{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "restart:ps:=proc(V,W ) add(V[i]*W[i],i=1..3) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p:=proc(m,v) [seq(simplify(m*v[i]),i=1..3)] end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "u:=[1,0,a];k:=[0,0,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%\"\"\"\"\"!%\"aG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG7%\"\"!F&\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "M:=[cos(t),sin(t),a];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG7%-%$cosG6#%\"tG-%$sinGF(%\"aG" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 52 "z:='z':K:=[0,0,z]:z:=simplify(solve(ps(M-K,M)=0,z)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG*&,&\"\"\"F'*$)%\"aG\"\"#F' F'F'F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n:=M-K;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG7%-%$cosG6#%\"tG-%$sinGF(,&*&,& \"\"\"F/*$)%\"aG\"\"#F/F/F/F2!\"\"F4F2F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "v:=simplify(p((1+a^2)*ps(u,n)/ps(n,n),n)-p((1+a^2)/2, u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%,*#!\"\"\"\"#\"\"\"*&# F*F)F**$)%\"aGF)F*F*F(*&)-%$cosG6#%\"tGF)F*F.F*F**&F2F*F.F*F(*(-%$sinG F4F*F.F*,&F2F*F*F(F*,$*&F/F*,(F(F*F-F**&F)F*F2F*F*F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "p(ps(v,k),M);p(ps(M,k),v);w:=p(ps(v ,k),M)-p(ps(M,k),v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*(%\"aG\" \"\",(!\"\"F'*$)F&\"\"#F'F'*&F,F'-%$cosG6#%\"tGF'F'F'F.F'#F)F,,$*(F&F' F(F'-%$sinGF0F'F2,$*&F+F'F(F'F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%, $*&%\"aG\"\"\",*!\"\"F'*$)F&\"\"#F'F)*(F,F')-%$cosG6#%\"tGF,F'F+F'F'*( F,F'F/F'F+F'F)F'#F'F,*()F&\"\"$F'-%$sinGF1F',&F/F'F'F)F',$*&F+F',(F)F' F*F'*&F,F'F/F'F'F'#F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG7%,&* &%\"aG\"\"\",*!\"\"F)*$)F(\"\"#F)F+*(F.F))-%$cosG6#%\"tGF.F)F-F)F)*(F. F)F1F)F-F)F+F)#F+F.*&#F)F.F)*(F(F),(F+F)F,F)*&F.F)F1F)F)F)F1F)F)F+,&*( )F(\"\"$F)-%$sinGF3F),&F1F)F)F+F)F+*&#F)F.F)*(F(F)F:F)F@F)F)F+\"\"!" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "N:=M+p(l,w);dn1:=diff(N,t) ;dn2:=diff(N,l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG7%,&*(%\"lG \"\"\"%\"aGF),.!\"\"F)*$)F*\"\"#F)F,*(F/F))-%$cosG6#%\"tGF/F)F.F)F)*&F 2F)F.F)F,F2F,*&F/F)F1F)F)F)#F,F/F2F),&**F(F)F*F)-%$sinGF4F),*F6F/F-F,F )F,*&F/F)F2F)F)F)F8F;F)F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$dn1G7% ,&*(%\"lG\"\"\"%\"aGF),**(-%$cosG6#%\"tGF))F*\"\"#F)-%$sinGF/F)!\"%*&F 3F)F1F)F)F3F)*(\"\"%F)F-F)F3F)!\"\"F)#F9F2F3F9,(**F(F)F*F)F-F),**&F-F) F1F)F2*$F1F)F9F)F9*&F2F)F-F)F)F)F:*&#F)F2F)**F(F)F*F)F3F),&F6!\"#*&F2F )F3F)F9F)F)F9F-F)\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$dn2G7%,$* &%\"aG\"\"\",.!\"\"F)*$)F(\"\"#F)F+*(F.F))-%$cosG6#%\"tGF.F)F-F)F)*&F1 F)F-F)F+F1F+*&F.F)F0F)F)F)#F+F.,$*(F(F)-%$sinGF3F),*F5F.F,F+F)F+*&F.F) F1F)F)F)F7\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "solu:=si mplify(solve(dn1[1]*dn2[2]-dn1[2]*dn2[1],l));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%soluG,$*&\"\"\"F'*&,&F'F'*$)%\"aG\"\"#F'F'F'F,F'!\" \"#!\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "x:=subs(l=s olu,N[1]):x:=-1/3+factor(x+1/3);y:=factor(subs(l=solu,N[2]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG,&#!\"\"\"\"$\"\"\"*(#\"\"#F(F), &F)F)-%$cosG6#%\"tGF)F)F.F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"y G,$*&,&\"\"\"F(-%$cosG6#%\"tGF(F(-%$sinGF+F(#\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([x,y,t=0..2*Pi],scaling=constr ained);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7`o7$$\"\"\"\"\"!$F*F*7$$\"3#pp;u(G;`**!#=$ \"3?G*QibXD6*!#>7$$\"3kI&*e\"G4L\")*F/$\"3e\"R]&\\\"[=\"=F/7$$\"3pK(> \\&=W<'*F/$\"3OGYn$y-'yDF/7$$\"3Uu$G!pypa$*F/$\"3+q-ow$GEK$F/7$$\"3=lQ #\\6.G)*)F/$\"3L<&4G&G@CTF/7$$\"38l(QIW9[`)F/$\"3Ah(4^')*H!)[F/7$$\"3& eBgKA:I,)F/$\"3'RN&[i1U(e&F/7$$\"3<.J\"4q*3FuF/$\"3g'eKs@OJB'F/7$$\"33 DU1%4ozy'F/$\"3>L$RgcN&3oF/7$$\"3%z*R'=q985'F/$\"3i;R+z>g6tF/7$$\"3\\S U\"[R?*HaF/$\"3?pRIxI)F/7$$\"3Bo]KdDPQKF/$\"3$o63rbh,])F/7$$\"3[4rp' \\vnX#F/$\"3boVk!*3')>')F/7$$\"3Ud#QoZO5o\"F/$\"3?u^UF1Cg')F/7$$\"3'[- E\"Q$[JA*F2$\"3Am%3Ne]Ri)F/7$$\"3O:$HVBUK'=F2$\"3l))[@$R[Y^)F/7$$!3UDr %3n5(*Q&F2$\"30orwA'>1L)F/7$$!3wtrdf2dC7F/$\"3y#y5G1*fz!)F/7$$!3#pd>oC \\fJ#F/$\"3#H\\,A:'o%\\(F/7$$!3R6Wgp;JPLF/$\"33$pPdw%oimF/7$$!3&)z8@%p (*y6%F/$\"3&*Rm&R'pk/dF/7$$!3&*[o/-hjIYF/$\"3h^!>HX&yFZF/7$$!33n5L?#\\ '**[F/$\"3:38]TrUWQF/7$$!3m;7%yi!))**\\F/$\"3K6s!fE+U&GF/7$$!3Fm'p&3T' z#\\F/$\"3%)Hf>'*>T/@F/7$$!3^8Ha8Q)*3ZF/$\"3q!)R2B#f&o8F/7$$!3,Y+$)Glq PWF/$\"3]/A(3NP(e&)F27$$!3yO>$*y#)H2TF/$\"3'y8+YsV/Z%F27$$!3!ya5N)=h1Q F/$\"3eEZZ:Fvr>F27$$!3))yc$30T+b$F/$\"3Y]eQQ`Qnd!#?7$$!3%[PpD4PZR$F/$ \"3:M)*G?C\">V)!#@7$$!3[ypl$RlLL$F/$\"3o,z@Ui%H%**!#E7$$!3%3%pvZd\"QR$ F/$!3ay2e#=$QT#)F^v7$$!3\\b=_+3aVNF/$!3A8$=S!**Q-bFhu7$$!3iW#GNN1$)y$F /$!3'3[/T&RI]=F27$$!3N9J6^P-+TF/$!3U-%ftq\")zR%F27$$!3S\\]8z*y8U%F/$!3 LP[R[GG8$)F27$$!3CaE,&4MIq%F/$!3s>KUe5Wa8F/7$$!3$R^rlw(pE\\F/$!3U0Hb4. w(4#F/7$$!3e#pH'>U****\\F/$!3g3&*e90#R*GF/7$$!3%49Esk&f**[F/$!3E)*HL)o %oWQF/7$$!31<.')zzw=YF/$!3e=gS+!Rpv%F/7$$!3'*3q;,#oJ3%F/$!3KRB\"eT&*pv &F/7$$!3ud&RAd`ON$F/$!3#>VCIkBF/$!3se?+Ne\\iuF/7$$ !3e2OJ%ocZ>\"F/$!3?%)*o()H4A4)F/7$$\"3!y?=oSbv/#F2$!30!>HP26$=&)F/7$$ \"3MnI`k\"G+F*F2$!3S7*zz)*3Wi)F/7$$\"3U@.0h\"o5n\"F/$!3gWsczFDg')F/7$$ \"39+&3^\")pkW#F/$!30*R#frr!4i)F/7$$\"3u9t!*>W%yA$F/$!3GH7k!=3B])F/7$$ \"3='3**e`@(**RF/$!3&=[$e6740$)F/7$$\"3]2]Y*)z`fZF/$!3=//;qYPG!)F/7$$ \"3?+Q1>AKRaF/$!3lzm'4qTZq(F/7$$\"3C?KKRoI%4'F/$!3(*GTg=p?;tF/7$$\"3)) >w$*QFt0oF/$!3A,LRr]0%z'F/7$$\"3?1)[rwLeY(F/$!3i=vzL8J%>'F/7$$\"3y&)*f fS-_+)F/$!3%*o)[UFVpf&F/7$$\"3mC%*fuM/!\\)F/$!3=+'QWI$>Z\\F/7$$\"3$\\: t\"\\MtS*)F/$!3\\&Q.)fe7.UF/7$$\"3Vs\\\"p0FtJ*F/$!3#)y$Hx=0OT$F/7$$\"3 DLGkWY#Qg*F/$!3QL4uFr-BEF/7$$\"3M))pTB#)[9)*F/$!3ux8HV5=1=F/7$$\"3]w9[ !pfM&**F/$!3ET`IA[q$3*F27$F($\"39m<3_\\)Q4\"Fdv-%'COLOURG6&%$RGBG$\"#5 !\"\"F+F+-%+AXESLABELSG6$Q!6\"Fbal-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6 $%(DEFAULTGF[bl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "n1:=[n[1], n[2],0];w1:=factor(simplify(p(2*ps(w,n1)/ps(n1,n1),n1)-w));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#n1G7%-%$cosG6#%\"tG-%$sinGF(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#w1G7%,$*(%\"aG\"\"\",&F)F)*$)F(\"\" #F)F)F),&-%$cosG6#%\"tGF)F)!\"\"F)#F)F-,$*(F(F)-%$sinGF1F)F*F)F4\"\"! 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