{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 20 "Matrices de rotation" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart:with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, n ew definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new de finition for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "gen_ rotation:=proc(v0,theta)\nlocal v,nv0,x,y,z;\nv:=vector([x,y,z]):nv0:= sqrt(dotprod(v0,v0)): RETURN(genmatrix(convert(evalm(dotprod(v,v0)/nv0 ^2*v0+cos(theta)*(v-dotprod(v,v0)/nv0^2*v0)+sin(theta)*crossprod(v0,v) /nv0),list),[x,y,z]))\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "M:=gen_rotation(vector([1,-1,1]),Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'MATRIXG6#7%7%#\"\"#\"\"$#!\"#F,#!\"\"F,7%#\"\" \"F,F*F-7%F*F2F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(M); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "transpose(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'MATRIXG6#7%7%#\"\"#\"\"$#\"\"\"F*F(7%#!\"#F*F(F+7%#!\"\"F*F.F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(1/M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%#\"\"#\"\"$#\"\"\"F*F(7%#!\"#F*F(F +7%#!\"\"F*F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "angle:=p roc(A)\nRETURN(arccos((trace(A)-1)/2));\nend:\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 230 "direction:=proc(A)\nlocal v0, w;\nv0:=op(kern el(A-Matrix(3,shape=identity))):\nif v0[1]=0 then w:=vector([1,0,0]) e lse w:=vector([v0[2],-v0[1],0]) fi:\nif evalf(dotprod(A&*w,crossprod(v 0,w)))>0 then RETURN(v0) else RETURN(-v0) fi\nend:" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 257 22 "Projection perspective" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "projeteP:=p roc(p,d)\nRETURN([p[1]*(p[3]+d)/p[3],p[2]*(p[3]+d)/p[3]]);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "projete:=proc(s,d)\nRETURN([ seq(projeteP(s[i],d),i=1..nops(s))]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "translateP:=proc(p,v)\nRETURN([p[1]+v[1],p[2]+v[2] ,p[3]+v[3]]);\nend:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "translate:=p roc(s,d)\nRETURN([seq(translateP(s[i],d),i=1..nops(s))]);\nend:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "rotationP:=proc(p,M)\nlocal v;\nv:=multiply(M,vector(p));\nRETURN( [v[1],v[2],v[3]]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "rotation:=proc(s,M)\nRETURN([seq(rotationP(s[i],M),i=1..nops(s))]);\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "cube:=[[0,0,0],[1, 0,0],[1,1,0],[0,1,0],[0,0,0],[0,0,1],[1,0,1],[1,0,0],[1,1,0],[1,1,1],[ 1,0,1],[1,1,1],[0,1,1],[0,1,0],[0,1,1],[0,0,1],[0,0,0]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot(projete(translate(rotation(cub e,M),[1,-1,40]),2),x=-5..5,y=-5..5);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$737$$\"1++++++]5!#:$ !1++++++]5F*7$$\"1@9yzQj[b`%*p!#;7$$\"1y[!y[!y[5F*\"\"!7$$ \"1!oJ&*eA')\\$F2$!1!oJ&*eA')\\$F2F'7$$\"1$oD\">b`%*pF2$!1@9yzQj[ " 0 "" {MPLTEXT 1 0 46 "triang le:=[[0,0],[1,0],[0.5,sqrt(3)/2],[0,0]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)triangleG7&7$\"\"!F'7$\"\"\"F'7$$\"\"&!\"\",$*$\"\"$#F)\"\"#F 1F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "translationPoint:=pr oc(p,v)\nRETURN([p[1]+v[1],p[2]+v[2]]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "translation:=proc(s,v)\nRETURN([seq(translationP oint(s[i],v),i=1..nops(s))]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "reduirePoint:=proc(p,scal)\nRETURN([scal*p[1],scal*p[ 2]]);\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "reduire:=pro c(s)\nRETURN([seq(reduirePoint(s[i],0.5),i=1..nops(s))]);\nend:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "suivant:=proc(s)\nlocal t1, t2;\nt1:=translation(s,[1,0]);\nt2:=translation(s,[0.5,sqrt(3)/2]);\nR ETURN(reduire([op(s),op(t2),op(t1),[0,0]]));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sierpinski:=proc(n)\nRETURN(plot((suivant@@ n)(triangle)));\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "si erpinski(5);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7a_o7$\"\"!F( 7$$\"1++++++DJ!#k_fZKFEFgqFfq7 $FcpFcqFjqFfqFbqFeq7$FcpF\\o7$F`pF_qFeqF`r7$FhpF_qF^rF`rF_r7$F[qF\\oFa rF_rFeqF[oF[r7$FcpF\\r7$F`p$\"1#GuG?G#=NFEF[rFdr7$FhpFer7$Fcp$\"1>pbT6 '))y$FEFdrFcr7$F[qF\\rFgrFcrF[rFhr7$F[qFir7$Fhp$\"1c&R-3%\\fSFEFhrF]s7 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F6F[[mFijlFgjlFhjl7$FjilF(7$FgilF0FhjlF_[m7$F_jlF0F][mF_[mF^[m7$FbjlF( F`[mF^[mFhjlF\\glF\\[m7$FjilFC7$FgilFSF\\[mFc[m7$F_jlFS7$FjilFWFc[mFb[ m7$FbjlFCFd[mFb[mF\\[mFe[m7$FbjlFW7$F_jlFfnFe[mFh[m7$$\"1+++++D1*)FEFf nFajlFh[mFg[m7$$\"1+++++]i!*FEFWFi[mFg[mFe[mFf[m7$F]\\mFC7$Fj[mFSFf[mF `\\m7$$\"1+++++v=#*FEFSF\\\\mF`\\mF_\\m7$$FHFEFCFa\\mF_\\mFf[mF\\[mFa[ m7$F]\\mF(7$Fj[mF0Fa[mFg\\m7$Fb\\mF07$F]\\mF6Fg\\mFf\\m7$Fe\\mF(Fh\\mF f\\mFa[mFi\\m7$Fe\\mF67$Fb\\mF=Fi\\mF\\]m7$$\"1+++++DJ&*FEF=Fd\\mF\\]m F[]m7$$\"1+++++](o*FEF6F]]mF[]mFi\\mFj\\m7$Fa]mF(7$F^]mF0Fj\\mFd]m7$$ \"1+++++vV)*FEF0F`]mFd]mFc]m7$$\"\"\"F(F(Fe]mFc]mFj\\mFa[mF\\glFixF'-% 'COLOURG6&%$RGBG$\"#5!\"\"F(F(" 2 624 624 624 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 -16411 0 0 0 0 0 0 0 }}}{EXCHG {PARA 259 "" 0 "" {TEXT 259 12 "Gra m-Schmidt" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "f:=(x,y)-> x[1]*y[1] +x[2]*y[2]+x[3]*y[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6$%\"x G%\"yG6\"6$%)operatorG%&arrowGF),(*&&9$6#\"\"\"F2&9%F1F2F2*&&F06#\"\"# F2&F4F7F2F2*&&F06#\"\"$F2&F4F " 0 "" {MPLTEXT 1 0 19 "N:=x->sqrt(f(x,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"NG:6#%\"xG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#-%\"fG6$9$F2F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 253 "schmidt:=proc(v)\nlocal \+ i,j, somme, u;\nu := [scalarmul(v[1], 1/N(v[1]))];\nfor j from 2 to no ps(v) do\n somme := v[j];\n for i from 1 to j-1 do\n somme:=matad d(somme, u[i],1,-f(u[i],v[j]));\n od;\n u:=[op(u),scalarmul(somme,1/ N(somme))];\nod;\nop(u);\nend;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "schmidt([[1,1,1],[1,1,0],[1,0,1]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "15 2 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }