{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 54 "T.Pajdla: Inverse Kinemat ics of a 6-DOF Manipulator. \n" }{TEXT -1 153 "[1] D.Manocha, J.F.Cann y. Efficient Inverse Kinematics for General 6R Manipulators. IEEE Tran s. on Robotics and Automation, 10(5), pp. 648-657, Oct. 2004" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 257 20 " Packages & settings" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "restart:\nwith(linalg):\nwith(Linea rAlgebra):\nwith(PolynomialTools):\ninterface(rtablesize=20):" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, the assigned name GramSchmidt now has a global binding\n " }}{PARA 7 "" 1 "" {TEXT -1 53 "Warning, the name MinimalPolynomial h as been rebound\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 13 "DH-Kinema tics" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Joint transformations:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4295 "# Two one-parametric moti ons transformatin in DH-convention(phi, theta, a,d) indexed by i\n# c \+ = cos(phi), s = sin(phi), P = cos(alpha), R = sin(alpha)\ndhTs := proc (i)\nlocal M1, M2;\n M1:=Matrix(4,4,[[+cat(`c`,i),-cat(`s`,i),0, \+ 0],\n [ +cat(`s`,i),+cat(`c`,i),0, 0],\n [ 0, 0,1,cat(`D`,i)],\n \+ [ 0, 0,0, 1]]);\n M2:=Matrix(4,4, [[1, 0, 0,cat(`A`,i)], \n [ 0,+cat(` P`,i),-cat(`R`,i), 0],\n [ 0,+cat(`R`,i),+cat (`P`,i), 0],\n [ 0, 0, 0, \+ 1]]);\n [M1,M2];\nend proc:\n#\n# Inverse of the DH-conventio n for one-aprametric DH rigid motion transformations\ndhInvs := proc(M )\n local M1, M2;\n M1 := M[1];\n M2 := M[2];\n [simplify(Matr ixInverse(M2),\{M2[3,2]^2+M2[3,3]^2=1\}),\n simplify(MatrixInverse( M1),\{M1[1,1]^2+M1[2,1]^2=1\})];\nend proc:\n#\n# Rigid motion transfo rmatin in DH-convention(phi, theta, a,d) indexed by i\n# c = cos(phi), s = sin(phi), P = cos(alpha), R = sin(alpha)\ndhT := proc(i)\nlocal M ;\n M:=Matrix(4,4,[[+cat(`c`,i),-cat(`s`,i)*cat(`P`,i),+cat(`s`,i)*c at(`R`,i),+cat(`A`,i)*cat(`c`,i)],\n [+cat(`s`,i),+ca t(`c`,i)*cat(`P`,i),-cat(`c`,i)*cat(`R`,i),+cat(`A`,i)*cat(`s`,i)],\n \+ [ 0, cat(`R`,i), cat(` P`,i), cat(`D`,i)],\n [ 0, \+ 0, 0, 1]]);\nend proc:\n#\n# Inverse of the DH-convention rigid motion transformation \ndhInv := proc(M)\n simplify(MatrixInverse(M),\{M[1,1]^2+M[2,1]^2=1 ,M[3,2]^2+M[3,3]^2=1\});\nend proc:\n#\n# Simplify using trigonometric indentities c^2+s^2=1 & P^2+R^2=1\ndhSimpl := proc(M,i)\n simplify( M,\{cat(`c`,i)^2+cat(`s`,i)^2=1,cat(`P`,i)^2+cat(`R`,i)^2=1\});\nend p roc:\n#\n# Simplify using Rotation matrin in Mh\nMhSimpl := proc(M)\n \+ simplify(\n simplify(\n simplify(\n simplify(M,\n \{L x^2+Ly^2+Lz^2=1,Mx^2+My^2+Mz^2=1,Nx^2+Ny^2+Nz^2=1\}),\n \{Lx *Mx+My*My+Lz*Mz=0,Lx*Nx+Ly*Ny+Lz*Nz=0,Mx*Nx+My*Ny+Mz*Nz=0\}),\n \+ \{Lx^2+Mx^2+Nx^2=1,Ly^2+My^2+Ny^2=1,Lz^2+Mz^2+Nz^2=1\}),\n \+ \{Lx*Ly+Mx*My+Nx*Ny=0,Lx*Lz+Mx*Mz+Nx*Nz=0,Lz*Ly+Mz*My+Nz*Ny=0\});\nend proc:\n#\n# Simplify a general motion matrix using rotation matrix id entities in columns\nrcSimp := proc(M,R)\n simplify(\n si mplify(\n simplify(\n simplify(\n simplify( \n simplify(M,\{R[1,1]*R[1,1]+R[2,1]*R[2,1]+R[3,1]*R[3,1]=1 \}),\n \{R[1,1]*R[1,2]+R[2,1]*R[2,2]+R[3,1]*R[3,2]=0\}) ,\n \{R[1,1]*R[1,3]+R[2,1]*R[2,3]+R[3,1]*R[3,3]=0\}),\n \+ \{R[1,2]*R[1,2]+R[2,2]*R[2,2]+R[3,2]*R[3,2]=1\}),\n \+ \{R[1,2]*R[1,3]+R[2,2]*R[2,3]+R[3,2]*R[3,3]=0\}),\n \+ \{R[1,3]*R[1,3]+R[2,3]*R[2,3]+R[3,3]*R[3,3]=1\});\nend proc:\n#\n# Sim plify a general motion matrix using rotation matrix identities in rows \nrrSimp := proc(M,R)\n simplify(\n simplify(\n s implify(\n simplify(\n simplify(\n simpl ify(M,\{R[1,1]*R[1,1]+R[1,2]*R[1,2]+R[1,3]*R[1,3]=1\}),\n \+ \{R[1,1]*R[2,1]+R[1,2]*R[2,2]+R[1,3]*R[2,3]=0\}),\n \+ \{R[1,1]*R[3,1]+R[1,2]*R[3,2]+R[1,3]*R[3,3]=0\}),\n \+ \{R[2,1]*R[2,1]+R[2,2]*R[2,2]+R[2,3]*R[2,1]=1\}),\n \+ \{R[2,1]*R[3,1]+R[2,2]*R[3,2]+R[2,3]*R[3,3]=0\}) ,\n \{R[3,1]*R[3,1]+R[3,2]*R[3,2]+R[3,3]*R[3,3]= 1\});\nend proc:\n#\n# Matrix representation of a set of polynomials \+ \nPolyCoeffMatrix:=proc(S,m,Ord::\{ShortTermOrder, TermOrder\}) \nloca l A,v,i,j,k,c,q;\n A:=Matrix(nops(S),nops(m),storage=sparse);\n v:=indets(m);\n for i from 1 to nops(S) do\n \+ c:=[coeffs(expand(S[i]),v,'q')];\n q:=[q];\n \+ \011 for j from 1 to nops(m) do\n for k \+ from 1 to nops(q) do\n if (m[j]=q[k]) t hen A[i,j]:=c[k] end if\n end do\n \011 end do\n end do;\n Matrix(A);\nend proc:\n#\n## Ca rtesian product of a two lists\n#\nLxL:=proc(X::list,Y::list)\n Fl atten(map(x->(map(y->Flatten([x,y]),Y)),X),1);\nend proc:\n#\n## n x 1 matrix to a list conversion\n#\nM2L:=proc(M) \n\011convert(convert(M, Vector),list);\nend proc:\n" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 259 27 "6-DOF Robot IK formulation " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 189 "G iven ai, di, i = 1...6, and Ah, find parameters ci, si, pi, ri subject to\n\n(1) M1 * M2 * M3 * \+ M4 * M5 * M6 = Mh " }}{PARA 0 "" 0 "" {TEXT -1 71 "\n(2) (M11*M12)*(M21*M22)*(M31*M32)*(M41*M42)*(M51*M52) *(M61*M62) = Mh" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "(3) \011ci^2 + si^2 = 1 \011i = 1...6\n(4) \011pi^2 + ri^ 2 = 1 \011i = 1...6" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "[2] M. Raghavan, B. Roth. Kinematic Analysis o f the 6R Manipulator of General Geometry. \n Int. Symposium on R obotic Research. pp. 264-269, Tokyo 1990." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Write (1) equivalently as\n\n(5) \+ M3 * M4 * M5 = M2^\{-1\} * M1^\{-1\} * Mh * M6^\{-1\}" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "(6) M31*M32*M41 *M42*M51*M52 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 " = M22^\{-1\}*M21^\{ -1\}*M12^\{-1\}*M11^\{-1\}* Ah * M62^\{-1\}*M61^\{-1\}" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 260 39 "Raghavan and Roth construction of P & Q " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "iM12:=dhInvs(dhTs(1))[1 ]:iM11:=dhInvs(dhTs(1))[2]:\niM22:=dhInvs(dhTs(2))[1]:iM21:=dhInvs(dhT s(2))[2]:\nM31:=dhTs(3)[1]:M32:=dhTs(3)[2]:\nM41:=dhTs(4)[1]:M42:=dhTs (4)[2]:\nM51:=dhTs(5)[1]:M52:=dhTs(5)[2]:\niM62:=dhInvs(dhTs(6))[1]:iM 61:=dhInvs(dhTs(6))[2]:\nMh:=Matrix(4,4,[[Lx,Mx,Nx,Qx],[Ly,My,Ny,Qy],[ Lz,Mz,Nz,Qz],[0,0,0,1]]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Let \+ us first inspect the matrices. \nLeft hand side:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 26 "[M31,M32,M41,M42,M51,M52];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(-%'RTABLEG6%\")+%fO\"-%'MATRIXG6#7&7&%#c3G,$%#s3G! \"\"\"\"!F17&F/F-F1F17&F1F1\"\"\"%#D3G7&F1F1F1F4%'MatrixG-F%6%\")WyG9- F)6#7&7&F4F1F1%#A3G7&F1%#P3G,$%#R3GF0F17&F1FCFAF1F6F7-F%6%\");,v:-F)6# 7&7&%#c4G,$%#s4GF0F1F17&FNFLF1F17&F1F1F4%#D4GF6F7-F%6%\")#fad\"-F)6#7& 7&F4F1F1%#A4G7&F1%#P4G,$%#R4GF0F17&F1FgnFenF1F6F7-F%6%\")Kpv:-F)6#7&7& %#c5G,$%#s5GF0F1F17&FboF`oF1F17&F1F1F4%#D5GF6F7-F%6%\")[%fd\"-F)6#7&7& F4F1F1%#A5G7&F1%#P5G,$%#R5GF0F17&F1FapF_pF1F6F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The right hand side" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "[iM22,iM21,iM12,iM11,Mh,iM62,iM61];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)-%'RTABLEG6%\")!)Rw:-%'MATRIXG6#7&7&\"\"\"\"\"!F., $%#A2G!\"\"7&F.%#P2G%#R2GF.7&F.,$F4F1F3F.7&F.F.F.F-%'MatrixG-F%6%\")%= md\"-F)6#7&7&%#c2G%#s2GF.F.7&,$FAF1F@F.F.7&F.F.F-,$%#D2GF1F7F8-F%6%\") W'od\"-F)6#7&7&F-F.F.,$%#A1GF17&F.%#P1G%#R1GF.7&F.,$FRF1FQF.F7F8-F%6% \")_g79-F)6#7&7&%#c1G%#s1GF.F.7&,$FgnF1FfnF.F.7&F.F.F-,$%#D1GF1F7F8-F% 6%\")CQQ8-F)6#7&7&%#LxG%#MxG%#NxG%#QxG7&%#LyG%#MyG%#NyG%#QyG7&%#LzG%#M zG%#NzG%#QzGF7F8-F%6%\")3mx:-F)6#7&7&F-F.F.,$%#A6GF17&F.%#P6G%#R6GF.7& F.,$F]qF1F\\qF.F7F8-F%6%\")kp'[\"-F)6#7&7&%#c6G%#s6GF.F.7&,$FhqF1FgqF. F.7&F.F.F-,$%#D6GF1F7F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "Notic e that the the two last columns of iM61 are free of c6, s6 and so we c an get six equations without the sixth variable." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "M31,M32,M41,M42,M51,M52[1..4,3..4],\"=\",iM22 ,iM21,iM12,iM11,Mh,iM62,iM61[1..4,3..4];" }}{PARA 12 "" 1 "" {XPPMATH 20 "60-%'RTABLEG6%\")+%fO\"-%'MATRIXG6#7&7&%#c3G,$%#s3G!\"\"\"\"!F07&F .F,F0F07&F0F0\"\"\"%#D3G7&F0F0F0F3%'MatrixG-F$6%\")WyG9-F(6#7&7&F3F0F0 %#A3G7&F0%#P3G,$%#R3GF/F07&F0FBF@F0F5F6-F$6%\");,v:-F(6#7&7&%#c4G,$%#s 4GF/F0F07&FMFKF0F07&F0F0F3%#D4GF5F6-F$6%\")#fad\"-F(6#7&7&F3F0F0%#A4G7 &F0%#P4G,$%#R4GF/F07&F0FfnFZF0F5F6-F$6%\")Kpv:-F(6#7&7&%#c5G,$%#s5GF/F 0F07&FaoF_oF0F07&F0F0F3%#D5GF5F6-F$6%\")?:y:-F(6#7&7$F0%#A5G7$,$%#R5GF /F07$%#P5GF07$F0F3F6Q\"=6\"-F$6%\")!)Rw:-F(6#7&7&F3F0F0,$%#A2GF/7&F0%# P2G%#R2GF07&F0,$F`qF/F_qF0F5F6-F$6%\")%=md\"-F(6#7&7&%#c2G%#s2GF0F07&, $F[rF/FjqF0F07&F0F0F3,$%#D2GF/F5F6-F$6%\")W'od\"-F(6#7&7&F3F0F0,$%#A1G F/7&F0%#P1G%#R1GF07&F0,$F\\sF/F[sF0F5F6-F$6%\")_g79-F(6#7&7&%#c1G%#s1G F0F07&,$FgsF/FfsF0F07&F0F0F3,$%#D1GF/F5F6-F$6%\")CQQ8-F(6#7&7&%#LxG%#M xG%#NxG%#QxG7&%#LyG%#MyG%#NyG%#QyG7&%#LzG%#MzG%#NzG%#QzGF5F6-F$6%\")3m x:-F(6#7&7&F3F0F0,$%#A6GF/7&F0%#P6G%#R6GF07&F0,$F]vF/F\\vF0F5F6-F$6%\" )?)*3:-F(6#7&7$F0F0Ffv7$F3,$%#D6GF/FbpF6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "D3 from M31 and A2 from iM22 can be moved to M32 and iM21 respectively:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "M31,M32, \"=\",M31.M32;\nM32[3,4]:=D3:M31[3,4]:=0:\nM31,M32,\"=\",M31.M32;\n\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")+%fO\"-%'MATRIXG6#7 &7&%#c3G,$%#s3G!\"\"\"\"!F07&F.F,F0F07&F0F0\"\"\"%#D3G7&F0F0F0F3%'Matr ixG-F$6%\")WyG9-F(6#7&7&F3F0F0%#A3G7&F0%#P3G,$%#R3GF/F07&F0FBF@F0F5F6Q \"=6\"-F$6%\")Kyy:-F(6#7&7&F,,$*&F.F3F@F3F/*&F.F3FBF3*&F,F3F>F37&F.*&F ,F3F@F3,$*&F,F3FBF3F/*&F.F3F>F37&F0FBF@F4F5F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")+%fO\"-%'MATRIXG6#7&7&%#c3G,$%#s3G!\"\" \"\"!F07&F.F,F0F07&F0F0\"\"\"F07&F0F0F0F3%'MatrixG-F$6%\")WyG9-F(6#7&7 &F3F0F0%#A3G7&F0%#P3G,$%#R3GF/F07&F0FAF?%#D3GF4F5Q\"=6\"-F$6%\")g')y:- F(6#7&7&F,,$*&F.F3F?F3F/*&F.F3FAF3*&F,F3F=F37&F.*&F,F3F?F3,$*&F,F3FAF3 F/*&F.F3F=F3FBF4F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "iM22, iM21,\"=\",iM22.iM21;\niM21[1,4]:=-A2:iM22[1,4]:=0:\niM22,iM21,\"=\",i M22.iM21;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")!)Rw:-%' MATRIXG6#7&7&\"\"\"\"\"!F-,$%#A2G!\"\"7&F-%#P2G%#R2GF-7&F-,$F3F0F2F-7& F-F-F-F,%'MatrixG-F$6%\")%=md\"-F(6#7&7&%#c2G%#s2GF-F-7&,$F@F0F?F-F-7& F-F-F,,$%#D2GF0F6F7Q\"=6\"-F$6%\");Kz:-F(6#7&7&F?F@F-F.7&,$*&F2F,F@F,F 0*&F2F,F?F,F3,$*&F3F,FEF,F07&*&F3F,F@F,,$*&F3F,F?F,F0F2,$*&F2F,FEF,F0F 6F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\")!)Rw:-%'MATRIXG6 #7&7&\"\"\"\"\"!F-F-7&F-%#P2G%#R2GF-7&F-,$F0!\"\"F/F-7&F-F-F-F,%'Matri xG-F$6%\")%=md\"-F(6#7&7&%#c2G%#s2GF-,$%#A2GF37&,$F>F3F=F-F-7&F-F-F,,$ %#D2GF3F4F5Q\"=6\"-F$6%\"(_qH$-F(6#7&F<7&,$*&F/F,F>F,F3*&F/F,F=F,F0,$* &F0F,FEF,F37&*&F0F,F>F,,$*&F0F,F=F,F3F/,$*&F/F,FEF,F3F4F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Look at the 6 equations again" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "M31,M32,M41,M42,M51,M52[1..4,3..4], \"=\",iM22,iM21,iM12,iM11,Mh,iM62,iM61[1..4,3..4];" }}{PARA 12 "" 1 " " {XPPMATH 20 "60-%'RTABLEG6%\")+%fO\"-%'MATRIXG6#7&7&%#c3G,$%#s3G!\" \"\"\"!F07&F.F,F0F07&F0F0\"\"\"F07&F0F0F0F3%'MatrixG-F$6%\")WyG9-F(6#7 &7&F3F0F0%#A3G7&F0%#P3G,$%#R3GF/F07&F0FAF?%#D3GF4F5-F$6%\");,v:-F(6#7& 7&%#c4G,$%#s4GF/F0F07&FMFKF0F07&F0F0F3%#D4GF4F5-F$6%\")#fad\"-F(6#7&7& F3F0F0%#A4G7&F0%#P4G,$%#R4GF/F07&F0FfnFZF0F4F5-F$6%\")Kpv:-F(6#7&7&%#c 5G,$%#s5GF/F0F07&FaoF_oF0F07&F0F0F3%#D5GF4F5-F$6%\"()G*H$-F(6#7&7$F0%# A5G7$,$%#R5GF/F07$%#P5GF07$F0F3F5Q\"=6\"-F$6%\")!)Rw:-F(6#7&7&F3F0F0F0 7&F0%#P2G%#R2GF07&F0,$F^qF/F]qF0F4F5-F$6%\")%=md\"-F(6#7&7&%#c2G%#s2GF 0,$%#A2GF/7&,$FiqF/FhqF0F07&F0F0F3,$%#D2GF/F4F5-F$6%\")W'od\"-F(6#7&7& F3F0F0,$%#A1GF/7&F0%#P1G%#R1GF07&F0,$F\\sF/F[sF0F4F5-F$6%\")_g79-F(6#7 &7&%#c1G%#s1GF0F07&,$FgsF/FfsF0F07&F0F0F3,$%#D1GF/F4F5-F$6%\")CQQ8-F(6 #7&7&%#LxG%#MxG%#NxG%#QxG7&%#LyG%#MyG%#NyG%#QyG7&%#LzG%#MzG%#NzG%#QzGF 4F5-F$6%\")3mx:-F(6#7&7&F3F0F0,$%#A6GF/7&F0%#P6G%#R6GF07&F0,$F]vF/F\\v F0F4F5-F$6%\"(?8I$-F(6#7&7$F0F0Ffv7$F3,$%#D6GF/FbpF5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "Multiply the second column of M31 and the seco nd row of M32 by -1 and analogically multiply the second column of iM2 2 and the second row of iM21 by -1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "M31:=M31.DiagonalMatrix(<1,-1,1,1>):M32:=DiagonalMat rix(<1,-1,1,1>).M32:iM22:=iM22.DiagonalMatrix(<1,-1,1,1>):iM21:=Diagon alMatrix(<1,-1,1,1>).iM21:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "R ealize, that the product remains the same because." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "DiagonalMatrix(<1,-1,1,1>),DiagonalMatrix( <1,-1,1,1>),\"=\",DiagonalMatrix(<1,-1,1,1>).DiagonalMatrix(<1,-1,1,1> );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&-%'RTABLEG6%\"(obI$-%'MATRIXG6#7 &7&\"\"\"\"\"!F-F-7&F-!\"\"F-F-7&F-F-F,F-7&F-F-F-F,%'MatrixG-F$6%\")[' zd\"F'F2Q\"=6\"-F$6%\"()[3L-F(6#7&F+7&F-F,F-F-F0F1F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Look at the 6 equations again. They correspond \+ to equations (11) and (12) in [2]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "M31,M32,M41,M42,M51,M52[1..4,3..4],\"=\",iM22,iM21,iM 12,iM11,Mh,iM62,iM61[1..4,3..4];" }}{PARA 12 "" 1 "" {XPPMATH 20 "60-% 'RTABLEG6%\"(oaJ$-%'MATRIXG6#7&7&%#c3G%#s3G\"\"!F.7&F-,$F,!\"\"F.F.7&F .F.\"\"\"F.7&F.F.F.F3%'MatrixG-F$6%\"(CwJ$-F(6#7&7&F3F.F.%#A3G7&F.,$%# P3GF1%#R3GF.7&F.FAF@%#D3GF4F5-F$6%\");,v:-F(6#7&7&%#c4G,$%#s4GF1F.F.7& FMFKF.F.7&F.F.F3%#D4GF4F5-F$6%\")#fad\"-F(6#7&7&F3F.F.%#A4G7&F.%#P4G,$ %#R4GF1F.7&F.FfnFZF.F4F5-F$6%\")Kpv:-F(6#7&7&%#c5G,$%#s5GF1F.F.7&FaoF_ oF.F.7&F.F.F3%#D5GF4F5-F$6%\"(s*>L-F(6#7&7$F.%#A5G7$,$%#R5GF1F.7$%#P5G F.7$F.F3F5Q\"=6\"-F$6%\"(%GAL-F(6#7&7&F3F.F.F.7&F.,$%#P2GF1%#R2GF.7&F. F_qF^qF.F4F5-F$6%\"(olK$-F(6#7&7&%#c2G%#s2GF.,$%#A2GF17&Fiq,$FhqF1F.F. 7&F.F.F3,$%#D2GF1F4F5-F$6%\")W'od\"-F(6#7&7&F3F.F.,$%#A1GF17&F.%#P1G%# R1GF.7&F.,$F\\sF1F[sF.F4F5-F$6%\")_g79-F(6#7&7&%#c1G%#s1GF.F.7&,$FgsF1 FfsF.F.7&F.F.F3,$%#D1GF1F4F5-F$6%\")CQQ8-F(6#7&7&%#LxG%#MxG%#NxG%#QxG7 &%#LyG%#MyG%#NyG%#QyG7&%#LzG%#MzG%#NzG%#QzGF4F5-F$6%\")3mx:-F(6#7&7&F3 F.F.,$%#A6GF17&F.%#P6G%#R6GF.7&F.,$F]vF1F\\vF.F4F5-F$6%\"([(GL-F(6#7&7 $F.F.Ffv7$F3,$%#D6GF1FbpF5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Mu ltiply both sides from the lest by M22 = inv(iM22) to get equations co rresponding to (12), (13) in [2]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "dhInv(iM22),M31,M32,M41,M42,M51,M52[1..4,3..4],\"=\", iM21,iM12,iM11,Mh,iM62,iM61[1..4,3..4];" }}{PARA 12 "" 1 "" {XPPMATH 20 "60-%'RTABLEG6%\"(!yIL-%'MATRIXG6#7&7&\"\"\"\"\"!F-F-7&F-,$%#P2G!\" \"%#R2GF-7&F-F2F0F-7&F-F-F-F,%'MatrixG-F$6%\"(oaJ$-F(6#7&7&%#c3G%#s3GF -F-7&F>,$F=F1F-F-7&F-F-F,F-F4F5-F$6%\"(CwJ$-F(6#7&7&F,F-F-%#A3G7&F-,$% #P3GF1%#R3GF-7&F-FMFL%#D3GF4F5-F$6%\");,v:-F(6#7&7&%#c4G,$%#s4GF1F-F-7 &FYFWF-F-7&F-F-F,%#D4GF4F5-F$6%\")#fad\"-F(6#7&7&F,F-F-%#A4G7&F-%#P4G, $%#R4GF1F-7&F-FboF`oF-F4F5-F$6%\")Kpv:-F(6#7&7&%#c5G,$%#s5GF1F-F-7&F]p F[pF-F-7&F-F-F,%#D5GF4F5-F$6%\"(G^L$-F(6#7&7$F-%#A5G7$,$%#R5GF1F-7$%#P 5GF-7$F-F,F5Q\"=6\"-F$6%\"(olK$-F(6#7&7&%#c2G%#s2GF-,$%#A2GF17&Fiq,$Fh qF1F-F-7&F-F-F,,$%#D2GF1F4F5-F$6%\")W'od\"-F(6#7&7&F,F-F-,$%#A1GF17&F- %#P1G%#R1GF-7&F-,$F\\sF1F[sF-F4F5-F$6%\")_g79-F(6#7&7&%#c1G%#s1GF-F-7& ,$FgsF1FfsF-F-7&F-F-F,,$%#D1GF1F4F5-F$6%\")CQQ8-F(6#7&7&%#LxG%#MxG%#Nx G%#QxG7&%#LyG%#MyG%#NyG%#QyG7&%#LzG%#MzG%#NzG%#QzGF4F5-F$6%\")3mx:-F(6 #7&7&F,F-F-,$%#A6GF17&F-%#P6G%#R6GF-7&F-,$F]vF1F\\vF-F4F5-F$6%\"(/fL$- F(6#7&7$F-F-Ffv7$F,,$%#D6GF1F^qF5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "Let us denote the left and the right hand side of the above marix equation by \ne1 and e2, respectively and select the 6 non-tautologic al equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "e1 := (dh Inv(iM22).M31.M32.M41.M42.M51.M52)[1..3,3..4];\ne2 := (iM21.iM12.iM11. Mh.iM62.iM61)[1..3,3..4];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#e1G-%' RTABLEG6%\")cRE8-%'MATRIXG6#7%7$,&*&,&*&,&*&%#c3G\"\"\"%#c4GF5F5*(%#s3 GF5%#P3GF5%#s4GF5!\"\"F5%#s5GF5F;*&,&*&,&*&F4F5F:F5F;*(F8F5F9F5F6F5F;F 5%#P4GF5F5*(F8F5%#R3GF5%#R4GF5F5F5%#c5GF5F5F5%#R5GF5F;*&,&*&F@F5FFF5F; *(F8F5FEF5FCF5F5F5%#P5GF5F5,,*&,&*&F2F5FGF5F5*&F>F5F%#e2G-%'RTABLEG6%\"(g@M$-%'MATRIXG6#7%7$,&*&,(*&,&*&%# c2G\"\"\"%#c1GF5F5*(%#s2GF5%#P1GF5%#s1GF5!\"\"F5%#MxGF5F5*&,&*&F4F5F:F 5F5*(F8F5F9F5F6F5F5F5%#MyGF5F5*(F8F5%#R1GF5%#MzGF5F5F5%#R6GF5F5*&,(*&F 2F5%#NxGF5F5*&F>F5%#NyGF5F5*(F8F5FCF5%#NzGF5F5F5%#P6GF5F5,2*&F.F5%#D6G F5F;*&,(*&F2F5%#LxGF5F5*&F>F5%#LyGF5F5*(F8F5FCF5%#LzGF5F5F5%#A6GF5F;*& F2F5%#QxGF5F5*&F>F5%#QyGF5F5*(F8F5FCF5%#QzGF5F5*(F8F5FCF5%#D1GF5F;*&F4 F5%#A1GF5F;%#A2GF;7$,&*&,(*&,&*&F8F5F6F5F5*(F4F5F9F5F:F5F5F5F " 0 "" {MPLTEXT 1 0 15 "Dimensions(e1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\" \"$\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Convert e1, e2 to the two lists of the corresponding elements:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 85 "E1:=M2L():nops(E1);\nE2:=M2L ():nops(E2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "The terms can be read out from the last matrix equa tion above. We will look at c3, s3 as on a coefficinet for a while, \n to get only 8 unknowns." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 210 "tE1:=<>:\ntE2:=<>:\nmE1:=PolyCoeffMatrix(E1,M2L(tE1),plex(op( indets(tE1)))):\nmE2:=PolyCoeffMatrix(E2,M2L(tE2),plex(op(indets(tE2)) )):\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Check it and see that it works!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "simplify(Transp ose((convert(E1,Matrix)))-mE1.tE1),\nsimplify(Transpose((convert(E2,Ma trix)))-mE2.tE2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"(%y ]L-%'MATRIXG6#7(7#\"\"!F+F+F+F+F+%'MatrixG-F$6%\"(3IN$F'F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "We can therefore conclude that tE1, tE2 \+ are our terms and that we thus would have 2 * 8 = 16 \"unknowns\" vers us 6 equations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Let us see wh at will happen if we construct the equation e1^T.e1 = e2^T.e2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "e1e1:=map(x->expand(x),Trans pose(e1).e1):\ne2e2:=map(x->expand(x),Transpose(e2).e2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Elements [1,1] lead to a tautology:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "dhSimpl(dhSimpl(dhSimpl(dhS impl(e1e1[1,1],2),3),4),5),\"=\",MhSimpl(dhSimpl(dhSimpl(dhSimpl(e2e2[ 1,1],2),1),6));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"Q\"=6\"F#" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Elements [1,2] seem to lead to a \+ new equation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "dhSimpl(d hSimpl(dhSimpl(dhSimpl(e1e1[1,2],2),3),4),5);\"=\";MhSimpl(dhSimpl(dhS impl(dhSimpl(e2e2[1,2],2),1),6));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(*(,***%#P4G\"\"\"%#c4GF(%#R3GF(%#D3GF(!\"\"*(F'F(%#s4GF(%#A3GF(F(*& %#R4GF(%#D4GF(F,*(F1F(%#P3GF(F+F(F,F(%#R5GF(%#c5GF(F(*(,(*&F)F(F/F(F(* (F.F(F*F(F+F(F(%#A4GF(F(F5F(%#s5GF(F(*&,,*(F/F(F.F(F1F(F(**F+F(F*F(F)F (F1F(F,*(F+F(F4F(F'F(F(*&F2F(F'F(F(%#D5GF(F(%#P5GF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\"=6\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4%#D6G !\"\"*&,,*&%#R6G\"\"\"%#QxGF*F**,F)F*%#s2GF*%#P1GF*%#s1GF*%#A2GF*F**(F )F*%#c1GF*%#A1GF*F%**F)F*%#c2GF*F2F*F0F*F%**F)F*%#R1GF*F/F*%#D2GF*F%F* %#MxGF*F**&,,*(F)F*F/F*F3F*F%*&F)F*%#QyGF*F**,F)F*F-F*F.F*F2F*F0F*F%** F)F*F5F*F/F*F0F*F%**F)F*F7F*F2F*F8F*F*F*%#MyGF*F**&,**(F)F*F.F*F8F*F%* &F)F*%#D1GF*F%**F)F*F-F*F7F*F0F*F%*&F)F*%#QzGF*F*F*%#MzGF*F**&,,*(F3F* F/F*%#P6GF*F%**F8F*F2F*F7F*FOF*F**,F0F*F2F*F.F*F-F*FOF*F%*&FOF*F>F*F** *F0F*F/F*F5F*FOF*F%F*%#NyGF*F**&,***F0F*F7F*F-F*FOF*F%*&FJF*FOF*F**&FG F*FOF*F%*(F8F*F.F*FOF*F%F*%#NzGF*F**&,,*,F0F*F/F*F.F*F-F*FOF*F**(F3F*F 2F*FOF*F%**F8F*F/F*F7F*FOF*F%*&F+F*FOF*F***F0F*F2F*F5F*FOF*F%F*%#NxGF* F**(F)F*%#A6GF*)FB\"\"#F*F***F)F*FBF*F_oF*%#LyGF*F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Elements [2,2] seem to lead to a new equation, \+ too:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "dhSimpl(dhSimpl(dh Simpl(dhSimpl(e1e1[2,2],2),3),4),5);\"=\";MhSimpl(dhSimpl(dhSimpl(dhSi mpl(e2e2[2,2],2),1),6));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,@*,\"\"# \"\"\"%#D5GF&%#R4GF&%#s4GF&%#A3GF&F&*,F%F&%#A4GF&F)F&%#R3GF&%#D3GF&F&* .F%F&F'F&F(F&%#c4GF&F-F&F.F&!\"\"**F%F&F,F&F0F&F*F&F&*,F%F&F'F&%#P4GF& %#P3GF&F.F&F&**F%F&F'F&F4F&%#D4GF&F&**F%F&F7F&F5F&F.F&F&*$)F*F%F&F&*$) F.F%F&F&*$)F,F%F&F&*$)F7F%F&F&*$)F'F%F&F&*$)%#A5GF%F&F&*&,**,F%F&FEF&F 4F&F)F&F*F&F1*.F%F&FEF&F4F&F0F&F-F&F.F&F&**F%F&FEF&F(F&F7F&F&*,F%F&FEF &F(F&F5F&F.F&F&F&%#s5GF&F&*&,(*(F%F&FEF&F,F&F&**F%F&FEF&F0F&F*F&F&*,F% F&FEF&F)F&F-F&F.F&F&F&%#c5GF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#Q\" =6\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,`o**\"\"#\"\"\"%#P1GF&%#D1GF& %#D2GF&F&*(F%F&%#QzGF&F(F&!\"\"*&,,*.F%F&%#A6GF&%#s2GF&F'F&%#c1GF&%#A2 GF&F&**F%F&F0F&%#s1GF&%#A1GF&F&*,F%F&F0F&%#R1GF&F2F&F)F&F,*(F%F&F0F&%# QyGF&F,*,F%F&F0F&%#c2GF&F5F&F3F&F&F&%#LyGF&F&*.F%F&F:F&F1F&F'F&F2F&F3F &F,*,F%F&F1F&F8F&F(F&F3F&F&*,F%F&%#QxGF&F " 0 "" {MPLTEXT 1 0 404 "mq1:=<>:\nmq2:=<>:\nMq1:=PolyCoeffMatrix(M2L(mq1),M2L(tE1),plex(op(indets(tE1)))): \nMq2:=PolyCoeffMatrix(M2L(mq2),M2L(tE2),plex(op(indets(tE2)))):\n\nsi mplify(mq1-Mq1.tE1);\nsimplify(mq2-Mq2.tE2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(7^N$-%'MATRIXG6#7$7#\"\"!F+%'MatrixG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(ccN$-%'MATRIXG6#7$7#\" \"!F+%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Add the equatio ns to the matrices mE1, mE2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "mt1:=:mt2:=:\nunassign('mE1','mE2');\nmE1:=mt 1:Dimensions(mE1);\nmE2:=mt2:Dimensions(mE2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\")\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\")\"\" *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "mE1;mE2;" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(_*fL-%'MATRIXG6#7*7+,$*(%#R5G\"\" \"%#s3GF/%#P3GF/!\"\"*(F.F/%#P4GF/%#c3GF/*&F.F/F5F/**F.F/F4F/F0F/F1F/* (%#P5GF/%#R4GF/F5F/**F9F/F:F/F0F/F1F/\"\"!,$**F.F/F0F/%#R3GF/F:F/F2**F 9F/F0F/F?F/F4F/7+,&*(F.F/%#R2GF/F?F/F/**F.F/%#P2GF/F5F/F1F/F2,$**F.F/F 4F/FFF/F0F/F2,$*(F.F/FFF/F0F/F2,&**F.F/F4F/FDF/F?F/F2*,F.F/F4F/FFF/F5F /F1F/F/,$**F9F/F:F/FFF/F0F/F2,&**F9F/F:F/FDF/F?F/F2*,F9F/F:F/FFF/F5F/F 1F/F/F<,&*,F.F/F:F/FFF/F5F/F?F/F2**F.F/F:F/FDF/F1F/F2,&**F9F/F4F/FDF/F 1F/F/*,F9F/F4F/FFF/F5F/F?F/F/7+,&*(F.F/FFF/F?F/F/**F.F/FDF/F5F/F1F/F/* *F.F/F4F/FDF/F0F/*(F.F/FDF/F0F/,&**F.F/F4F/FFF/F?F/F2*,F.F/F4F/FDF/F5F /F1F/F2**F9F/F:F/FDF/F0F/,&**F9F/F:F/FFF/F?F/F2*,F9F/F:F/FDF/F5F/F1F/F 2F<,&*,F.F/F:F/FDF/F5F/F?F/F/**F.F/F:F/FFF/F1F/F2,&**F9F/F4F/FFF/F1F/F /*,F9F/F4F/FDF/F5F/F?F/F27+,$*(%#A5GF/F4F/F5F/F2,$*(FioF/F0F/F1F/F2,$* *FioF/F4F/F0F/F1F/F2*&FioF/F5F/,&*(%#D5GF/F:F/F5F/F/*(%#A4GF/F0F/F1F/F 2,&*&FcpF/F5F/F/**FapF/F:F/F0F/F1F/F/**FioF/F0F/F?F/F:F/F<,(*(F0F/F?F/ %#D4GF/F/**FapF/F0F/F?F/F4F/F/*&F5F/%#A3GF/F/7+**FioF/F4F/FFF/F0F/,&** FioF/FFF/F5F/F1F/F2*(FioF/FDF/F?F/F/,&*,FioF/F4F/FFF/F5F/F1F/F2**FioF/ F4F/FDF/F?F/F/,$*(FioF/FFF/F0F/F2,(*(FcpF/FDF/F?F/F/**FcpF/FFF/F5F/F1F /F2**FapF/F:F/FFF/F0F/F2,(*,FapF/F:F/FFF/F5F/F1F/F/*(FcpF/FFF/F0F/F2** FapF/F:F/FDF/F?F/F2,&**FioF/F:F/FDF/F1F/F/*,FioF/F:F/FFF/F5F/F?F/F/F<, .*(FjpF/FDF/F1F/F/*(FFF/F0F/F]qF/F2*&FDF/%#D3GF/F/**FjpF/FFF/F5F/F?F/F /**FapF/F4F/FDF/F1F/F/*,FapF/F4F/FFF/F5F/F?F/F/7+,$**FioF/F4F/FDF/F0F/ F2,&**FioF/FDF/F5F/F1F/F/*(FioF/FFF/F?F/F/,&*,FioF/F4F/FDF/F5F/F1F/F/* *FioF/F4F/FFF/F?F/F/*(FioF/FDF/F0F/,(*(FcpF/FFF/F?F/F/**FcpF/FDF/F5F/F 1F/F/**FapF/F:F/FDF/F0F/F/,(*,FapF/F:F/FDF/F5F/F1F/F2*(FcpF/FDF/F0F/F/ **FapF/F:F/FFF/F?F/F2,&**FioF/F:F/FFF/F1F/F/*,FioF/F:F/FDF/F5F/F?F/F2F <,.*(FjpF/FFF/F1F/F/*(FDF/F0F/F]qF/F/*&FFF/FgrF/F/**FjpF/FDF/F5F/F?F/F 2**FapF/F4F/FFF/F1F/F/*,FapF/F4F/FDF/F5F/F?F/F27+*(F.F/F?F/FgrF/*(F.F/ F4F/F]qF/*&F.F/F]qF/,$**F.F/F4F/F?F/FgrF/F2*(F9F/F]qF/F:F/,$**F9F/FgrF /F?F/F:F/F2*&F.F/FcpF/,&**F.F/F:F/F1F/FgrF/F2*(F.F/F:F/FjpF/F2,(*(F9F/ FjpF/F4F/F/**F9F/FgrF/F1F/F4F/F/*&F9F/FapF/F/7+,$**\"\"#F/FioF/F4F/F]q F/F2,$**F[vF/FioF/F?F/FgrF/F/,$*,F[vF/FioF/F4F/F?F/FgrF/F/,$*(F[vF/Fio F/F]qF/F/,&**F[vF/FapF/F:F/F]qF/F/**F[vF/FcpF/F?F/FgrF/F/,&*(F[vF/FcpF /F]qF/F/*,F[vF/FapF/F:F/F?F/FgrF/F2,&*,F[vF/FioF/F:F/F1F/FgrF/F/**F[vF /FioF/F:F/FjpF/F/,$*(F[vF/FioF/FcpF/F/,4*,F[vF/FapF/F4F/F1F/FgrF/F/**F [vF/FapF/F4F/FjpF/F/**F[vF/FjpF/F1F/FgrF/F/*$)F]qF[vF/F/*$)FgrF[vF/F/* $)FcpF[vF/F/*$)FjpF[vF/F/*$)FapF[vF/F/*$)FioF[vF/F/%'MatrixG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")C&*z:-%'MATRIXG6#7*7+,&*(%#R6 G\"\"\"%#MxGF/%#P1GF/!\"\"*(%#P6GF/%#NxGF/F1F/F2,&*&F.F/%#MyGF/F/*&F4F /%#NyGF/F/,&*(F4F/F:F/F1F/F/*(F.F/F8F/F1F/F/,&*&F.F/F0F/F/*&F4F/F5F/F/ \"\"!FA,&*(F.F/%#R1GF/%#MzGF/F/*(F4F/FDF/%#NzGF/F/FAFA7+F6,&F-F/F3F/F> ,&F " 0 "" {MPLTEXT 1 0 357 "e1x:=map(x-> expand(x),convert(CrossProduct(convert(e1[1..3,1],Vector),convert(e1[1 ..3,2],Vector)),Matrix)):\ne2x:=map(x->expand(x),convert(CrossProduct( convert(e2[1..3,1],Vector),convert(e2[1..3,2],Vector)),Matrix)):\nm1x: =map(x->expand(x),dhSimpl(dhSimpl(dhSimpl(dhSimpl(e1x,2),3),4),5)):\nm 2x:=map(x->expand(x),MhSimpl(dhSimpl(dhSimpl(dhSimpl(e2x,2),1),6))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Construct the linear representa tion, check it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "Mx1:=Po lyCoeffMatrix(M2L(m1x),M2L(tE1),plex(op(indets(tE1)))):\nMx2:=PolyCoef fMatrix(M2L(m2x),M2L(tE2),plex(op(indets(tE2)))):\n\nsimplify(m1x-Mx1. tE1);\nsimplify(m2x-Mx2.tE2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RT ABLEG6%\")37\"e\"-%'MATRIXG6#7%7#\"\"!F+F+%'MatrixG" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RTABLEG6%\")%o*R9-%'MATRIXG6#7%7#\"\"!F+F+%'Matrix G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Add the equations to the mat rices mE1, mE2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "mt1:=:mt2:=:\nunassign('mE1','mE2');\nmE1:=mt1:Dimensions(m E1);\nmE2:=mt2:Dimensions(mE2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ \"#6\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The last three equations will be generate d from columns e1, e2 as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "(pl.pl)*ll - (2*pl.ll)*pl = (pr.pr)*lr - (2*pr.lr )*pr" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "w hich can be derived from" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 80 " A x (B x C) = B (A . C) - C (A . B)\n\nusing th e substitution p = A = C, l = B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "B (A . C) - C (A . B) = \+ A x (B x C) " }}{PARA 0 "" 0 "" {TEXT -1 67 " l (p . p) - p (p . l \+ ) - p (p . l) = p x (l x p) - p (p . l)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "ll:=convert(e1[1 ..3,1],Matrix):pl:=convert(e1[1..3,2],Matrix):\nlr:=convert(e2[1..3,1] ,Matrix):pr:=convert(e2[1..3,2],Matrix):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Construct the new equations and simplify them:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 622 "plpl:=dhSimpl(dhSimpl(dhSim pl(dhSimpl(simplify((Transpose(pl).pl)[1,1]),2),3),4),5):\nplll:=dhSim pl(dhSimpl(dhSimpl(dhSimpl(simplify((2*Transpose(pl).ll)[1,1]),2),3),4 ),5):\nprpr:=MhSimpl(dhSimpl(dhSimpl(dhSimpl(simplify((Transpose(pr).p r)[1,1]),2),1),6)):\nprlr:=MhSimpl(dhSimpl(dhSimpl(dhSimpl(simplify((2 *Transpose(pr).lr)[1,1]),2),1),6)):\n\npl1 := map(x->expand(x),ScalarM ultiply(ll,plpl) - ScalarMultiply(pl,plll)):\npl2 := map(x->expand(x), ScalarMultiply(lr,prpr) - ScalarMultiply(pr,prlr)):\n\nm1pl:=dhSimpl(d hSimpl(dhSimpl(dhSimpl(simplify(pl1),2),3),4),5):\nm2pl:=MhSimpl(dhSim pl(dhSimpl(dhSimpl(simplify(pl2),2),1),6)):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "map(x->collect(x,\{c4,c5,s4,s5\},'distributed'), map(x->expand(x),m2pl)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Const ruct the linear representation, check it:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 180 "Mpl1:=PolyCoeffMatrix(M2L(m1pl),M2L(tE1),plex(op(i ndets(tE1)))):\nMpl2:=PolyCoeffMatrix(M2L(m2pl),M2L(tE2),plex(op(indet s(tE2)))):\n\nsimplify(m1pl-Mpl1.tE1);\nsimplify(m2pl-Mpl2.tE2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")7x\"e\"-%'MATRIXG6#7%7# \"\"!F+F+%'MatrixG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\"(g m!G-%'MATRIXG6#7%7#\"\"!F+F+%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 178 "We see that the resulting equations are linear combinati ons of the selected terms. \nWhat if we try to drop the factor 2 above ? Well, that does not work (One can easily check it)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Add the equations to the matrices mE1, mE 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "mt1:=:mt2: =:\nunassign('mE1','mE2');\nmE1:=mt1:Dimensions(mE1);\nmE2:= mt2:Dimensions(mE2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#9\"\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#9\"\"*" }}}{EXCHG {PARA 11 "" 1 " " {XPPMATH 20 "6$-%'RTABLEG6%\")+k]9-%'MATRIXG6#707+\"\"\"F,F,F,F,F,\" \"!F,F,F+F+7+F,F,F,F,F,F,F,F-F,F.F.7+F,F,F,F,F,F,F,F,F,F/F/F/F/F/F/F/% 'MatrixG-F$6%\")CUh9-F(6#707+F,F,F,F,F-F-F,F-F-7+F,F,F,F,F-F-F-F,F-7+F -F-F-F-F,F,F-F-F,7+F,F,F,F,F-F-F,F,F,7+F,F,F,F,F-F-F,F,F-F9F.F/F;F/F+F /F/F/F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 "Move all constants to the left hand side of the equations and construct P and Q matrices fr om (16) in [2] and the corresponding terms:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "PP:= : \nQQ:= mE2[1..14,1..8]:\npp:= tE1:\nqq:= :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Dimensions(PP);Dimensions(QQ);\nDim ensions(pp);Dimensions(qq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#9\" \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#9\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"*\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\")\" \"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Check it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Transpose((PP.pp-QQ.qq)-(mE1.tE1-mE 2.tE2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")%=Ce\"-%'MA TRIXG6#7#70\"\"!F,F,F,F,F,F,F,F,F,F,F,F,F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "OK." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 74 " \+ Manocha & Canny solution of the rest, see [1, pp. 4-5] for the algorti hm." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The structure of the matrices:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "sPP:=map(x->`if`(simplify(x)=0,0,1),PP):\nsQQ:=ma p(x->`if`(simplify(x)=0,0,1),QQ):\n,;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\")[l#e\"-%'MATRI XG6#717+*&%#s4G\"\"\"%#s5GF.*&F-F.%#c5GF.*&%#c4GF.F/F.*&F3F.F1F.F-F3F/ F1F.7+F.F.F.F.F.F.\"\"!F.F.F5F57+F.F.F.F.F.F.F.F6F.F7F77+F.F.F.F.F.F.F .F.F.F8F8F8F8F8F8F8%'MatrixG-F$6%\")!yFe\"-F(6#717**&%#s2GF.%#s1GF.*&% #c2GF.FCF.*&FBF.%#c1GF.*&FEF.FGF.FCFGFBFE7*F.F.F.F.F6F6F.F67*F.F.F.F.F 6F6F6F.7*F6F6F6F6F.F.F6F67*F.F.F.F.F6F6F.F.FLFK7*F.F.F.F.F.F.F.F67*F.F .F.F.F.F.F.F.FLFN7*F.F.F.F.F.F.F6F.FNFNFNF9" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 132 "We see that s1, c1 can be computed linearly from s4, c 4, s5, c5. by the fact that\n\nPP[[3,6],1..9].pp = QQ[[3,6],[5,6]] . < >\n" }}{PARA 0 "" 0 "" {TEXT -1 40 "and therefore we shall eli minate s1, c1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "pq:=: \nPQ:=: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "Dimensions( pq);\nDimensions(PQ);\nsPQ:=map(x->`if`(simplify(x)=0,0,1),PQ):\n;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#<\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$\"#9\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")%G)>8-%'MATRIXG6#7173*&%#s4G\"\"\"%#s5GF.*&F-F.%# c5GF.*&%#c4GF.F/F.*&F3F.F1F.*&%#s2GF.%#s1GF.*&%#c2GF.F7F.*&F6F.%#c1GF. *&F9F.F;F.F-F3F/F1F7F;F6F9F.73F.F.F.F.F.F.F.F.F.F.\"\"!F.F>F>F.F>F.73F .F.F.F.F.F.F.F.F.F.F>F.F>F>F>F.F.73F.F.F.F.F>F>F>F>F.F.F>F.F.F.F>F>F.7 3F.F.F.F.F.F.F.F.F.F.F.F>F>F>F.F.F.FA73F.F.F.F.F>F>F>F>F.F.F.F>F.F.F>F >F.73F.F.F.F.F.F.F.F.F.F.F.F.F.F.F.F>F.73F.F.F.F.F.F.F.F.F.F.F.F.F.F.F .F.F.73F.F.F.F.F.F.F.F.F.F.F.F.F>F>F.F.F.FD73F.F.F.F.F.F.F.F.F.F.F.F.F .F.F>F.F.FDFDFD%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rPQ:=GaussianElimination(PQ):" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning , computation interrupted\n" }}}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 3 4 1802 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 13659400 14287844 15750116 15754592 15756932 15759448 15763980 15766184 15768644 14126052 13383824 15776608 14866964 15781520 15089820 15787832 15788660 15793216 3297052 3299288 3301320 3305568 15779648 3308488 3315468 3317624 3319972 3322284 3326568 3328748 3330780 3335128 3335904 13263956 3342160 3350784 3353008 3355112 3355656 3359952 15799524 15811208 14399684 15817712 2806660 15824184 15826548 15827780 13198284 }{RTABLE M7R0 I5RTABLE_SAVE/13659400X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c3G%#s3G""!F),$F(!""F'F )F)F)F)"""F)F)F)F)F,6" } {RTABLE M7R0 I5RTABLE_SAVE/14287844X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(%#P3G%#R3GF(F (,$F*!""F)F(%#A3GF(%#D3GF'6" } {RTABLE M7R0 I5RTABLE_SAVE/15750116X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c4G%#s4G""!F),$F(!""F'F )F)F)F)"""F)F)F)%#D4GF,6" } {RTABLE M7R0 I5RTABLE_SAVE/15754592X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(%#P4G%#R4GF(F (,$F*!""F)F(%#A4GF(F(F'6" } {RTABLE M7R0 I5RTABLE_SAVE/15756932X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c5G%#s5G""!F),$F(!""F'F )F)F)F)"""F)F)F)%#D5GF,6" } {RTABLE M7R0 I5RTABLE_SAVE/15759448X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(%#P5G%#R5GF(F (,$F*!""F)F(%#A5GF(F(F'6" } {RTABLE M7R0 I5RTABLE_SAVE/15763980X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(%#P2G,$%#R2G! ""F(F(F+F)F(F(F(F(F'6" } {RTABLE M7R0 I5RTABLE_SAVE/15766184X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c2G,$%#s2G!""""!F+F)F'F +F+F+F+"""F+,$%#A2GF*F+,$%#D2GF*F,6" } {RTABLE M7R0 I5RTABLE_SAVE/15768644X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(%#P1G,$%#R1G! ""F(F(F+F)F(,$%#A1GF,F(F(F'6" } {RTABLE M7R0 I5RTABLE_SAVE/14126052X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c1G,$%#s1G!""""!F+F)F'F +F+F+F+"""F+F+F+,$%#D1GF*F,6" } {RTABLE M7R0 I5RTABLE_SAVE/13383824X,%)anythingG6"6"[gl!"%!!!#1"%"%%#LxG%#LyG%#LzG""!%#MxG%# MyG%#MzGF*%#NxG%#NyG%#NzGF*%#QxG%#QyG%#QzG"""6" } {RTABLE M7R0 I5RTABLE_SAVE/15776608X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(%#P6G,$%#R6G! ""F(F(F+F)F(,$%#A6GF,F(F(F'6" } {RTABLE M7R0 I5RTABLE_SAVE/14866964X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c6G,$%#s6G!""""!F+F)F'F +F+F+F+"""F+F+F+,$%#D6GF*F,6" } {RTABLE M7R0 I5RTABLE_SAVE/15781520X,%)anythingG6"6"[gl!"%!!!#)"%"#""!,$%#R5G!""%#P5GF'%#A5G F'F'"""6" } {RTABLE M7R0 I5RTABLE_SAVE/15089820X,%)anythingG6"6"[gl!"%!!!#)"%"#""!F'"""F'F'F',$%#D6G!""F (6" } {RTABLE M7R0 I5RTABLE_SAVE/15787832X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c3G%#s3G""!F),$*&F("""% #P3GF,!""*&F'F,F-F,%#R3GF)*&F(F,F0F,,$*&F'F,F0F,F.F-F)*&F'F,%#A3GF,*&F(F,F5F,%# D3GF,6" } {RTABLE M7R0 I5RTABLE_SAVE/15788660X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c3G%#s3G""!F),$*&F("""% #P3GF,!""*&F'F,F-F,%#R3GF)*&F(F,F0F,,$*&F'F,F0F,F.F-F)*&F'F,%#A3GF,*&F(F,F5F,%# D3GF,6" } {RTABLE M7R0 I5RTABLE_SAVE/15793216X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c2G,$*&%#P2G"""%#s2GF+! ""*&%#R2GF+F,F+""!F,*&F*F+F'F+,$*&F/F+F'F+F-F0F0F/F*F0,$%#A2GF-,$*&F/F+%#D2GF+F -,$*&F*F+F8F+F-F+6" } {RTABLE M7R0 I4RTABLE_SAVE/3297052X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c2G,$*&%#P2G"""%#s2GF+!" "*&%#R2GF+F,F+""!F,*&F*F+F'F+,$*&F/F+F'F+F-F0F0F/F*F0,$%#A2GF-,$*&F/F+%#D2GF+F- ,$*&F*F+F8F+F-F+6" } {RTABLE M7R0 I4RTABLE_SAVE/3299288X,%)anythingG6"6"[gl!"%!!!#)"%"#""!,$%#R5G!""%#P5GF'%#A5GF 'F'"""6" } {RTABLE M7R0 I4RTABLE_SAVE/3301320X,%)anythingG6"6"[gl!"%!!!#)"%"#""!F'"""F'F'F',$%#D6G!""F( 6" } {RTABLE M7R0 I4RTABLE_SAVE/3305568X,%)anythingG6#%)diagonalG6"[gl!"#!!!#%"%"%"""!""F(F(6" } {RTABLE M7R0 I5RTABLE_SAVE/15779648X,%)anythingG6#%)diagonalG6"[gl!"#!!!#%"%"%"""!""F(F(6" } {RTABLE M7R0 I4RTABLE_SAVE/3308488X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(F'F(F(F(F(F'F( F(F(F(F'6" } {RTABLE M7R0 I4RTABLE_SAVE/3315468X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c3G%#s3G""!F)F(,$F'!""F) F)F)F)"""F)F)F)F)F,6" } {RTABLE M7R0 I4RTABLE_SAVE/3317624X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(,$%#P3G!""%#R3 GF(F(F,F*F(%#A3GF(%#D3GF'6" } {RTABLE M7R0 I4RTABLE_SAVE/3319972X,%)anythingG6"6"[gl!"%!!!#)"%"#""!,$%#R5G!""%#P5GF'%#A5GF 'F'"""6" } {RTABLE M7R0 I4RTABLE_SAVE/3322284X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(,$%#P2G!""%#R2 GF(F(F,F*F(F(F(F(F'6" } {RTABLE M7R0 I4RTABLE_SAVE/3326568X,%)anythingG6"6"[gl!"%!!!#1"%"%%#c2G%#s2G""!F)F(,$F'!""F) F)F)F)"""F),$%#A2GF+F),$%#D2GF+F,6" } {RTABLE M7R0 I4RTABLE_SAVE/3328748X,%)anythingG6"6"[gl!"%!!!#)"%"#""!F'"""F'F'F',$%#D6G!""F( 6" } {RTABLE M7R0 I4RTABLE_SAVE/3330780X,%)anythingG6"6"[gl!"%!!!#1"%"%"""""!F(F(F(,$%#P2G!""%#R2 GF(F(F,F*F(F(F(F(F'6" } {RTABLE M7R0 I4RTABLE_SAVE/3335128X,%)anythingG6"6"[gl!"%!!!#)"%"#""!,$%#R5G!""%#P5GF'%#A5GF 'F'"""6" } {RTABLE M7R0 I4RTABLE_SAVE/3335904X,%)anythingG6"6"[gl!"%!!!#)"%"#""!F'"""F'F'F',$%#D6G!""F( 6" } {RTABLE M7R0 I5RTABLE_SAVE/13263956X,%)anythingG6"6"[gl!"%!!!#'"$"#,&*&,&*&,&*&%#c3G"""%#c4G F.F.*(%#s3GF.%#P3GF.%#s4GF.!""F.%#s5GF.F4*&,&*&,&*&F-F.F3F.F4*(F1F.F2F.F/F.F4F. %#P4GF.F.*(F1F.%#R3GF.%#R4GF.F.F.%#c5GF.F.F.%#R5GF.F4*&,&*&F9F.F?F.F4*(F1F.F>F. FF. F.F.F3F.F.F.F5F.F4*&,&*&,&*(FMF.F1F.F3F.F.*&FOF.F/F.F.F.FF .F.*&FRF.F2F.F.F.F?F.F.F.F@F.F.F.FAF.F4*&,&*&FVF.F?F.F4*&FZF.FF.F.F.F3F.F.F.F5F.F4*&,&*&,&* (FRF.F1F.F3F.F4*&FboF.F/F.F.F.FF.F4*&FMF.F2F.F.F.F?F.F.F.F @F.F.F.FAF.F4*&,&*&FhoF.F?F.F4*&F\pF.FF.%#D4GF.F.*&F-F.%#A3GF.F.,. *&,&*&FKF.F@F.F.*&FTF.F5F.F.F.FhpF.F.*&FhnF.FjpF.F.*&FKF.F\qF.F.*&FZF.F^qF.F.*( FMF.F1F.F`qF.F4*&FRF.%#D3GF.F.,.*&,&*&F_oF.F@F.F.*&FfoF.F5F.F.F.FhpF.F.*&F`pF.F jpF.F.*&F_oF.F\qF.F.*&F\pF.F^qF.F.*(FRF.F1F.F`qF.F.*&FMF.F[rF.F.6" } {RTABLE M7R0 I4RTABLE_SAVE/3342160X,%)anythingG6"6"[gl!"%!!!#'"$"#,&*&,(*&,&*&%#c2G"""%#c1GF .F.*(%#s2GF.%#P1GF.%#s1GF.!""F.%#MxGF.F.*&,&*&F-F.F3F.F.*(F1F.F2F.F/F.F.F.%#MyG F.F.*(F1F.%#R1GF.%#MzGF.F.F.%#R6GF.F.*&,(*&F+F.%#NxGF.F.*&F7F.%#NyGF.F.*(F1F.F< F.%#NzGF.F.F.%#P6GF.F.,&*&,(*&,&*&F1F.F/F.F.*(F-F.F2F.F3F.F.F.F5F.F.*&,&*&F1F.F 3F.F.*(F-F.F2F.F/F.F4F.F:F.F.*(F-F.FF.F.*&,(*&FLF.FBF.F.*&FPF.FDF. F.*(F-F.FF.F.*&,(*(FF*F6,&*(F)F*F3F*FHF*F-*(F/F* F5F*FHF*F-,***F9F*F)F*F3F*FHF*FL*&FBF*FHF*FN**F9F*F/F*F5F*FHF*FL*(F