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0 0 0 1 } {CSTYLE "Caption Text" -1 213 "SimSun" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times New Roman" 1 24 0 0 255 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times New Roman" 1 18 0 0 255 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 210 "" 0 "" {TEXT 209 25 "A Demo for using Impli cit" }}{PARA 0 "" 0 "" {TEXT 214 0 "" }}{PARA 211 "" 0 "" {TEXT 209 83 "First, invoke Implicit. Of course, you should change the directory to your own one." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart;\n" }{MPLTEXT 1 0 84 "march('open',\"D:\\\\a\3 27\300\303\346\261\270\267\335\\\\implicitization\\\\implicit-3\\\\fin al version\\\\Implicit.mla\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q@D: \\Program~Files\\Maple~2016\\lib6\"QaoD:\\a|ay|\\u|gs|dy|ht|]u|`y|_u|b s|_y|fv|hv\\implicitization\\implicit-3\\final~version\\Implicit.mlaF$ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Implicit);" }} {PARA 11 "" 1 "" {XPPMATH 20 "7-I)HowToUseG6\"I)ImpCurveGF$I*ImpDegree GF$I*ImpMatrixGF$I)ImpRuledGF$I+ImpSurfaceGF$I)Impby3mpGF$I,ImpbyDixon AGF$I*ImpbympmqGF$I(ThreempGF$I'polyBPGF$" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 9 "ImpDegree" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "How ToUse" }{MPLTEXT 1 0 12 "(ImpDegree);" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling Sequence" }}{PARA 6 "" 1 "" {TEXT 216 15 " ImpDegree(P);" } {TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " P: P is a rational parametrization of a rational surface in affine space or a homogeneous parametrization in projective space" } {TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 59 " Degree of an implicit equation of the rational surface P;" } {TEXT 216 1 "\n" }{TEXT 216 11 "Description" }{TEXT 216 1 "\n" }{TEXT 216 82 " Compute the implicit degree without resorting to computing th e implicit equation;" }{TEXT 216 1 "\n" }{TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 45 " ImpDegree( [s^2*t+t, s*t+t+1, s+2, t^2+s] \+ )" }{TEXT 216 1 "\n" }{TEXT 216 69 " ImpDegree( [(s^2*t+t)/(t^2+s), ( s*t+t+1)/(t^2+s), (s+2))/(t^2+s)] )" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 217 7 "Example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "P1:=[s^2*t +t, s*t+t+1, s+2, t^2+s];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&,&*&)I\"s G6\"\"\"#\"\"\"I\"tGF'F)F)F*F),(*&F&F)F*F)F)F*F)F)F),&F&F)F(F),&*$)F*F (F)F)F&F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ImpDegree(P1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 8 "ImpCurve" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "How ToUse(ImpCurve);" }{MPLTEXT 1 0 0 "" }}{PARA 6 "" 1 "" {TEXT 216 16 "C alling Sequence" }}{PARA 6 "" 1 "" {TEXT 216 13 " ImpCurve(P)" }{TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 102 " P: P is a rational planar curve in affine space or a homogeneou s parametrization in projective space" }{TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 55 " The implicit equation of t he rational planar curve P." }{TEXT 216 1 "\n" }{TEXT 216 11 "Descript ion" }{TEXT 216 1 "\n" }{TEXT 216 62 " Implicitization using the mu-b asis of rational planar curve." }{TEXT 216 1 "\n" }{TEXT 216 7 "Exampl e" }{TEXT 216 1 "\n" }{TEXT 216 37 " ImpCurve( [t^5+4*t^3, t^2+1, t+4 ] )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "P:=[t^5+4*t^3, t^2+1 , t+4];\n" }{MPLTEXT 1 0 12 "ImpCurve(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%,&*$)I\"tG6\"\"\"&\"\"\"F)*&\"\"%F))F&\"\"$F)F),&*$)F& \"\"#F)F)F)F),&F&F)F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",8*&\"\"*\"\" \")I\"wG6\"\"\"&F%!\"\"*(\"\"'F%)F'\"\"%F%I\"xGF(F%F**(\"#%)F%F-F%I\"y GF(F%F%*(\"# " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 38 "Im pbyDixonA and Impbympmq and Impby3mp" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "HowToUse(" }{MPLTEXT 1 0 11 "ImpbyDixonA" }{MPLTEXT 1 0 2 ");" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling Sequence" }}{PARA 6 " " 1 "" {TEXT 216 40 " ImpbyDixonA(P, mode=\{\"ploy\",\"matrix\"\})" } {TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " P: P is a rational parametrization of a rational surface in affine space or a homogeneous parametrization in projective space" } {TEXT 216 1 "\n" }{TEXT 216 66 " mode(optional) : choose the expressi on of output,defualt is ploy" }{TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 73 " The implicit equation of the ration al surface P or the implicit matrix." }{TEXT 216 1 "\n" }{TEXT 216 11 "Description" }{TEXT 216 1 "\n" }{TEXT 216 72 " Implicitization using the Dixon A-resultant of three obvious syzygies." }{TEXT 216 1 "\n" } {TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 36 " ImpbyDixonA( \+ [s^2*t,s+t,t^3,s*t] )" }{TEXT 216 1 "\n" }{TEXT 216 53 " ImpbyDixonA( [s^2*t,s+t,t^3,s*t] , mode = \"matrix\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ImpbyDixonA" }{MPLTEXT 1 0 6 "(P1);\n" }{MPLTEXT 1 0 35 "ImpbyDixonA( P1 , mode = \"matrix\");" }}{PARA 11 "" 1 "" {XPPMATH 20 ",^o**#\"\"\"\"\"%F%I\"wG6\"F%I\"xGF(F%)I\"yGF(\"\"$F%F%*( F$F%F'F%)I\"zGF(F&F%F%*(#\"\"'\"\"&F%)F'\"\"#F%)F/F,F%!\"\"*&)F'F,F%)F /F5F%F%**#\"#>\"#?F%)F)F5F%F+F%F:F%F%**#F3F&F%F)F%F*F%F/F%F7**#\"#LF>F %F)F%)F+F5F%F:F%F%**#\"#JF>F%F)F%F+F%F6F%F7*&#F%F>F%)F/F3F%F7*(FJF%)F) F,F%F:F%F%*(FJF%F?F%F*F%F%*(#\"#6\"#5F%F?F%F6F%F7*(FJF%F)F%F.F%F7*(#\" #@F&F%)F+F&F%F/F%F7*(#\"#F%F'F%F?F%F:F%F%**FCF%F'F%F)F%F6F%F7* *F&F%F'F%F*F%F/F%F7**#FZFRF%F'F%FEF%F:F%F%**F$F%F4F%F?F%F/F%F7**FfoF%F 4F%F)F%F:F%F%*(F4F%FEF%F/F%F7*,#FVF>F%F'F%F?F%F+F%F/F%F7*,#\"#pF>F%F'F %F)F%FEF%F/F%F7*,#\"#(*F>F%F'F%F)F%F+F%F:F%F%*,#F,F5F%F4F%F)F%F+F%F/F% F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'7',**&\"\"#\"\"\"I\"xG6\"F'!\"\" *&F&F'I\"yGF)F'F'I\"zGF)F*F'F',$F(F*,$F%F*\"\"!,(F+F'F-F*F'F'7',&F(F*F &F*,&F(F'F,F',&F-F'F'F',&F+F*F-F',&F-F*F'F'7',$F%F',$F-F*,(F(F'F+F*F-F 'F-,$F,F*7'F&F6,&F-F*\"\"$F'F " 0 "" {MPLTEXT 1 0 8 "HowToUse" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 9 "Imp bympmq" }{MPLTEXT 1 0 2 ");" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling S equence" }}{PARA 6 "" 1 "" {TEXT 216 38 " Impbympmq(P, mode=\{\"ploy \",\"matrix\"\})" }{TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " P: P is a rational parametrization of a r ational surface in affine space or a homogeneous parametrization in pr ojective space" }{TEXT 216 1 "\n" }{TEXT 216 66 " mode(optional) : ch oose the expression of output,defualt is ploy" }{TEXT 216 1 "\n" } {TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 73 " The implicit e quation of the rational surface P or the implicit matrix." }{TEXT 216 1 "\n" }{TEXT 216 11 "Description" }{TEXT 216 1 "\n" }{TEXT 216 62 " \+ Implicitization using the moving planes and moving quadrics." }{TEXT 216 1 "\n" }{TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 34 " I mpbympmq( [s^2*t,s+t,t^3,s*t] )" }{TEXT 216 1 "\n" }{TEXT 216 51 " Im pbympmq( [s^2*t,s+t,t^3,s*t] , mode = \"matrix\")" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 4 "P2:=" }{MPLTEXT 1 0 20 "[s^2*t,s+t,t^3,s*t];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "Impbympmq" }{MPLTEXT 1 0 6 "(P2 );\n" }{MPLTEXT 1 0 33 " Impbympmq(P2 , mode = \"matrix\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&*&)I\"sG6\"\"\"#\"\"\"I\"tGF&F(,&F%F(F)F(*$) F)\"\"$F(*&F%F(F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&)I\"wG6\"\"\" %\"\"\"I\"xGF&F(F(*&F$F(I\"zGF&F(!\"\"*,\"\"#F()F%F.F(F)F(I\"yGF&F(F+F (F(*()F)F.F()F0F.F(F+F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "7%7%I\"zG6 \"I\"wGF%,$I\"yGF%!\"\"7%,$*&F&\"\"\"I\"xGF%F-F),&*$)F&\"\"#F-F)*&F.F- F(F-F-\"\"!7%F4F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "HowTo Use" }{MPLTEXT 1 0 11 "(Impby3mp);" }{MPLTEXT 1 0 0 "" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling Sequence" }}{PARA 6 "" 1 "" {TEXT 216 38 " I mpby3mp(P , mode=\{\"ploy\",\"matrix\"\})" }{TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " P: P is a rati onal parametrization of a rational surface in affine space or a homoge neous parametrization in projective space" }{TEXT 216 1 "\n" }{TEXT 216 66 " mode(optional) : choose the expression of output,defualt is \+ ploy" }{TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 73 " The implicit equation of the rational surface P or the impl icit matrix." }{TEXT 216 1 "\n" }{TEXT 216 11 "Description" }{TEXT 216 1 "\n" }{TEXT 216 81 " Implicitization using the resultant matrix of three low bidegree moving planes." }{TEXT 216 1 "\n" }{TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 44 " Impby3mp( [s^2*t+t, s*t+t +1, s+2, t^2+s] )" }{TEXT 216 1 "\n" }{TEXT 216 61 " Impby3mp( [s^2*t +t, s*t+t+1, s+2, t^2+s] , mode = \"matrix\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Impby3mp(P1);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "Impby3mp( P1 , mode = \"matrix\");" }}{PARA 11 "" 1 "" {XPPMATH 20 ",^o*(\"#?\"\"\")I\"wG6\"\"\"$F%)I\"zGF(\"\"#F%F%**\"\"&F%)F'F,F%)I \"xGF(F,F%F+F%!\"\"*,\"#IF%F/F%F1F%I\"yGF(F%F+F%F2**\"#MF%F/F%F1F%F*F% F%**F$F%F/F%)F5F,F%F+F%F2**\"#SF%F/F%F5F%F*F%F%*(\"#CF%F/F%)F+F)F%F2*( F'F%)F1F)F%F+F%F2*,\"#@F%F'F%F0F%F5F%F+F%F2**\"#FF%F'F%F0F%F*F%F%**F.F %F'F%F1F%)F5F)F%F%*,\"#pF%F'F%F1F%F9F%F+F%F2*,\"#(*F%F'F%F1F%F5F%F*F%F %**\"#LF%F'F%F1F%F>F%F2*(\"#DF%F'F%)F5\"\"%F%F%**\"#!)F%F'F%FFF%F+F%F2 **F7F%F'F%F9F%F*F%F%*(F.F%F'F%)F+FPF%F%*&F@F%F*F%F%*&F0F%FFF%F%**\"#>F %F0F%F5F%F*F%F%*(\"#AF%F0F%F>F%F2*(F$F%F1F%FOF%F%**FNF%F1F%FFF%F+F%F2* *FLF%F1F%F9F%F*F%F%**\"#JF%F1F%F5F%F>F%F2*&F1F%FUF%F2*&\"#]F%)F5F.F%F% *(\"$0\"F%FOF%F+F%F2*(\"#oF%FFF%F*F%F%*(F,F%F9F%F>F%F%*(\"\"'F%F5F%FUF %F2*$)F+F.F%F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7(7(,**&\"#5\"\"\"I \"wG6\"F(F(*&\"\"(F(I\"xGF*F(F(*&\"\"#F(I\"yGF*F(!\"\"I\"zGF*F(,&*&\" \"&F(F-F(F1*&F5F(F0F(F1\"\"!,$*&\"#6F(F0F(F1,&F6F(*&\"\"%F(F2F(F(F77(F 7F%F3F7F8F;7(,&F&F1F+F1,(F4F(F6F(F2F(F7,$*&F'F(F0F(F(,&F6F1*&\"\"$F(F2 F(F1F77(F7F@FAF7FBFD7(,&*&F,F(F)F(F(*&F=F(F-F(F(,&*&FFF(F-F(F1*&F=F(F0 F(F1F7,(F)F(F6F1F2F1,&*&FFF(F0F(F(F2F(F77(F7FIFLF7FOFP<#,&F0F(*&#F,F5F (F2F(F1" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 10 "ImpSurface" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "HowToUse" }{MPLTEXT 1 0 12 "( ImpSurface)" }{MPLTEXT 1 0 1 ";" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calli ng Sequence" }}{PARA 6 "" 1 "" {TEXT 216 15 " ImpSurface(P)" }{TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " P: P is a rational parametrization of a rational surface in aff ine space or a homogeneous parametrization in projective space" }{TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 50 " T he implicit equation of the rational surface P." }{TEXT 216 1 "\n" } {TEXT 216 11 "Description" }{TEXT 216 1 "\n" }{TEXT 216 83 " Implicit ization by combining three functions ImpbyDixonA, Impbympmq and Impby3 mp." }{TEXT 216 1 "\n" }{TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 35 " ImpSurface( [s^2*t,s+t,t^3,s*t] )" }{TEXT 216 1 "\n" }{TEXT 216 60 " ImpSurface( [s^2*t,s+t,t^3] ) #The default last item is 1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ImpSurface(P1);" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",^o*(\"#?\"\"\")I\"w G6\"\"\"$F%)I\"zGF(\"\"#F%F%**\"\"&F%)F'F,F%)I\"xGF(F,F%F+F%!\"\"*,\"# IF%F/F%F1F%I\"yGF(F%F+F%F2**\"#MF%F/F%F1F%F*F%F%**F$F%F/F%)F5F,F%F+F%F 2**\"#SF%F/F%F5F%F*F%F%*(\"#CF%F/F%)F+F)F%F2*(F'F%)F1F)F%F+F%F2*,\"#@F %F'F%F0F%F5F%F+F%F2**\"#FF%F'F%F0F%F*F%F%**F.F%F'F%F1F%)F5F)F%F%*,\"#p F%F'F%F1F%F9F%F+F%F2*,\"#(*F%F'F%F1F%F5F%F*F%F%**\"#LF%F'F%F1F%F>F%F2* (\"#DF%F'F%)F5\"\"%F%F%**\"#!)F%F'F%FFF%F+F%F2**F7F%F'F%F9F%F*F%F%*(F. F%F'F%)F+FPF%F%*&F@F%F*F%F%*&F0F%FFF%F%**\"#>F%F0F%F5F%F*F%F%*(\"#AF%F 0F%F>F%F2*(F$F%F1F%FOF%F%**FNF%F1F%FFF%F+F%F2**FLF%F1F%F9F%F*F%F%**\"# JF%F1F%F5F%F>F%F2*&F1F%FUF%F2*&\"#]F%)F5F.F%F%*(\"$0\"F%FOF%F+F%F2*(\" #oF%FFF%F*F%F%*(F,F%F9F%F>F%F%*(\"\"'F%F5F%FUF%F2*$)F+F.F%F2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ImpSurface(P2);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&)I\"wG6\"\"\"%\"\"\"I\"xG F&F(!\"\"*&F$F(I\"zGF&F(F(*,\"\"#F()F%F.F(F)F(I\"yGF&F(F,F(F**()F)F.F( )F0F.F(F,F(F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 8 "ImpRuled" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "HowToUse(ImpRuled)" } {MPLTEXT 1 0 1 ";" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling Sequence" } }{PARA 6 "" 1 "" {TEXT 216 13 " ImpRuled(P)" }{TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " P: P is a rat ional parametrization of a rational surface in affine space or a homog eneous parametrization in projective space" }{TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 55 " The implicit equatio n of the rational ruled surface P" }{TEXT 216 1 "\n" }{TEXT 216 11 "De scription" }{TEXT 216 1 "\n" }{TEXT 216 45 " Implicitization using un ivariate resultant." }{TEXT 216 1 "\n" }{TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 31 " ImpRuled( [s^2*t,s+t,t,s*t] )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "P3:=[s^2*t,s+t,t,s*t];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&*&)I\"sG6\"\"\"#\"\"\"I\"tGF&F(,&F%F(F)F(F)*&F% F(F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "ImpRuled(P3);" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I\"xG6\"\"\"\"I \"zGF%F&F&F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT 215 9 "ImpMatrix" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "HowToUse(" }{MPLTEXT 1 0 9 "ImpMatrix" }{MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 0 "" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling \+ Sequence" }}{PARA 6 "" 1 "" {TEXT 216 14 " ImpMatrix(P)" }{TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " \+ P: P is a rational parametrization of a rational surface in affine spa ce or a homogeneous parametrization in projective space" }{TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 127 " The imp licit matrix of the rational surface P and extraneous factors associat ed to bad base points and anomalies at infinity." }{TEXT 216 1 "\n" } {TEXT 216 11 "Description" }{TEXT 216 1 "\n" }{TEXT 216 56 " Compute \+ the implicit matrix of the rational surface P." }{TEXT 216 1 "\n" } {TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 61 " ImpMatrix( [s ^2*t,s+t,t,s*t] ) #The default last item is 1" }{TEXT 216 1 "\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ImpMatrix(P3);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$7$I\"xG6\"!\"\"7$F'I\"zGF&< \"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 7 "Threemp" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "HowToUse(" }{MPLTEXT 1 0 7 "Threemp" } {MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 0 "" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling Sequence" }{TEXT 216 1 "\n" }{TEXT 216 12 " Threemp(P)" }} {PARA 6 "" 1 "" {TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 127 " P: P is a rational parametrization of a rational surface in affine space or a homogeneous parametrization in projective space" } {TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" }{TEXT 216 68 " Three moving planes which follow the surface P with the low degr ee" }{TEXT 216 1 "\n" }{TEXT 216 11 "Description" }{TEXT 216 1 "\n" } {TEXT 216 61 " Get three moving planes that used in the function Impb y3mp." }{TEXT 216 1 "\n" }{TEXT 216 7 "Example" }{TEXT 216 1 "\n" } {TEXT 216 32 " Threemp( [s^2*t,s+t,t^3,s*t] )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Threemp(P2);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%,(*(I\"sG6\"\"\"\"I\"tGF&F'I\"yGF&F'!