// xcas version=1.1.2-10 fontsize=14 font=0 currentlevel=0 // fltk 7Fl_Tile 10 64 1036 57 14 0 [ // fltk N4xcas23Comment_Multiline_InputE 10 64 1036 56 14 0 Repere de Frenet, calcul de la courbure et du cercle osculateur en un point de la parabole, developpee£faire bouger le curseur pour voir que la developpee est l'enveloppe des normales.£Observer que la courbe traverse le cercle osculateur (sauf au sommet de la parabole) , // fltk N4xcas10Log_OutputE 10 120 1036 1 14 0 ] , // fltk 7Fl_Tile 10 123 1036 368 14 0 [ // fltk N4xcas6FigureE 10 123 1036 367 14 0 landscape=0 history=0.32915 geo=0.5444 mouse_param=0.12645 // fltk N4xcas12History_PackE 10 -89 321 879 14 0 [ // fltk 7Fl_Tile 27 -89 304 48 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 -89 304 26 14 0 25 , x:=t; y:=t^2; xy:=[x,y]; , // fltk N4xcas10Log_OutputE 27 -63 304 1 14 0 , // fltk N4xcas10Gen_OutputE 27 -62 304 21 14 0 t,t^2,[t,t^2] ] , // fltk 7Fl_Tile 27 -39 304 48 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 -39 304 26 14 0 33 , xy1:=diff(xy,t); xy2:=diff(xy1,t), // fltk N4xcas10Log_OutputE 27 -13 304 1 14 0 , // fltk N4xcas10Gen_OutputE 27 -12 304 21 14 0 [1,2*t],[0,2] ] , // fltk 7Fl_Tile 27 11 304 64 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 11 304 26 14 0 18 , G:=plotparam(xy,t), // fltk N4xcas10Log_OutputE 27 37 304 1 14 0 , // fltk 9Fl_Scroll 27 38 304 37 14 0 [ // fltk N4xcas10Gen_OutputE 27 38 343 21 14 0 pnt(pnt[curve(group[pnt[t+(i)*t^2,t,-10.0,10.0,0,undef,undef],group[-10.0+100.0*(i),-9.95+99.0025*(i),-9.9+98.01*(i),-9.85+97.0225*(i),-9.8+96.04*(i),-9.75+95.0625*(i),-9.7+94.09*(i),-9.65+93.1225*(i),-9.6+92.16*(i),-9.55+91.2025*(i),-9.5+90.25*(i),-9.45+89.3025*(i),-9.4+88.36*(i),-9.35+87.4225*(i),-9.3+86.49*(i),-9.25+85.5625*(i),-9.2+84.64*(i),-9.15+83.7225*(i),-9.1+82.81*(i),-9.05+81.9025*(i),-9.0+81.0*(i),-8.95+80.1025*(i),-8.9+79.21*(i),-8.85+78.3225*(i),-8.8+77.44*(i),-8.75+76.5625*(i),-8.7+75.69*(i),-8.65+74.8225*(i),-8.6+73.96*(i),-8.55+73.1025*(i),-8.5+72.25*(i),-8.45+71.4025*(i),-8.4+70.56*(i),-8.35+69.7225*(i),-8.3+68.89*(i),-8.25+68.0625*(i),-8.2+67.24*(i),-8.15+66.4225*(i),-8.1+65.61*(i),-8.05+64.8025*(i),-8.0+64.0*(i),-7.95+63.2025*(i),-7.9+62.41*(i),-7.85+61.6225*(i),-7.8+60.84*(i),-7.75+60.0625*(i),-7.7+59.29*(i),-7.65+58.5225*(i),-7.6+57.76*(i),-7.55+57.0025*(i),-7.5+56.25*(i),-7.45+55.5025*(i),-7.4+54.76*(i),-7.35+54.0225*(i),-7.3+53.29*(i),-7.25+52.5625*(i),-7.2+51.84*(i),-7.15+51.1225*(i),-7.1+50.41*(i),-7.05+49.7025*(i),-7.0+49.0*(i),-6.95+48.3025*(i),-6.9+47.61*(i),-6.85+46.9225*(i),-6.8+46.24*(i),-6.75+45.5625*(i),-6.7+44.89*(i),-6.65+44.2225*(i),-6.6+43.56*(i),-6.55+42.9025*(i),-6.5+42.25*(i),-6.45+41.6025*(i),-6.4+40.96*(i),-6.35+40.3225*(i),-6.3+39.69*(i),-6.25+39.0625*(i),-6.2+38.44*(i),-6.15+37.8225*(i),-6.1+37.21*(i),-6.05+36.6025*(i),-6.0+36.0*(i),-5.95+35.4025*(i),-5.9+34.81*(i),-5.85+34.2225*(i),-5.8+33.64*(i),-5.75+33.0625*(i),-5.7+32.49*(i),-5.65+31.9225*(i),-5.6+31.36*(i),-5.55+30.8025*(i),-5.5+30.25*(i),-5.45+29.7025*(i),-5.4+29.16*(i),-5.35+28.6225*(i),-5.3+28.09*(i),-5.25+27.5625*(i),-5.2+27.04*(i),-5.15+26.5225*(i),-5.1+26.01*(i),-5.05+25.5025*(i),-5.0+25.0*(i),-4.95+24.5025*(i),-4.9+24.01*(i),-4.85+23.5225*(i),-4.8+23.04*(i),-4.75+22.5625*(i),-4.7+22.09*(i),-4.65+21.6225*(i),-4.6+21.16*(i),-4.55+20.7025*(i),-4.5+20.25*(i),-4.45+19.8025*(i),-4.4+19.36*(i),-4.35+18.9225*(i),-4.3+18.49*(i),-4.25+18.0625*(i),-4.2+17.64*(i),-4.15+17.2225*(i),-4.1+16.81*(i),-4.05+16.4025*(i),-4.0+16.0*(i),-3.95+15.6025*(i),-3.9+15.21*(i),-3.85+14.8225*(i),-3.8+14.44*(i),-3.75+14.0625*(i),-3.7+13.69*(i),-3.65+13.3225*(i),-3.6+12.96*(i),-3.55+12.6025*(i),-3.5+12.25*(i),-3.45+11.9025*(i),-3.4+11.56*(i),-3.35+11.2225*(i),-3.3+10.89*(i),-3.25+10.5625*(i),-3.2+10.24*(i),-3.15+9.9225*(i),-3.1+9.61*(i),-3.05+9.3025*(i),-3.0+9.0*(i),-2.95+8.7025*(i),-2.9+8.41*(i),-2.85+8.1225*(i),-2.8+7.84*(i),-2.75+7.5625*(i),-2.7+7.29*(i),-2.65+7.0225*(i),-2.6+6.76*(i),-2.55+6.5025*(i),-2.5+6.25*(i),-2.45+6.0025*(i),-2.4+5.76*(i),-2.35+5.5225*(i),-2.3+5.29*(i),-2.25+5.0625*(i),-2.2+4.84*(i),-2.15+4.6225*(i),-2.1+4.41*(i),-2.05+4.2025*(i),-2.0+4.0*(i),-1.95+3.8025*(i),-1.9+3.61*(i),-1.85+3.4225*(i),-1.8+3.24*(i),-1.75+3.0625*(i),-1.7+2.89*(i),-1.65+2.7225*(i),-1.6+2.56*(i),-1.55+2.4025*(i),-1.5+2.25*(i),-1.45+2.1025*(i),-1.4+1.96*(i),-1.35+1.8225*(i),-1.3+1.69*(i),-1.25+1.5625*(i),-1.2+1.44*(i),-1.15+1.3225*(i),-1.1+1.21*(i),-1.05+1.1025*(i),-1.0+1.0*(i),-0.95+0.9025*(i),-0.9+0.81*(i),-0.85+0.7225*(i),-0.8+0.64*(i),-0.75+0.5625*(i),-0.7+0.49*(i),-0.65+0.4225*(i),-0.6+0.36*(i),-0.55+0.3025*(i),-0.5+0.25*(i),-0.45+0.2025*(i),-0.4+0.16*(i),-0.35+0.1225*(i),-0.3+0.09*(i),-0.25+0.0625*(i),-0.2+0.04*(i),-0.15+0.0225*(i),-0.1+0.01*(i),-0.05+0.0025*(i),7.64666108211e-15,0.05+0.0025*(i),0.1+0.01*(i),0.15+0.0225*(i),0.2+0.04*(i),0.25+0.0625*(i),0.3+0.09*(i),0.35+0.1225*(i),0.4+0.16*(i),0.45+0.2025*(i),0.5+0.25*(i),0.55+0.3025*(i),0.6+0.36*(i),0.65+0.4225*(i),0.7+0.49*(i),0.75+0.5625*(i),0.8+0.64*(i),0.85+0.7225*(i),0.9+0.81*(i),0.95+0.9025*(i),1.0+i,1.05+1.1025*(i),1.1+1.21*(i),1.15+1.3225*(i),1.2+1.44*(i),1.25+1.5625*(i),1.3+1.69*(i),1.35+1.8225*(i),1.4+1.96*(i),1.45+2.1025*(i),1.5+2.25*(i),1.55+2.4025*(i),1.6+2.56*(i),1.65+2.7225*(i),1.7+2.89*(i),1.75+3.0625*(i),1.8+3.24*(i),1.85+3.4225*(i),1.9+3.61*(i),1.95+3.8025*(i),2.0+4.0*(i),2.05+4.2025*(i),2.1+4.41*(i),2.15+4.6225*(i),2.2+4.84*(i),2.25+5.0625*(i),2.3+5.29*(i),2.35+5.5225*(i),2.4+5.76*(i),2.45+6.0025*(i),2.5+6.25*(i),2.55+6.5025*(i),2.6+6.76*(i),2.65+7.0225*(i),2.7+7.29*(i),2.75+7.5625*(i),2.8+7.84*(i),2.85+8.1225*(i),2.9+8.41*(i),2.95+8.7025*(i),3.0+9.0*(i),3.05+9.3025*(i),3.1+9.61*(i),3.15+9.9225*(i),3.2+10.24*(i),3.25+10.5625*(i),3.3+10.89*(i),3.35+11.2225*(i),3.4+11.56*(i),3.45+11.9025*(i),3.5+12.25*(i),3.55+12.6025*(i),3.6+12.96*(i),3.65+13.3225*(i),3.7+13.69*(i),3.75+14.0625*(i),3.8+14.44*(i),3.85+14.8225*(i),3.9+15.21*(i),3.95+15.6025*(i),4.0+16.0*(i),4.05+16.4025*(i),4.1+16.81*(i),4.15+17.2225*(i),4.2+17.64*(i),4.25+18.0625*(i),4.3+18.49*(i),4.35+18.9225*(i),4.4+19.36*(i),4.45+19.8025*(i),4.5+20.25*(i),4.55+20.7025*(i),4.6+21.16*(i),4.65+21.6225*(i),4.7+22.09*(i),4.75+22.5625*(i),4.8+23.04*(i),4.85+23.5225*(i),4.9+24.01*(i),4.95+24.5025*(i),5.0+25.0*(i),5.05+25.5025*(i),5.1+26.01*(i),5.15+26.5225*(i),5.2+27.04*(i),5.25+27.5625*(i),5.3+28.09*(i),5.35+28.6225*(i),5.4+29.16*(i),5.45+29.7025*(i),5.5+30.25*(i),5.55+30.8025*(i),5.6+31.36*(i),5.65+31.9225*(i),5.7+32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, // fltk 12Fl_Scrollbar 27 59 304 16 14 0 [] , // fltk 12Fl_Scrollbar 315 38 16 21 14 0 [] ] ] , // fltk 7Fl_Tile 27 77 304 64 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 77 304 26 14 0 28 , assume(a=[0.125,-5,5,0.025]), // fltk N4xcas10Log_OutputE 27 103 304 1 14 0 , // fltk 9Fl_Scroll 27 104 304 37 14 0 [ // fltk N4xcas10Gen_OutputE 27 104 343 21 14 0 parameter([a,-5.0,5.0,-0.475,0.025]) , // fltk 12Fl_Scrollbar 27 125 304 16 14 0 [] , // fltk 12Fl_Scrollbar 315 104 16 21 14 0 [] ] ] , // fltk 7Fl_Tile 27 143 304 48 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 143 304 26 14 0 15 , M:=element(G,a), // fltk N4xcas10Log_OutputE 27 169 304 1 14 0 , // fltk N4xcas10Gen_OutputE 27 170 304 21 14 0 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5+63.2025*i,8.0+64.0*i,8.05+64.8025*i,8.1+65.61*i,8.15+66.4225*i,8.2+67.24*i,8.25+68.0625*i,8.3+68.89*i,8.35+69.7225*i,8.4+70.56*i,8.45+71.4025*i,8.5+72.25*i,8.55+73.1025*i,8.6+73.96*i,8.65+74.8225*i,8.7+75.69*i,8.75+76.5625*i,8.8+77.44*i,8.85+78.3225*i,8.9+79.21*i,8.95+80.1025*i,9.0+81.0*i,9.05+81.9025*i,9.1+82.81*i,9.15+83.7225*i,9.2+84.64*i,9.25+85.5625*i,9.3+86.49*i,9.35+87.4225*i,9.4+88.36*i,9.45+89.3025*i,9.5+90.25*i,9.55+91.2025*i,9.6+92.16*i,9.65+93.1225*i,9.7+94.09*i,9.75+95.0625*i,9.8+96.04*i,9.85+97.0225*i,9.9+98.01*i,9.95+99.0025*i,10.0+100.0*i]]),56]),a]],M]) ] , // fltk 7Fl_Tile 27 193 304 48 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 193 304 26 14 0 28 , T:=normalize(subst(xy1,t=a)), // fltk N4xcas10Log_OutputE 27 219 304 1 14 0 , // fltk N4xcas10Gen_OutputE 27 220 304 21 14 0 [1/sqrt(4*a^2+1),(2*a)/(sqrt(4*a^2+1))] ] , // fltk 7Fl_Tile 27 243 304 64 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 243 304 26 14 0 23 , vector(M,M+T,color=red), // fltk N4xcas10Log_OutputE 27 269 304 1 14 0 , // fltk 9Fl_Scroll 27 270 304 37 14 0 [ // fltk N4xcas10Gen_OutputE 27 270 547 21 14 0 pnt(pnt[vector[(i)*a^2+a,1/sqrt(1+2*a*2*a)+((i)*2*a)/(sqrt(1+2*a*2*a))+(i)*a^2+a],1]) , // fltk 12Fl_Scrollbar 27 291 304 16 14 0 [] , // fltk 12Fl_Scrollbar 315 270 16 21 14 0 [] ] ] , // fltk 7Fl_Tile 27 309 304 64 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 309 304 26 14 0 27 , vector(M,M+i*T,color=green), // fltk N4xcas10Log_OutputE 27 335 304 1 14 0 , // fltk 9Fl_Scroll 27 336 304 37 14 0 [ // fltk N4xcas10Gen_OutputE 27 336 564 21 14 0 pnt(pnt[vector[(i)*a^2+a,1/sqrt(1+2*a*2*a)*(i)+(i)*2*a/sqrt(1+2*a*2*a)*(i)+(i)*a^2+a],2]) , // fltk 12Fl_Scrollbar 27 357 304 16 14 0 [] , // fltk 12Fl_Scrollbar 315 336 16 21 14 0 [] ] ] , // fltk 7Fl_Tile 27 375 304 92 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 375 304 70 14 0 52 , R:=subst( l2norm(xy1)^3/l2norm(cross(xy1,xy2)), t=a), // fltk N4xcas10Log_OutputE 27 445 304 1 14 0 , // fltk N4xcas10Gen_OutputE 27 446 304 21 14 0 (sqrt(4*a^2+1)*(4*a^2+1))/2 ] , // fltk 7Fl_Tile 27 469 304 48 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 469 304 26 14 0 28 , circle(M+i*T*R,R,color=blue), // fltk N4xcas10Log_OutputE 27 495 304 1 14 0 , // fltk N4xcas10Gen_OutputE 27 496 304 21 14 0 Done ] , // fltk 7Fl_Tile 27 519 304 64 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 519 304 26 14 0 14 , trace(M+i*T*R), // fltk N4xcas10Log_OutputE 27 545 304 1 14 0 , // fltk 9Fl_Scroll 27 546 304 37 14 0 [ // fltk N4xcas10Gen_OutputE 27 546 876 21 14 0 trace(pnt(pnt[(1/sqrt(1+2*a*2*a)*(i)*sqrt(1+2*a*2*a)*(1+2*a*2*a))/2+((i)*2*a/sqrt(1+2*a*2*a)*(i)*sqrt(1+2*a*2*a)*(1+2*a*2*a))/2+(i)*a^2+a,56])) , // fltk 12Fl_Scrollbar 27 567 304 16 14 0 [] , // fltk 12Fl_Scrollbar 315 546 16 21 14 0 [] ] ] , // fltk 7Fl_Tile 27 585 304 64 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 585 304 26 14 0 25 , trace(half_line(M,M+i*T)), // fltk N4xcas10Log_OutputE 27 611 304 1 14 0 , // fltk 9Fl_Scroll 27 612 304 37 14 0 [ // fltk N4xcas10Gen_OutputE 27 612 542 21 14 0 trace(pnt(pnt[[(i)*a^2+a,1/sqrt(1+2*a*2*a)*(i)+(i)*2*a/sqrt(1+2*a*2*a)*(i)+(i)*a^2+a],56])) , // fltk 12Fl_Scrollbar 27 633 304 16 14 0 [] , // fltk 12Fl_Scrollbar 315 612 16 21 14 0 [] ] ] , // fltk 7Fl_Tile 27 651 304 108 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 651 304 70 14 0 46 , evolute(G, display=magenta+ epaisseur_ligne_3), // fltk N4xcas10Log_OutputE 27 721 304 1 14 0 , // fltk 9Fl_Scroll 27 722 304 37 14 0 [ // fltk N4xcas10Gen_OutputE 27 722 327 21 14 0 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, // fltk 12Fl_Scrollbar 27 745 288 14 14 0 [] , // fltk 12Fl_Scrollbar 315 722 16 21 14 0 [] ] ] , // fltk 7Fl_Tile 27 761 304 27 14 0 [ // fltk N4xcas16Xcas_Text_EditorE 27 761 304 26 14 0 0 , , // fltk N4xcas10Log_OutputE 27 787 304 1 14 0 ] ] // fltk N4xcas5Geo2dE 351 143 564 347 14 0 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