1、rXYXC(0,r)M(tCXMCC0,0)CMMCMNM=(),()xtyt()xtMCOCMCMCsinrtrt()ytCNCCcosrrtM(sin,cosrtrtrrt)2()(3,6,6)rttt2*21()(1,2,2)12rtttt*()rt123(,)ccCc2*12321cos(),)()(22)12rtCrtCcctctata2221231ccc1322,02ccac()rt22(,0,22)4clPllcPc()rrtP0ttcl1/ll1011()|,lcrrtttttt0121,)il/lli(idistldistl,)il010()|,iicrrtttttt2il2
2、,3,4,i12/nllllnll0(rtt20112 )()/nnnnrttlttP0()rt0201012()()()nnnnrttrttrttt0/()lrt/lcP()rt0t()x00t0t()yfx()xzg0()0xt0()0xt0t0()t()xxt0()t0()t()xxt()xth0()t()()()yytyhxfx()()()zztzhxgx0t()yfx()zgx2221xyz22xyx0z222211x22222211()24xyzyzxyxxy1sin2yt11cos22xt11cos22zt02t11111()(,s)ttcos,sinco22222rtt2.21
3、.,0a1,.()(,)rtachtashtat0tb2xyacha,0,x.3,()(cos,ln(sectan)sin)rtatattat0,t.1222()(,),|()|(1)2.rtashtachtartashtchtacht00|()|22.bbsrtdtachtdtashb2,()(,),()(1,tttxtyachrttachrtshaaa).200|()|1xxtxsrtdtshdtashaa.31()(sin,(cos)cosrtatatt22222001|()|sin(cos2)lncos.costtsrtdtatatdtatt2.drds1()(cos,sin,),0.
4、rtatatbta233()(cos,sin,cos2).rtttt1()(sin,cos,)rtatatb2211(sin,cos,).|()|drdrdtdratatbdsdtdsdtabrt222()(3cossin,3sincos,2sin2)rtttttt22()1(3cossin,3sincos,2sin2).|5sincos|()|drrttttttdsttrt3.c22221,0.xyzxachababa(,c0,0)a(,)xyz.,zt,ttxachybshaa()(,)ttrtachbshtaac()(,1)tbtrtshchaaa222200|()|zzabtzsrtd
5、tchdtabshaaate4.,.()rrt2(0)(1,0,5),()(,)trrtt2()(,)trttte3211()(,)32trtttec,c.,0t(0)(0,0,1)(1,0,5)rc(1,0,6).c3211()(1,6).32trttte2.3Frenet1.1=,2ln,.0aratatat2323=3,3,3.rttttt3=sin,1cos,.rattatbta0t433=cos,sin,cos2.rtt12222()(,),()0,aaaartarttttt32422432222()(),.aaartrtttt2223|()()|2.1|()|rtrttatrt22
6、2()(33,6,33),()6,6,6,rttttrttt22()()81,2,1.rtrtttt223|()()|1.31|()|rtrttrt3()(1cos,sin,),()sin,cos,0,rtatatbrtatat2()()cos,sin,cos1.rtrtabtabtat2243322224sin|()()|2.|()|4sin2tabartrtrttba422()(3cossin,3sincos,2sin2),rtttttt2323()6cossin3cos,6sincos3sin,4cos2,rtttttttt233222()()12sincos,12sincos,9sin
7、cos.rtrttttttt3|()()|3.25|sincos|()|rtrtttrt2.122()=cos,sin,0.rtatatbtab2,()=cos,sin,trtatbte0t,0ab.1()sin,cos,()=cos,sin,0,rtatatbrtatat2()()sin,cos,.rtrtabtabta=0Xr,()()=0Xrrtrt.sincos0btxbtyazabt2()sin,cos,()=cos,sin,ttrtatbtertatbte()()cossin,sincos,.ttrtrtbettaettab=0Xr,()()=0Xrrtrt,=0t,0,1,.ra
8、rrbaab,2xyzab.3.,sin1ln1zxshxyyzexx0,0,0Frenet.,xxsyyszzs,.s0s0,0,0,xsyszs222sin1(1)ln1213zxshxyyzexxxyz1,2,scos451zxchxxyyyxzezxx,0s10003xyz.3330(0)(,333r).3,4,5,s222220sincos*11zzxxyyzzxshxxchxxyyyyyxxzezezxxx,0s121100,0,0,9999xyzr2.96|0|9r,0666220,000,66322|0|rr,0.4.22222+=9=3xyzxz2,2,1,xxsyyszzs
9、,.s0s2,2,1,xsyszs22222222913213xyzxzxyz1,2,s0405xxyyzzxxzz,0s120,0,033xyz2.312200,333r.3,4,5,22222000xxyyzzxxxyyyzzzxxxzzz,0s1100,0,0,33xyz1100,33r022|0|,00,32|0|rrr2,222220=00=.36622222210366xyz0,49xyz.5.2211,0,0()0,0,0,00,0ttettrtttet,0t00t0t21320,(),1,0ttrtet21320,()0,1,ttrtet0,(0)0,1,0,(0)0,1,0,
10、(0)0,1,0.trrr,()0.trt.2164460,(),0,0ttrtett2164460,()0,0,ttrtett0,(0)0,0,0,(0)0,0,0,(0)0,0,0.trrr0.0t,221316()12(),1,04|()|1ttrttetrtet0t2()()()=sgn460,0,1|()()|rtrtttrtrt2221316sgn462()()()1,041ttttttetet,22131612()0,1,41tttetet0t2()sgn461,0,0tt2221316sgn461()0,141ttttetet0,0,1,0,0,0,1,1,0,0t0,0,1,
11、0,1,0,0,0,0,1t2.4Frenet1.31.1=,2ln,.0aratatat2323=3,3,3.rttttt3=sin,1cos,.rattatbta0t433=cos,sin,cos2.rtt12222()(,),()0,aaaartarttttt32342222226()=0,(),(),()21|()()|aartttrtrtrttatrtrt222()(33,6,33),()6,6,6,rttttrttt222()=6,0,6(),(),()131|()()|rtrtrtrttrtrt3()(1cos,sin,),()sin,cos,0,rtatatbrtatat222
12、2()=cos,sin,0(),(),()cos1|()()|rtatatrtrtrtbbatrtrt422()(3cossin,3sincos,2sin2),rtttttt32322()=6sin21cossin,6cos21sincos,8sin2(),(),()825sin2|()()|rttttttttrtrtrttrtrt2323()6cossin3cos,6sincos3sin,4cos2,rtttttttt2.33.0,0,0sin1ln1zxshxyyzexx333112(0)(,),0,.333999rr,*222332332403cos3sin1212311zzzxxxyy
13、yzzzcos1xchxxshxxxchxxyyyyyyyyxxxxxxxxxzezezzezxxx,0s8303,818181r320,0,0102|0|rrrr3.,()rrs1()()rssds.1r,21r,212r32123522,rrrrr232,|,.|rrrrrrrr4.2211+dds.11111ddddrdsdsdsds2211+dds,s2121110dddddsdsdsds2,11110dddddsdsdsds10dds,110dddsds,1110ddrdsds0111drrds0r,.0|rrc,.c10dds,5.()s,()()(),()()(),()()()s
14、ssssssss()s,()()()()()()()sassbsscsss,()()()()()()sabcsbcsabcsacsabcsab,0,()()()()()abcssss6.1,2,3,.1,c()()0rscrs()()0rscs.,()()0rscrs,()0rsc.,.()0rsc02c,.0rc,0rc.01,.0rc3c,0rc0rcr,0drcrcds,|rcc,.7.123();(),(),()rssss()rs,31iijjjdds,1,3i0ijji.0()()jiijjiijjiddssdsds.8.33221+1()=,11332sssrss,s111(),|
15、()|1222ssrsrss.2112(),0,|()|.414141rsrssss332222,112()1,1,0,.8841|rrrrssssr111()(),222sssrs()11(),022|()|rssssrs111()()(),.222sssss9.()s()rrs21,21 dds,Frenet.2,32322322|1.|,.|1.|dds21.|121.110.rrt;(),(),()rtttt33,|sgn2222,|,|rrrrrr23|r,222,|,|rrrrr223|r,|rr|sgn|rr,.|33,|,sgn.2.51.,0.,.()rrss,.ast()s
16、aconst.,()0sa()0sa0,()s.220100,020dds,const.c,c,cc.,scaa.,0ca,ac,()rs.2.cc,0()()stssds()dtsds.cddsdtdtdcdtdcdt2221dddccdtdtdt,22cos1sin1ActBct.2|122222222cos1sin12cos1sin11ctActBABctct22110ABAB12AeBe,22cos1,sin1,0ctctddtt,2221sin1,cos1,01ctctac2|1222122|c212sin1cos1111actecteac0t21tc,22|0,|1ceaac221
17、tc2321tc,10ea,a1e2e321caec2221sin1,cos1,1ctctcc,s222001sin1(),cos1(),1ssrcsdsdscsdsdscsc3.()3sin,2cos,3sinrtttttt12cos,2sin,22 uuruu.()13cos,2sin,3cosrtttt,()3sin,2cos,sinrtttt()3cos,2sin,cosrtttt,()()23cos2,4sin,2cos23rtrtttt,8rrr.,rrr32|11,44|rrrrr.1()sin,cos,122 uuru,111()cos,sin,02222uuru,111()s
18、in,cos,04242uuru,1111()()sin,cos,22222uururu1,8rrr.111111132111,|11,44|rrrrrrrr.11,.rt.1()ru4.1:,crchtshtt,2:,22uueecru1,.3E1c2c,s,22tttteeeechthttt11022110,0,1,102222001tteet11022110220011c2,c.2.61.,.,Bertrand.,.12,cc1c1111();(),(),()rssss1()s2c1cs21()()rss()rss221111111|1r110(s)()sc21|()()|()|rsrs
19、sc1c2c121212dds11222121221|0/,r12()()ssc1c2c01stcrk:(rs)s()rs1()()()rsrsss11r112221111|rr112221cos,cos10001ctg01ctg0ctg1()()()rsrss1rr1/1222111122sgn1,s11122sgn|r1/1()rs()rs()rs1c2c1c2c2c1c1c2c1c1()rrss1cc2211()()()rsrscssc1()sc12211:()()()()crsrssss1r2r1212r1211r112110rsc1c1()rrs1()s1c1)s2c2c21()()
20、()()(rsrsss1rs2()/122111:()()()()()()crsrsssss12111111-+-+r22c1c11()()()()/()ssssrs1120r22211111110,1-,-,+,0221111-0,012211,arctan111tan()sds211111111()()()tan()()()()rsrsssdssss=()rrss11()()()()rsrssss11()()()()()()srsssss()rs()0s11()()()rsss)1()rs10()/(rss0s0r()rrs0s222000111lim00|0srcs|()rs0c220(
21、)0002srsrsos0100r0c222000111lim|()000|srsrccs222020011lim|000|2ssscsc222202000111lim2sssccs22202022220004lim0112sssscc()rrss()rrss1123()()()()()()()()rsrsssssss222123()()()()Rssss22130()()lim0srssrsRssrss2311()()()()6srssrsssss232()()()()26ssssss333()()()()6sssssos22112()2()1()()rssrssssss2332312123
22、()()()()()()()()()()333ssssssssosss22s231212()()()2()0,1()()0,()()()()333ssssssssss12321()()0,(),()()()()sssssss1221()()()()()()()()()()srsrssssssss2.71.r1cos,sin,02ratbtttsin,cos,cos,sinratbtratb3322222222sincosrxyxyabxyatbt2,rachtbsht,rashtbchtrachtbsht322222rabashtbcht32,rtt,1,2,0,2rtr322214rt4si
23、n,1cosrattat1cos,sin,sin,cosratatratat1222cosrat5,trtacha11,0,ttrshrchaaa,21rtchaa6cos,lnsectansin,02raaa2sinsin,cos,cos,sincoscosaraaraaatanra2.,=,.223222cos,sinsincos,cossincos2sin,sin2cos2r rrr3.211rss,.s2020202200,00,0010arctan10cosarctanln|1|0sinarctan11ln|1|,11sssxysdsssxsxsdsssysysdssrssss4.1
24、.1=cos,sin,02ratbtt.2222332211,csincos|1cossin,rtyxbtaatbtrttrttababos,sint2.,rachtbsht222222223311,|1,rtyxbchtabchtashtrabbarttchtshtabsht3.2,rtt23211,21+4|114,32rtyxtrrtttt,1t4.sin,1cosrattat211,sin,22cos|1sin,1cosrtyxataatrrttattat1cost5,trtacha.211,|112,22rttyxshatrshaattrtttshachaa16cos,lnsecta
25、nsin,02raaa.1cos,sincos|1sec,lnsectanraaarraa5.1222xya.02cos,sin|,()cos,sin,()()sin,cos()=()+()cossin,sincostrtatatssrtdtattassssrsaasrsaaaasssrsrscssacsacsaaasa2xyacha.02,|()=()+,|ttrttachatsrtdtashattcashcashrttaartrtcashtttachchrtaa3sin,1cosrtttt.021cos,sin|4cos2()=()+4cossinsin,3coscos22|trtttts
26、rtdtrtttrtrtcttctcrt2 t3.12222222222222222221.1,coscos,cossin,sin,22+=1,seccos,secsin,tan,22+=xyzabcrabcxyzabcrabcxyzabc2222222221,tancos,tansin,sec,0,2+=21,cos,sin,2=2,2rabcxyzabruvauvbuvuxyzabruvauvbuvuv222222222222.+=1N=001,=001=,0,2211,.11123xyzSuvuvuvxyzuvuvuv22222222.3,21,1.11211,1,0NP/NPuvuvt
27、xtuytvztuvxyztuvtuvtutvt222222222212P=,11uvuvuvuvuv,12,2,2222,2,2xyy2222,2222uy22vv11N=001,=001001,=0S22222222222uuxyxy222yzy2222222=2zz,2u2222222222223,1,*112212,11*,uvuvxyuvzzuvuxyzuvuvuvuvuvuvuv,1v2222223.+=111+=11111,222xyzxzxzyyabcacacbbxzyuacbxzyuacbyaaccvxuvuzuvbuuu2222121111,0,22222,+22,22uu
28、acacruuvubuuuuuxyxyxyzzabababxycabxyczabuavaubvbvzucxyuur,0,122uaubabvuu312341213241213244.14,OP,1OPOP1PPPP,OPOPPPPPiiiitixyzttrtrtr13241213122413241PPPP=1PPPP+PPPPPPPP0trrtybcvv11c1uuu2uu2u22+22222zz22czz22uavauvauucxyy,2,r11uu11uuuu1111111111cc1u,u,u,u,uu22u22,222,u22xyxy+ababababxyxyxyyyabbabbxy,
29、uc,uaubab0,0ua0,0,0222u,0,22,0,34.4.OPOPrr33111111zz,uu,zz,zz,uu222yybb22,1111111111uu222,2yy22,22222222212111222225.=,cos,sin,cossin2cos0,2cos,0,0,042cos,2cossin,4rruvuvuvbvxayauvauvauauvuuavccrvbvarvavavvbvab222 2,4bab3.2221.00000,0rcrdrdrrrrdrrc2.,cos,sin,sin,cos,0,cos,sin,cos,sin,cossincossinuvu
30、vruvfvufvugvrfvufvurfvufvugvrrfvgvufvgvufvfvxfvuyfvuzgvfvgvufvgvu0,0,0,0,0,0,/,00uvfvfvfvfvgvZgvZZhuvruvxuvyuvzuvruvhuvrrxyzxyzhuuuxyzxyzhvvvS222222222222220,.=cos,sin,xyzzhuuxyzzhvvxyxyuvzzuvxyzfstxyfzSrfzfzzS3.,X=+=0,=,X000,uvruvavlurvlurluuvavluvluvarxyfxy000,uzvv zZZZZZ,222200vzvvxyzy222SS2uvvuv
31、zzuvuvyy22zzz3.3zgzzzff1,0,0,1,0,10010,xxyyxyxyxyxrfrfrrrrrrf0yffxyxfyfxyfxyxfxyFy2222220000004.1,.,xyzabcabccabccbbaxyzxyzfab1232221112222222231,.,0,0,0,0,1,2,311icccbbafccbbaffcfcfbfbfaccbbafixyzabcxyzabcxyab2330000002220002222220001,1,2,3,10,iiiiijijijijijijijijzcxyzxyzniabcxyznnaabbccxxyyzzijaab
32、bcc5.cos,sin,2,.1,2:.tuvuvrvuvuvcutvecrrcrr22200.yzz00ab0000b1222222222231,1,111cczyzy222xyyab31c2bcb22231,1,cyybcb2233211ijjiijijzzxx0ni33ccijjjc00jj00njj22xx0x000aa11111122sin,cos,cos,sin,122cos,(),cos,()22|,()tuvuvuvuvuvuvudrdudvrtrrrerdtdtdtrvuvuvruurtrtrtrtrtrtrtrtrtrtrtrtrtrt=,()=4vrtrt3.31.:(
33、1).(cos,sin,().(2).(cos,sin,(),ruvuvvruvuvuava222222222222(1).(cos,sin,0)(sin,cos,()cossin1,0,()2()uvuuvvrvvruvuvvErvvFrrGruVIEduFdudvGdvduuvdv2222222222(2).(cos,sin,()(sin,cos,)1(),(),2(1()2()(uvuuvvrvvuruvuvaEruFrrauGrua22)IEduFdudvGdvuduaududvuadv2.22222222222,auavuvaruvauvauva,.22222222222222222
34、222222222222222222222222222222222()44,()()()42()4,()()()44,0,()()4()uvuv auvaauvauruvauvauvaauvauvaavruvauvauvaaaErFGruvauvaaIuva22()dudv3.,2220PduQdudvRdv.:20ERFQGP222()(ua22)uadv2()(2)u2u2()(222avuvavuva2vv22,22avuvavuvavvvauvaauv2222222vauvauvau22222,2222222442()222224()(222u2222,()22222,u2222,2a
35、I2)444auvaa444)222222222222),2222222222222222)auva)444)()()()(22222222222222222222)()()()()()()(222222222222222222222(22222)2(222(222()(),2222222()()2()()()22222222222222222)()()()()(22222222222r22222,22222222)I22(3.22()(22()()()()(22()()2222222,vava2222vauvav222.222222222121212121212121212121212121
36、220202,()02()0duduPduQdudvRdvPQRdvdvdudududuQRdvdvPdvdvPdrdrdrdrEduduFdudvdvduGdvdvdudududuEFGdvdvdvdvRQEFGPPER20FQGP)4.222222220020222222222(coscos,cossin,sin)cos,0,coscoscoscosraaaEaFGaIadaddrraddrraaad222220022000001coscostancos10,tan,tanlnsectan2cos(,tanlnsectan)crc5.2222()Iduuadv(1).12:0:CuvCuv
37、0(2)22123:,:,:22 aaCuvCuvCv1.(3).,uavuav1v2v22222222222222222222os222dadd22adad22222222222d222222222222222222000cos1coscos0tanlnsectan)lnsectan)22000000010,0,tanlnsectanlnsec0tan00tanlnsectanlnsectantanlnsectanlnsec,tanlnsectanlnsec0I(1)1CC1:0:000(2)(2)11aCuCu11:a(3)(3)u222022raa22n,sn,n,222co22rcos
38、co2222221cos1cososs222222221222222222222222222()1,0,(1).0,00,()cos()()()()(2IduuadvEFGuauvdudvdudvuvuvCCdrrduuuadvvdrrduuadvuuavdvvuadvvdvuadvvua22222222222)(1)(0,0),cos1(1)(1)1:arccos,.1vadvvaaadvavaa1123(2).,(0,0),(,1),(,1)22 aaCCCOAB2112242007()44267,6aduOAuadvvvdvadvOBaABa11122233,CduadvCduadvCv
39、01322222421132121222222211222cos3()()1422arccos,arccos33cos10duduvAduuadvduvvABAduduadvdvOduadvduadvO123(3).,(0,0),(,1),(,1)CCCOAaBa1222220222ln(33auaDDAEGFdudvuadudvuadudva221)11),(,1)(,1)2),(a(1),)1424107702 advdv74444424dvv444442400 11222222222222CduadvCduadv22222222222222C2213322222221111221(112
40、arccos,3,cos1200duduudu133d222)2222112dudu1212OOOduu221dvdv21()222ABABA2arccos,aOO11(3).,(3).11,11),(,(a,(,(,ABaABa,()2222()()2222()2221advadv113.51.(cos,sin,),02(cos,sin,),02,ttrachachttaarvuvuauuv22222*22222222*:(),:ttIchdtachdaaIvadudvttachvauvashuIIaatuvasha,.2.:(cos),(sinsin),().raconuvauvbuv22
41、2222:(2coscos,2cossin,2()222222cos,(cossin22(4)()cos,()sin,(),()()()uvuvuvuvuvraabuvuvuvarvuvubuIvbdudvrfvufvugvIfvdufvgvdv222222222222222,2cos,22()4,()()1()4,()2ln4,.(4cos,4sin,2ln4:,2cos.22uvuvuuvvafvvbfvgvfvvbgvbvvbIIrvbuvbubvvbuvuvuva)3.:)2222,2cossin,2sin22222,(),v2i2i,(),(,dvdv2222()()()22222v
42、uvuuvuvu2iii()()22222222(coscosr(cos(coscosr),os,()sin,(,()i(222222222,2cos,2222,v,f2()4,)4,222222222f(),()()ru4,(4,(222222vv2222222222cos,2cosv222u222f()4,()()224,()()22)223.3.sinsisinsioscososcos22,22uvuvuvvdvd(uuos,s,vvuossr()c()()22*22*:(,)(,0),(,)(,0),(,),(,),TTruvuvIdudvruvuvIdudvufuvvguvfgdud
43、uduuuJIdudvdudvJJIdudvfgdvdvdvvvIIJJ1,1)1JJJ4.()cos,()sin,().rfuvfuvgu2222222222222222222222202()()()()()()()(,0)()()()(),()()()uIfugudufudvfugufududvfuruvIdudvfuguftgtdududvdvudtvvfuftIfuI,2220()(),.()uftgtudtvvft5.221122222221111222222211112221(,0),(2coscos,2cossin,2()22222,22(4cos4)4sinsin(4cos4)
44、cosruvIdudvuvuvuvuvuvraabuvuvuvIavbduavdvavavbdudvavb2222221112221sincosavdududvdvavb11.22222222dv()()22dudvd2d222u002Idu2(2222u00d2dvdv222222vdv222222222200)uf22()()()(2222uf()()f()()ff22(00uf()()112(,),(2coss22(urr(2cossuuII(42vvv(42(,0),ruv(,0),uv(0)(0)11uvuvuvu11vv2()()(22ff2()()II)()()22222dudv
45、dv2222u2()()222dudu()()222222dudu2222221,112lncoscosbuuvvva2221222(4cos4),lncoscos222IavbIuvuvuvbuva.3.6223422342321.1:=,2,3312=,2,332,6,46,02cossin,sincos,2=cosvuvruuuvuuruuuvuuvluuuuuuuluululuruuvvvuvvuvrv,sin,sin,cos,1sin,cos,1,0=.3:,2.:=,0,2,0,0,0,2,vvuvvvvtlvvvvlvululurvtvrrauvbuvuvraubuvabuuvl
46、uuabluululu40ab2.12:0=,.()()(),()()()()(),()()()()(),()()()(),(),()rrssrsrstsrsrstsrssssssssssrsss()0,(),(),()()0,.srssss3.12220.:,()()()()()()()=,rstrstssssastlsarlall2=0.404b4411(),(),()rr11(),),()()()()()()(),()()()()()()(),(),()()()()()()()(),()()()()()()()()()()s()00()0,)0,0,()00()00()0,0)0,0,(
47、),3.ara=0:rr210,=+1,=tan(),=tan().=0:,()()tan()()()sdssdsrstrstssdsss14.=(),()=()()+()(),(),()()()()()+()(),rrsslssssslsrstrstlsrstssssrlrll0,=0,()()=()(),()=()()+()()slssslsssss.12.:,=0,cSScSSSSruvruvlucrruruluruluSrutrutnunuScnurulurnnrr2,|,lrlrlrrlrlrlrlrrrlrlrlrrllSS5.(),(SccScSclllllllllllllllllrlrllrllrlllllrrlrrllllrlrllrlrllrrlr2llll|2l|llll|lrlrlrlrllrlllllllrlrll|rlrllrlrlrl2S)()()()()(SS4.12222221.1coscos,cossin,sin12,23,2:1sincos,sinsin,cos,cossin,coscos,0coscos,cossin,sincossincraabruvuvrauvauvuvraabraanbbabara2222222222222222oscos,cossin,sin,sinsin,sincos,0coscos,cossin,0c
