¤ ¶v je[f\emx`iRQW;Yfl>`iW?Y]LOR `iP\kj2RC[]RQje`iaeW(^a7eRA[ZR.m#n#YZLORP#a7;`iWOj~aexrsR.^_YAxKa~mxa#lZagh;YZLOR
a2xrsRA^YMlZLg\ewJR Lg\elYZac+R ¯nWOa7Wh;+RA^Q\eglZRCY]LORlZO[Z²\e^QRCWOa2[ZP\k
\
\ewOwJR.\k[fl`iWY]LOR¤ ¶v je[f\emx`iRQW;Y]l
`iW³®+g|OOKa\ea2`imYZLO`plQhgR\kwgwOb?q:RAW;YZp\kWgm\el]lZOP#wxYZ`iaeW(YZLg\kY
'"$ C # H3Q 9 =$9n4%3 Q# 5%
KMLg`il#[]RAl«YZ[]`i^Y]l~YZLgRYs;w+Rak;`pmxRAal]^RAWORAlcY]a¨lZ^QRQWOR.l~`bYZLg\ebbdHb`i¯eR$aexrsR.^_YflQ>µ lcYZLOR[]RAlZObY]l#`iW
lZRA^YZ`iaeW 2`ib:l«LOa7hnY]LO`plC\el]lZOP#wxYZ`iaeW\kiia7tlMRQeRAW$ae[tLnOP\kWLgRA\emglQ$I¬?YZLO`plC\2lZlZOP#wxYZ`iaeWuhxYZLOR
lZO[Z²\e^QR WOa2[ZP\e
\
X7/80
U
`c7/80
U 7/80
`E
! b `
9«.&;
mOR¡gWOR.mncYZLORl«p\kW;YM\kWgm#Y]`bbY\kWOj2bR.lcU
`
^Qae[][ZR.l«w+aeWgmO`bWOj&YZa `bW(¶`ijg02h2Lg\2l´Y]LOR wg[Za2g\kO`ii` Ys
mORQWglZ` YsOWg^_Y]`ba2W
U ` !
7/80
U U
`
9«AS;
lZRQR .A0HOKMLORi`bWOR.\k[P#RA\eW(aeY]LOR&¤¶:v j2[]\2mx`bRAW;Y]l¬`il
'
%
7%
' U ` ! e U e
9«2 ;
\eWgm^a2[Z[]RAlZwJa2Wgmx`iWOjeiae[
*
:KMLOR`iW2Y]RQj2[]\e´`bW³®+G2^a7eRA[]l Y]LORLOa2bRn`pl«`iOiRl«O[Z²\e^QR#akMYZLOR
a2xrsRA^YAh°LORA[ZR.\el³®+}$^QaeW;Y]\e`bWglY]LORlZ` Yf^fL
.
YZamxR.^`pmxRLORQYZLORA[\l«O[Z²\e^QR#`il`iibOP#`iWg\7Y]RAmae[
WgakYnYZLgRwJa2`bW;Yb`ijeL;Yfll«a2O[]^QRe#JRglZR Y]LnglMYZLgR&\el]l«OP#wxY]`ba2W?Y]Lg\7Y
"$&%& ' (- H5(=(&E
\eWgmlZRY
.
°KMLO`pl\el]lZOP#wxYZ`iaeW§Lg\el+RQRQWglZRAm§`bP#wOi`p^`bYZi\kpl«a(`bWL.i205.A30Eµ lZlZOP#`iWOj?g[«Y]LORQ[
Y]Lg\7Y
'"$2A$' :(6") B# / N
RAWg\kgbR.l#gl#YZa¢lZaeieR$Y]LOR`bW;Y]RQje[f\k`iW³®+CG2g>KMLO`plP#RA\eWgl#YZLJ\7Y
& /:\
LOaepmOl\eWgm¦YZLOR
mORQWOa2Pc`iWg\kYZae[fl¬`bW³®+g|#^A\kW$JR&mx[f\el«YZ`p^Q\ebilZ`bP#wOi` ¡JRAm°
KMLgR`imxR.\`ilYZLg\kYYZLORPcR.\kWje[f\emO`bRAW2Yae Y]LOR¤¶v`il[]RQp\7Y]RAm¦YZa¢YZLORY]LOR£ae[]`iRQW;Y]\kYZ`iaeWakYZLOR
wg[ZaersRA^_Y]RAmc[]akY]\kYZ`iaeW\7x`plQ2Kua&P#aeYZ`i\kYZRY]LO`pl>`pmxRA\gh2¶:`bjJ2lZLOa7tl´RQx\eP#wObR.lae[>YZLOR¤ ¶v;UHY^Q\eWJR
lZRQRAWYZLJ\7YYZLORk¤ wO[]akrsR.^_YZ`iaeWae>YZLgRY][ZORcSe¤ aeOrsRA^_Y[ZaeY]\kYZ`iaeW\kn`pll«R.\kPlY]a(+R~a2[«Y]LOaej2aeWg\e°YZa
Y]LORPcR.\kW(je[f\emx`iRQW;YaeuY]LOR¤¶vOKa~[]RA^Qa7eRQ[YZLORa2[Z`iRQW;Y]\kYZ`iaeWaeYZLgR P#RA\eW(¤¶v j2[]\2mx`bRAW;YAh;YZLOR
a
A rotation axis
B B
axis rotation
projectedB A B projected
rotation axis B
B B B
A
projected
B
axis rotation projected A
B B
(d) (c)
(b) (a)
0 BACE % / " $ ' ( / / # 9E % BA " ' (- #H%
3>G '> '"$ ' ( +A EK"U + $ " +A$' (1"# # E1 % " "
1 g CU
1 + 3H%
+ " 3 53 37 n " $ 6 '"$
_
" $
+
" $ + 3"%$
" = 3A 3 3 A$ " =5(&
AL '"$ & D 3 &%< K nn#='H3 5% - '"$.A$' 3(6") 9#0" " %& $
"$ 5(333 '"$ G$%')( & %&D % ' " B QQ')( H ' ( $ $ '"$
3 H3 YAH %3 %<*< : 3(A$
" &(&9#H%-(3FE 3Q]A$K%6
/3 4 3H
6 6 6 6 66 6 6 6
6 6 6 6 6 6 6 6 6 6 6
O e 4 h 4 4
U 1 e 4 h 4 4
!P /7/1=07 "!$#%&=0/'(')= *+-,.7/1=0G0/5'(1
2 2 2 2 2 2 2 2 2
2
3 ! 7/(47/1=05 6 7!$#8-990;:+<9'
= = = = = = = = =
=
XP! 7/(&7/1=05 6?>/(@6/A<917!$#8-990;:+<9'
B B B B B B B B B B
B
S
n Q '
K%& % % 1
O Q '
Q3 7 n 6 )Dnn#='H3 3< E
U 1
6 #%
U
+
+
3H%
6G(% 6
1 6 +
DC " -# 6 "$ " $ '"$ ' LQ3=' '"$ %&9C63
'"$QQ(1%<' ( Q " 3 " 6-9n & %' (7 9n )% #3AH $ " 5 $N " %$< "
Q ' Q3C 9Q n4 '"$ ' ( Q H% " $ " Q ' Q =
'"$ /' ( Y)')( (6") % " $
^QaeP#w+aeWORAW2Y
*
ak°Y]LOR P#R.\kWj2[]\2mx`bRAW;Y`il¬mx`in`imxR.mn#` Yfl
_
^a2P#wJa2WORQW;Y
'
je`i;`iWOjYZLOR[]\kYZ`ialZLOa7W
`iW³®+e\7YY]aew$ak:YZLORWgRnYtwg\kj2RexKa~¡JWgm(YZLOR&^Qba;l«R[]RQp\7YZ`iaeWJl«LO`iw(YZacYZLOR[]akYf\7Y]`ba2W$\kn`plAhn\eW
FE
nn# 'K 7' 1 ')( %' 3
`pl\2lZlZOP#RAmhe`ER2=U
1
76g2KMLORAWh2YZLOR []\kYZ`ia`bW³®JJeClZ`iPcwgb`b¡gRAl>YZa
*,+
'
+47/80
h;l«J^fL
Y]Lg\7YY]LOR[]akY]\kYZ`iaeW$\7x`il¬Y]`bbYC^Q\eW(+RRAl«YZ`iP\7Y]RAm$n
G
HII
JIIK LAMLAN
G* + G' !
LAMLAN G* + G'
!6
+
+ O f G
'&P
6
G
'&Q
6
G'
76 6
G* P 6
G'
76 6
G* Q 6
9«.}&;
KMLOR?RAl«YZ`iP#\kYZ`iaeWak¬YZLOR?[ZaeY]\7Y]`ba2W\kx`ilY]`bbY~^Q\eWJR?^A\k[][Z`iRAmaexY&n§³®+.}(glZ`iWOjYZLORP#R.\kWgl
G
6R
` G
S
aktY]LOR¤ ¶v[f\kP#R$[]\kYZ`ia£`iWgl«`pmxR(YZLORk¤¥O`iWg\e[ZaeOrsRA^_YcP\elZ¯+hYZLgR(`iP\kjeR([ZRAje`iaeW¨Y]Lg\7Y
^Qae[][ZR.l«w+aeWJmOl>Y]a#YZLORa2xrsRA^YA
Ka LJ\eR\ ^ia2lZRQ[:banae¯\kYuYZLgRw+RQ[Zae[]P\kWg^QRakxY]LOR¬RAl«YZ`iP#\kYZa2[AhAYZLOR¬RAl«YZ`iP\7Y]`ba2W&Lg\el+RQRQWc^Q\e[Z[]`bR.m
a2xYae[:YZLgRMlZ`iPcwgbRtl]^RAWOR¬mORAl]^[]`b+RAm~`bW#¶:`bjJexkKMLOR^QaeWgmx`bYZ`iaeWJlaegY]LORMRxwJRA[Z`iP#RQW;Y]l´\e[ZRM\ela2bia7tlQ¸
xrsRA^YCl«LJ\kw+Re¸n¡OxRAmhOaexrsR.^_Y`pl\l«wOLgRQ[]R
>Dx^RAWOR`ibiOP#`bWJ\7YZ`iaeW
UXW;Y]RQWglZ` Ys72
/
2 ¸:¡OxRAmY]a 2/ 2c
6 9²P\7x`bP~OPhP#a;lsY#^aeW;Y][]\2mx`i^YZa2[ZYZa\el]l«OP#wxY]`ba2W
µ}&;
Dxi\eW2YC\eWOjeiR U 1
¸x¡gnR.m$wJRA[CRAl«YZ`iP\7Y]`ba2WhglZRQeRA[]\e0RAl«YZ`iP#\kYZ`iaeWglt` Y]L\e[Zn`iWOjl«p\kW;Yfl 6
U 1 + hOl«RARLgae[]`E?AaeW;Y]\eCU
1
dH\kn`pl¬`bW¶:`ijgOS
·KM`ibY¬\kWgjeiR
1
¸n¶`bxRAmw+RQ[¬RAl«YZ`iP\7Y]`ba2Wh;lZRQ2RQ[f\kOR.lsY]`bP\kYZ`iaeWgl` Y]L7\k[];`iWOjY]`bbYAhnR.lsY]`bP\7d
Y]`ba2W?RA[Z[]ae[M\eRA[]\ejeRAmca72RQ[6
1 +
hxlsYf\kWgmg\k[fmmxRQn`p\7Y]`ba2W?akRAl«YZ`iP#\kYZ`iaeW(RQ[][Za2[]l`il
2RQ[ZYZ`p^Q\e0\kx`ilM`iW¶`ijgOS
xrsRA^YtP#akYZ`iaeW
·KM`ibY\kWgjeiR
¸+¶:`bxRAmY]a
+
9`bYZLgaexY[ZR.lsY][Z`p^_Y]`ba2Wak\kWn(jeRQWgRQ[f\ki` Ys2hOJR.^Q\egl«R
[]RAlZObY]l\e[ZR\eRA[]\ejeR.m#a7eRQ[
1 ;
Dxi\eW2Yt\eWOjeiR&U
¸g¶:` xR.mY]a U +
9[]RAl«YZ[]`p^_YZ`iaeW(YZa\c[]RQwO[]RAlZRQW;Y]\kYZ`ieR^Q\2l«R6;
µCbORQxwJRA[Z`iP#RQW;Y]l:Lg\2R+RQRQW^A\k[][Z`iRAma2xY´glZ`bWOj³®+i.}x,'CakYZR¬Y]Lg\7Y´Rxw+RQ[]`bP#RAW2Yfl:ae[´\kinwJa;lZlZ`iObR
`iibOP#`iWg\7Y]`ba2W(lZi\eW;Y]lcU
1
Lg\eRCJRARQW(mORQ[]`b2RAm°h;RQ2RQW`bYZLgR R.lsY]`bP\7Y]ae[M\el]l«gPcR.lcU
1 6 9²\2lZlZOP#wxY]`ba2W
µ |;'takY]R\kpl«aJh2Y]Lg\7YMY]LORwJa2`bW;Yb`ijeL;YtlZaeO[f^R`ilMWgakYR.\k¯0hnOxYt\7YP\7x`iPOP\kP#wOi` Y]gmxRehxRAeRAW?`b
Y]LORRAl«YZ`iP\7Y]ae[\el]l«gPcR.l\cR.\k¯w+ae`iW;YMi`ijeL;YCl«a2O[f^R 9²\el]lZOP#wxYZ`iaeWµ };_
¶:`ijgS$9²\ ;lZLOa7tlYZLgR[]RAlZObYAhn`b:\ki°glZRAm(\el]l«OP#wxY]`ba2Wgl¬LOaepm°OKMLO`plYZLORAae[]RY]`i^A\k°^Q\2l«RP#RA\kWJlY]Lg\7Y
Y]LOR?P#R.\kW¢¤¶:vje[f\emx`iRQW;Y&[]RA\ebi§+RQLJ\eRAl~\el&YZLOR$l«`iP#wOb`b¡gR.m¨³®+2?l]\nlAuKMLOR(RQ[][Za2[l«Y]\kWJmO\k[fm
mORQn`i\kYZ`iaeW`pl ?QRA[Zaa2[?\kW·`ibiOP#`iWg\7Y]`ba2W`iW·n`bRA`bWgj mx`i[ZR.^_YZ`iaeW 9XU
1 6gh\el]lZOP#wxYZ`iaeW·µC| ;\kWJm
`iWg^Q[ZR.\elZRAlMY]LOR~P#a2[ZRY]LORP#a2[ZRYZLOR~`bibgPc`iWg\kYZ`iaeWmx`i[]RA^_Y]`ba2W£ak´Y]LORw+ae`iW;Y i`bj2L;YlZaeO[f^R&`iWg^Qb`iWORAlCYZa
Y]LOR$`bP\kj2R(wgi\eWOR79à^a2W2Y][]\2mx`p^_YZa2[Z§YZa\el]lZOP#wxYZ`iaeWµC|;´KMLORP#\kx`bP~OP lsYf\kWgmg\k[fm¢mxRQn`p\7Y]`ba2W `iW
^A\elZR&ak\eW£`ibiOP#`bWJ\7YZ`iaeWmx`i[ZR.^_YZ`iaeWwg\k[f\kibRAY]aYZLgR`iP#\ejeR~wOi\eWOR`pl
O 6 e 4 h 4 4 L
LO`p^fL
`pl®;O`bYZR&\cbaeYA
¶:`ijg>S$9² ;~l«LOa7tlYZLgR$[]RAlZObY~a2[\P#ae[]R?[]RA\eb`pl«YZ`p^Rxw+RQ[]`bP#RAW2Y.¸>\kW¦RAl«YZ`iP#\kYZ`iaeW [Za2P \e^_Y]g\k
`iP\kj2RAlA:KMLOR$`iP#\ejeR.l~LJ\el~JRARQWl«nW;YZLOR.l«`@?QR.m \2l~mxR.lZ^Q[Z`i+RAm¨[]aeP \§P#a7n`bWOj§OWO`p^a2ba2[ZR.m lZwOLORQ[]Re
¶g[Za2PYZLOR#`iP\kjeR.lQh+Y]LOR¤ ¶v je[f\emx`iRQW;Y]l P#RA\2l«O[]RAm§\kWgm£YZLORAW§Jl«R.m£a2[R.lsY]`bP\kYZ`iaeW§\2^Q^Qae[fmx`bWgjYZa
³®Ji.}OQKMLOR>l«Y]\eWgmO\k[fmmxRQn`p\7YZ`iaeW`plLO`ijeLgRQ[0YZLJ\kW+Rae[]RehQJR.^Q\kJl«R´`iWYZLO`plP#ae[]R´[]RA\ki`plsY]`i^:Rxw+RQ[]`bP#RAW2Y
\wg\k[ZYCak´Y]LORa2xrsRA^Y `plClZRQbl«LJ\emxa7¬RAm>9²^QaeW;YZ[f\emO`i^YZae[]Y]a(\2lZlZOP#wxYZ`iaeW£µ2 ;C\kWgm$Y]LORw+ae`iW;Y i`bj2L2Y
lZaeg[]^QR(`plcWOakY¬RA\k¯ 9à^a2W2Y][]\2mx`p^_YZa2[ZY]a\2lZlZOP#wxY]`ba2Wµ}&;_>KMLgR$P\kn`iPgP lsYf\kWgmg\k[fm mxRA;`p\7Y]`ba2W
`iW^Q\2l«Rae\eW`bibgPc`iWg\kYZ`iaeWmx`i[ZR.^_Y]`ba2WwJ\k[f\kibRAYZaY]LOR£`iP#\ejeRwOp\kWgRwg\el]l«R.l`iWYZLO`pl?^A\elZR$RAeRAW
O P 6 e 4 h 4 4 L
Ka~[]R¡JWORtY]LOR R.lsY]`bP\kYZae[.hn\¬RQ`ijeL;YZR.mP#RA\eW?ak°Y]LOR¤ ¶v je[f\emx`iRQW;Y]l`plwO[]aew+a2lZRAm?\e^Q^Qae[fmx`iWOjYZa
GR
G
S
"! $#
XR4!
(
egfCh SRT!
9«A|;
`bYZL$Y]LORRA`bj2L;Y
B B
A axisrotation
A A
B
A B
B B
rotation axis
(b) (a)
2
&%3 " $ '
_
3H% "%$ '
a
%' '"$>%BACE %< 7 "
$
' ' ( 77,# 9E % BA "$ ' (:9#H%LF <' '"$ ' (
A EK"U + $ ' '" 3 (=E
+
" _ G
C
3 (=(3%$ = '
3 % ' " = '"$ " %& %-H%<'F " - % ' " '"$
A$Q " =1 $%3' ( & % "$ K ' ( 3$ /Q
#
SRT!
% 2( e fCh XR4!
2
9« 3;
KMLgRRA`bj2L;Y^Q\kW¨+R?P#aeYZ`i\kYZR.m;[]RQj2\e[]mO`bWOj$Y]LOR?¤¶:v je[f\emx`iRQW;Y]l&`bW¢¶:`ijg 2Jµ l^Q\eW¨JR(l«RARQWh
Y]LOR?+ae[fmxRA[~aeYZLOR$l«RA ClZLg\emOa7R.m[ZRAje`iaeW 9Y]LOR$[ZRAje`iaeWLO`p^fL `plWOaeY~`ibiOP#`iWg\7Y]RAm¨nYZLOR$w+ae`iW2Y
i`ijeL;Y#lZaeg[]^QR1;`pl#l«g[Z[]aeOWJmxRAm¨n¢l«YZ[]aeWOjje[f\emx`iRQW;Y]lYZLJ\7Y\k[]R(WOaeY#^QaeWglZ`pmxRQ[]RAm¨nYZLgR$R.lsY]`bP\kYZae[
9à\el]l«gPcwOYZ`iaeW¦µ2 ;¶OO[ZYZLgRQ[.hY]LORQ[]R\k[]R$l«YZ[]aeWgj§j2[]\2mx`bRAW;Y]lc\7Y#YZLOR+ae[fmxRA[caeCY]LORaexrsR.^_YmxOR$YZa
Ys¬a[]RA\elZaeWJlQx¶:`b[fl«YAhJ\e[ZR.\elMae:WORAylZ^QRQWOR&^QaeW;YZRAW2Yflt^Q\eWWOakYt+R&wO[]RAmO`i^YZRAmCDnR.^aeWJm°hgmx`pl«wgi\2^RQP#RAW2Y
R.lsY]`bP\kYZRAlM\kYY]LOR a2xrsRA^Y¬+ae[fmxRA[M\e[ZR RQ[][Za2WORQa2gl.ºkS0Ex¶Oa2[¬YZLOR.l«R[ZR.\elZaeWglAhkY]LORje[f\emx`iRQW;Yfl`iW?³®+°.|
\e[ZR RA`bj2L;YZRAm?n#YZLOR`iW;2RQ[fl«RCakuY]LORQ`i[\kP#wOi` Y]gmxR\kWJmYZLgRQ?\e[ZR\eRQ[f\kj2RAmcaeWgb`iWglZ`imxR\cl«Lg[Z`iWO¯eR.m
a2xrsRA^YP#\2l«¯0
UXW£¶:`ijggS 9²^G;M^A\kW$JR~l«RARQW$YZLg\kYYZLO`plP#axmx`b¡J^Q\kYZ`iaeW[ZR.mxg^QRAlMY]LOR`bW 8gORQWJ^RakP#akY]`ba2W^QaeP#w+RQWxd
l]\7Y]`ba2W#RQ[][Za2[]l\kWJm~WgRQjeiRA^YZR.ml«RA lZLg\emxa7^a2W2Y]aeO[flAkKMLORCl«Y]\kWJmO\k[fm~mORQn`i\kYZ`iaeW#`pl>WOa7^a2PcwJ\k[f\kOiR
Y]acYZLOR`pmxRA\e^Q\2l«R`iW¶:`ijgOS 9²\;O¶Oa2[MYZLOR¬ae[flsY^Q\2l«Rae:\kW$`ibiOP#`bWJ\7YZ`iaeWmx`i[]RA^_Y]`ba2W(wJ\k[f\kibRAJY]a#YZLOR
`iP\kj2R&wOp\kWOR-9SU
1 + !
Y]LOR[]RAlZObY `plCRAeRQWJRQY«YZRA[A°KMLOR~P\7x`bP~OPRQ[][Za2[Cl«Y]\kWJmO\k[fm£mxRA;`p\7Y]`ba2W
`plMWOa7
O e 4 h 4 4 L
,_ &$% ¬ Q&(
UXWYZLO`pllZRA^YZ`iaeWh.[]RAlZO Yfl°a2[uY]LORR.lsY]`bP\7Y]`ba2W&akxY]LOR[]akYf\7Y]`ba2W&\7x`iluYZ`ibYu[]aeP4[ZR.\k2;`pmxRAa lZRA®;ORQWJ^RAl
\e[ZRwO[]RAlZRQW;Y]RAm°>¶gO[«Y]LORQ[.h>`bY`il?l«LOa7W¦YZLJ\7YY]LOR£mxRQ2RQiaew+RAmRAl«YZ`iP\7Y]ae[mxRA[Z`ieR.l~[]RAlZObY]l`bYZL·\eW
\2^Q^QO[]\2^LO`ijeL$RQWgaeOj2L?Y]acRAWg\kgbRn`pmxRQa#`iWgmxRx`iWOj#g\elZRAm(aeW(YZLgRYZ`i YtR.lsY]`bP\7Y]RAlA
"!# $%$ &'(")
UXW¶:`ij2lAA}\eWgm|Oh.lZ\ePcwgbR´[]RAlZObY]l`bYZL[]RA\ek`iP#\ejeRlZRA®;ORAWg^R.l\k[]R>lZLOa7WQKMLOR`bP\kj2RlZRA®;ORQWJ^RAl
lZLOa7\kwOwg[Zax`iP#\kYZRAb&[]`ije`pm~a2xrsRA^Y]l` Y]L?\l«wgLORQ[]RdHb`i¯eRtlZLg\ewJR22KMLORClZRA®;ORAWg^R.l>\k[]R`iW
Us¶a2[ZP\kY
9àS2}2gk{2{&w+RQpl ;M\eWgmxº} ?[]\eP#R[f\7Y]ReOKMLORR.lsY]`bP\kYZRAm(aeOrsRA^_Yt[]akYf\7Y]`ba2W(\7x`pl¬YZ`i Yt`pln`pl«g\eb`@?QR.m
n\LO`bYZRc\k[][Za7gKMLgR&[ZR.l«g YflC\k[]R&a2xY]\e`bWOR.mn$\ ;t\kOYZaeP\kYZ`p^aexrsR.^_Y lZRQj2P#RQW;Y]\kYZ`iaeWglZ`bWOj?YZLOR
&DOK x2+*),.-0/°`iP\kj2R¬\eWg\kixl«`pluP#axmxRAM.10Hh7 ;n&\kxY]aeP\7Y]`i^RAl«YZ`iP\7Y]`ba2Wak0\ mO`ilZwOp\e^RAP#RQW;Y:2RA^YZae[
¡JRQpm(Jl«`iWOj#LO`iRQ[f\k[f^fLO`i^A\k+Oian^f¯?P#\kY]^fLO`iWOj#`bYZLLg\kbwJRA[ZR.l«a2bOYZ`iaeW .i0u\eWgm^G;¬\kwgwOb`p^Q\kYZ`iaeW$ak:YZLOR
mORQeRAba2wJR.m?[]akYf\7Y]`ba2W(\7x`pl¬YZ`i YtR.lsY]`bP\kYZae[M[]aeP DnRA^YZ`iaeWSO
KMLOR¬[]RAlZO YflulZLOa7¢\tj2ananmw+RQ[Zae[]P\kWg^QR>RQ2RQWh.`bOYZLOR¬aeOrsRA^_Yl«LJ\emx`iWOjt`pl:R.\k¯ 9²^a2W;YZ[f\emx`p^_Y]ae[]tYZa
\2lZlZOP#wxY]`ba2Wµ}`iW DnRA^YZ`iaeW&S ;°\kWgmY]LOR`iibOP#`iWg\7Y]`ba2W`ilWOaeYu`bWn`iRQ`iWOjCmx`i[]RA^_Y]`ba2W 9²^QaeW;YZ[f\emx`p^_Y]ae[]YZa
\2lZlZOP#wxY]`ba2WµC|`iWDnR.^_YZ`iaeWS ;_nµClba2WOj&\el>YZLORk¤ aexrsR.^_YlZ`ibLOa2ORYZYZR.l>¬RQ[]Rt[Za2OjeLgbaeOWgm 9[]aeP
\eJa2xY´YZLgR&AkYZL`iP#\ejeRa2W ;_h7Y]LORC\2^Q^g[]\2^a2[´YZLORl«Lga7W~RQnw+RQ[]`iPcRAW;Y]l>`pl>\eJa2xY21
6 mxRQj2[ZRARAln
n`pl«J\kO`iWglZwJR.^_YZ`iaeWu;KMLO`pllZ`@?QRtaeRQ[][Za2[]l´^Qae[][ZR.l«w+aeWgmglRAbgYZa&Y]LORtw+RQ[Zae[]P\kWg^QRt\kWg\ebxlZ`il´`iWDnR.^_Y]`ba2W
Sg
No. 21 No. 26 No. 31
No. 16 (a)
(b)
No. 36 No. 41 No. 46 No. 50
°} 53A 3 5(& $ /D"$ ( K) & B # 9n9#H%D/ "
$K43 5$Q 7698 $3 ! ( "$ ' G( E nn 4H% " %$ ' G( AE
nn ;: PA$Q'3A % '"$ Q ' % ! AH / '"$ & 3 53 37G&