\"\"*&F(F'I\" wGF&F'F'I\"xGF&F',&*(F%F'F(F'F,F'F'*&F(F'F-F'F*,(*&)F(\"\"#F'F)F'F*F+F 'I\"zGF&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 215 6 "polyBP" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "HowToUse(" }{MPLTEXT 1 0 6 "polyBP" }{MPLTEXT 1 0 2 ") ;" }{MPLTEXT 1 0 0 "" }}{PARA 6 "" 1 "" {TEXT 216 16 "Calling Sequence " }}{PARA 6 "" 1 "" {TEXT 216 21 " polyBP( [bp], m, n)" }{TEXT 216 1 "\n" }{TEXT 216 10 "Parameters" }{TEXT 216 1 "\n" }{TEXT 216 48 " bp: a series of points specified as base point" }{TEXT 216 1 "\n" }{TEXT 216 50 " m: the degree of rational surface in parameter s" }{TEXT 216 1 "\n" }{TEXT 216 50 " n: the degree of rational surface in param eter t" }{TEXT 216 1 "\n" }{TEXT 216 6 "Output" }{TEXT 216 1 "\n" } {TEXT 216 56 " A rational tensor product surface with base points bp. " }{TEXT 216 1 "\n" }{TEXT 216 11 "Description" }{TEXT 216 1 "\n" } {TEXT 216 76 " Generate a rational tensor product surface with the sp ecified base points." }{TEXT 216 1 "\n" }{TEXT 216 7 "Example" }{TEXT 216 1 "\n" }{TEXT 216 32 " polyBP( [[1, 1],[2,0]], 3, 4 )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "bp:=[1, 1],[2,0];\n" }{MPLTEXT 1 0 12 "m:=3; n:=4;\n" }{MPLTEXT 1 0 17 "polyBP([bp],m,n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$7$\"\"\"F$7$\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&,H*(\"\")\"\"\")I\"sG6\"\"\"$F&)I\"tGF)\"\"%F&! \"\"*(\"\"*F&F'F&)F,F*F&F&*(\"\"&F&)F(\"\"#F&F+F&F.*(F0F&F'F&)F,F5F&F. *(\"#5F&F4F&F1F&F&*(\"\"'F&F(F&F+F&F.*(F0F&F'F&F,F&F&*(F5F&F4F&F7F&F&* (\"\"(F&F(F&F1F&F&*&F0F&F+F&F.*&F-F&F'F&F&*(F-F&F4F&F,F&F.*&F(F&F7F&F& *&F9F&F1F&F.*$F4F&F.*(F%F&F(F&F,F&F.*&F0F&F7F&F.*&\"#bF&F(F&F.\"##)F&, J*(F0F&F'F&F+F&F&*(F9F&F'F&F1F&F.*(F0F&F4F&F+F&F.*&F'F&F7F&F&*(F*F&F4F &F1F&F&*(F-F&F(F&F+F&F.*(F;F&F'F&F,F&F.*(F-F&F4F&F7F&F.*&F(F&F1F&F&*&F 5F&F+F&F&*&F%F&F'F&F&*(F;F&F4F&F,F&F.*(F;F&F(F&F7F&F.*&F5F&F1F&F&*&F%F &F4F&F&*(F*F&F(F&F,F&F.*&F3F&F7F&F.*&\"$6\"F&F(F&F.*&F-F&F,F&F&\"$E\"F &,H*(F3F&F'F&F+F&F.FMF.*(F%F&F4F&F+F&F&*(F5F&F'F&F7F&F.*(F-F&F4F&F1F&F &*(F*F&F(F&F+F&F.*(F?F&F'F&F,F&F&*(F*F&F4F&F7F&F&*(F5F&F(F&F1F&F&*$F+F &F&*&F;F&F'F&F&*(F%F&F4F&F,F&F&*(F5F&F(F&F7F&F&FZF.*(F0F&F(F&F,F&F&*&F 9F&F7F&F.F(F&*&F3F&F,F&F&\"#=F.,J*(F*F&F'F&F+F&F.*(F5F&F'F&F1F&F.*(F;F &F4F&F+F&F.*(F9F&F'F&F7F&F&F_oF&*(F?F&F(F&F+F&F&*(F3F&F'F&F,F&F&*(F3F& F4F&F7F&F.FTF&F@F.FeoF.*(F0F&F4F&F,F&F.*(F*F&F(F&F7F&F&*&F3F&F1F&F.*&F ?F&F4F&F.*(F-F&F(F&F,F&F&*&F;F&F7F&F.*&\"#iF&F(F&F&*&F9F&F,F&F&\"#[F." }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 214 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }