%@AUTEUR: David Nivaud verbatimtex %&latex \documentclass{article} \usepackage{palatino} \newcommand{\vect}[1]{\overrightarrow{#1}} \newcommand{\Vect}[1]{\overrightarrow{\strut #1}} \begin{document} etex beginfig(1); %caracterisation vectorielle d'un plan numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2.5u); T = identity shifted t; S = identity shifted r; %on place les points du parallèlogramme représentant le plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; %on trace le parallelogramme draw z0--z2; draw z2--z3; %on donne un effet d'epaisseur en changeant l'epaisseur du stylo pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; %on rechange l'epaisseur du stylo pickup pencircle scaled 0.5pt; %on indique le nom du plan label.urt(btex $P$ etex, z0+(0.1u,0u)); %les vecteurs z4=(u,u); z5=(3u,0.5u);z6=(2u,1.8u); draw z4-0.2*(z5-z4)--z5+0.2*(z5-z4); draw z4-0.2*(z6-z4)--z6+0.2*(z6-z4); z7= z4 shifted ((0.7*(z5-z4))+(0.9*(z6-z4))); drawarrow z4--z7; z8 = z4 shifted (0.7*(z5-z4)); z9 = z4 shifted (0.9*(z6-z4)); draw z7--z8 dashed evenly; draw z7--z9 dashed evenly; z10 = z4 shifted (0.3*(z5-z4)); z11 = z4 shifted (0.5*(z6-z4)); drawarrow z4--z10; drawarrow z4--z11; label.bot(btex $A$ etex, z4-0.15*(z5-z4)); label.bot(btex $B$ etex, z10); label.lft(btex $C$ etex, z11); label.rt(btex $M$ etex, z7); label.bot(btex $x$ etex, z8); label.lft(btex $y$ etex, z9); label.rt(btex $y\Vect{AC}$ etex, 0.7[z7,z8]); label.top(btex $x\Vect{AB}$ etex, 0.2[z7,z9]); endfig; beginfig(2); %caracterisation vectorielle d'un plan numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2.5u); T = identity shifted t; S = identity shifted r; %on place les points du parallèlogramme représentant le plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; %on trace le parallelogramme draw z0--z2; draw z2--z3; %on donne un effet d'epaisseur en changeant l'epaisseur du stylo pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; %on rechange l'epaisseur du stylo pickup pencircle scaled 0.5pt; %on indique le nom du plan label.urt(btex $P$ etex, z0+(0.1u,0u)); %les vecteurs z4=(u,u); z5=(3u,0.5u);z6=(2u,1.8u); draw z4-0.2*(z5-z4)--z5+0.2*(z5-z4); draw z4-0.2*(z6-z4)--z6+0.2*(z6-z4); z10 = z4 shifted (0.5*(z5-z4)); z11 = z4 shifted (0.7*(z6-z4)); drawarrow z4--z10; drawarrow z4--z11; label.bot(btex $A$ etex, z4-0.15*(z5-z4)); label.bot(btex $\vect{u}$ etex, 0.5[z10,z4]); label.lft(btex $\vect{v}$ etex, 0.5[z11,z4]); endfig; beginfig(3); %caracterisation vectorielle d'un plan numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2.5u); T = identity shifted t; S = identity shifted r; %on place les points du parallelogramme représentant le plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; %on trace le parallelogramme draw z0--z2; draw z2--z3; %on donne un effet d'epaisseur en changeant l'epaisseur du stylo pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; %on rechange l'epaisseur du stylo pickup pencircle scaled 0.5pt; %on indique le nom du plan label.urt(btex $P$ etex, z0+(0.1u,0u)); %les vecteurs z4=(u,u); z5=(3u,0.5u);z6=(2u,1.8u); z7= z4 shifted ((0.7*(z5-z4))+(0.9*(z6-z4))); drawarrow z4--z7; z8 = z4 shifted (0.7*(z5-z4)); z9 = z4 shifted (0.9*(z6-z4)); drawarrow z4--z8; drawarrow z4--z9; label.bot(btex $O$ etex, z4-0.15*(z5-z4)); label.rt(btex $C$ etex, z7); label.bot(btex $A$ etex, z8); label.top(btex $B$ etex, z9); label.bot(btex $\vect{u}$ etex, 0.5[z8,z4]); label.lft(btex $\vect{v}$ etex, 0.5[z9,z4]); label.top(btex $\vect{w}$ etex, 0.5[z7,z4]); endfig; beginfig(4); % parallelisme plan et droite numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droite dans le plan z4 = (1u,.5u); z5 = (4u,1u); z10= z4 shifted (0.5*(z5-z4)); z11= z4 shifted (0.9*(z5-z4)); label.top(btex $\vect{u}$ etex, 0.5[z10,z11]); dotlabel.lft(btex $A$ etex, z10); label.rt(btex $B$ etex, z11); drawarrow z10--z11; %la droite en dehors du plan z6 = (1u,2.5u); z7 = z6 shifted z5-z4; draw z6--z7; z8= z6 shifted (0.2*(z5-z4)); z9= z6 shifted (0.6*(z5-z4)); drawarrow z8--z9; dotlabel(btex $$ etex, z8); label.top(btex $d$ etex, z6); label.top(btex $\vect{u}$ etex, 0.5[z8,z9]); endfig; beginfig(5); % parallelisme plan et droite numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droite dans le plan z4 = (1u,.5u); z5 = (4u,1u); z10= z4 shifted (0.3*(z5-z4)); z11= z4 shifted (0.7*(z5-z4)); drawarrow z10--z11; dotlabel(btex $$ etex, z10); label.top(btex $\vect{u}$ etex, 0.5[z10,z11]); draw z4--z5; label.top(btex $d'$ etex, z5); %la droite en dehors du plan z6 = (1u,2.5u); z7 = z6 shifted z5-z4; draw z6--z7; z8= z6 shifted (0.4*(z5-z4)); z9= z6 shifted (0.8*(z5-z4)); drawarrow z8--z9; dotlabel(btex $$ etex, z8); label.top(btex $d$ etex, z7); label.top(btex $\vect{u}$ etex, 0.5[z8,z9]); draw z10--z8 dashed evenly; draw z11--z9 dashed evenly; drawarrow (0.5u,0.5u)--(0.6u,1u); drawarrow (0.5u,0.5u)--(1.5u,0.1u); dotlabel(btex $$ etex, (0.5u,0.5u)); label.rt(btex $\vect{v}$ etex, (0.6u,1u)); label.top(btex $\vect{w}$ etex,(1.5u,0.1u)); endfig; beginfig(6); % plans paralleles a l'aide de droites secantes numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %premier plan P z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droites secantes z4 = (2u,1u); z5 =(3.8u,1.5u);drawarrow z4--z5; z6 = (2u,1u);z7=(3.6u,0.4u);drawarrow z6--z7; label.top(btex $\vect{u}$ etex, 0.5[z4,z5]); label.bot(btex $\vect{v}$ etex, 0.5[z6,z7]); %label.lft(btex $A$ etex, z4); %label.rt(btex $B$ etex, z5); %label.rt(btex $C$ etex, z7); %deuxieme plan Q z10=(0u,-2.5u); z11 = z10 transformed T; z12 = z10 transformed S; z13 = z10 transformed T transformed S; draw z10--z12; draw z12--z13; pickup pencircle scaled 2pt; draw z10--z11; draw z11--z13; pickup pencircle scaled 0.5pt; label.urt(btex $Q$ etex, z10+(0.1u,0u)); %droites secantes z14 = (2u,-1.5u); z15 =(3.8u,-1.5u);drawarrow z14--z15; z16 = (2u,-1.5u);z17=(1u,-2u);drawarrow z16--z17; label.top(btex $\vect{u}'$ etex, 0.5[z14,z15]); label.bot(btex $\vect{v}'$ etex, 0.5[z16,z17]); %label.top(btex $A'$ etex, z14); %label.rt(btex $B'$ etex, z15); %label.bot(btex $C'$ etex, z17); endfig; beginfig(7); %repere cartesien dans l'espace numeric u; pair t,s,q; u = 1cm; %definitions de l'origine et des vecteurs de base t=(2u,0u); s=(0u,2u); q=(-1u,-1.5u); z0 = (0,0); z1= z0 shifted t; z2 = z0 shifted s; z3 = z0 shifted q; %trace de l'origine et des vecteurs de base dotlabel.lft(btex $O$ etex, z0); label.bot(btex $\vect{\jmath}$ etex, z1); label.lft(btex $\vect{\imath}$ etex, z3); label.lft(btex $\vect{k}$ etex, z2); %trace des axes drawarrow z0--z1 ; drawarrow z0--z2 ; drawarrow z0--z3 ; endfig; beginfig(8); % coordonnées d'un point dans un repere de l'espace numeric u; pair t,s,q; u = 1cm; %definitions de l'origine et des vecteurs de base t=(1u,0u); s=(0u,1u); q=(-0.5u,-0.5u); z0 = (0,0); z1= z0 shifted t; z2 = z0 shifted s; z3 = z0 shifted q; %trace de l'origine et des vecteurs de base dotlabel.lft(btex $O$ etex, z0); label.bot(btex $\vect{\jmath}$ etex, z1); label.lft(btex $\vect{\imath}$ etex, z3); label.lft(btex $\vect{k}$ etex, z2); %trace des axes drawarrow z0--z1 ; drawarrow z0--z2 ; drawarrow z0--z3 ; %construction du point M z4 = z0 shifted 4t; z5 = z0 shifted 4s; z6 = z0 shifted 4q; drawarrow z0--z4 ; drawarrow z0--z5 ; drawarrow z0--z6 ; z7= 3.5t+2.5q; label.bot(btex $M'$ etex, z7); z8 = 3.5t ; z9 = 2.5q ; draw z7--z8 dashed evenly; draw z7--z9 dashed evenly; label.top(btex $y$ etex, z8); label.lft(btex $x$ etex, z9); z10 = z7 shifted 3s; draw z7--z10 dashed evenly; label.rt(btex $M$ etex, z10); z11 = z10 shifted z0-z7; draw z11--z10 dashed evenly; label.lft(btex $z$ etex, z11); drawarrow z0--z10; drawarrow z0--z7; endfig; beginfig(9); %repere cartesien orthonormal dans l'espace numeric u; pair t,s,q; u = 1cm; %definitions de l'origine et des vecteurs de base t=(2u,0u); s=(0u,2u); q=(-1u,-1.5u); z0 = (0,0); z1= z0 shifted t; z2 = z0 shifted s; z3 = z0 shifted q; %trace de l'origine et des vecteurs de base dotlabel.lft(btex $O$ etex, z0); label.bot(btex $\vect{\jmath}$ etex, z1); label.lft(btex $\vect{\imath}$ etex, z3); label.lft(btex $\vect{k}$ etex, z2); %trace des axes drawarrow z0--z1 ; drawarrow z0--z2 ; drawarrow z0--z3 ; %on marque les angles droits z4=0.1[z0,z1];z5=0.1[z0,z2]; z6= z5 shifted z4-z0; draw z4--z6; draw z5--z6; z14=0.1[z0,z1];z15=0.1[z0,z3]; z16= z15 shifted z14-z0; draw z14--z16; draw z15--z16; z24=0.1[z0,z2];z25=0.1[z0,z3]; z26= z25 shifted z24-z0; draw z24--z26; draw z25--z26; endfig; beginfig(10); %illustration du calcul de distance dans un repere orthonormal numeric u; pair t,s,q; u = 1cm; %definitions de l'origine et des vecteurs de base t=(1u,0u); s=(0u,1u); q=(-0.5u,-0.5u); %construction de l'origine et des vecteurs de base z0 = (0u,0u); z1= z0 shifted t; z2 = z0 shifted s; z3 = z0 shifted q; dotlabel.lft(btex $O$ etex, z0); label.bot(btex $\vect{\jmath}$ etex, z1); label.lft(btex $\vect{\imath}$ etex, z3); label.lft(btex $\vect{k}$ etex, z2); %trace des axes drawarrow z0--z1 ; drawarrow z0--z2 ; drawarrow z0--z3 ; %construction du point M et de ses projetes z4 = z0 shifted 4t; z5 = z0 shifted 4s; z6 = z0 shifted 4q; drawarrow z0--z4 ; drawarrow z0--z5 ; drawarrow z0--z6 ; z7= 3.5t+2.5q; label.bot(btex $m$ etex, z7); z8 = 3.5t ; z9 = 2.5q ; draw z7--z8 dashed evenly; draw z7--z9 dashed evenly; label.top(btex $b$ etex, z8); label.lft(btex $a$ etex, z9); z10 = z7 shifted 3s; draw z7--z10 dashed evenly; label.rt(btex $M$ etex, z10); z11 = z10 shifted z0-z7; draw z11--z10 dashed evenly; label.lft(btex $c$ etex, z11); drawarrow z0--z10; drawarrow z0--z7; %on marque l'angle droit en m z12=0.1[z7,z10];z13=0.1[z7,z0]; z14= z13 shifted z12-z7; draw z12--z14; draw z13--z14; endfig; beginfig(11); % parallelisme plan et droite numeric u; pair t,r; transform T,S; u= 1cm; t=(4u,0u); r=(1u,2u); T = identity shifted t; S = identity shifted r; %plan z0=(0u,0u); z1 = z0 transformed T; z2 = z0 transformed S; z3 = z0 transformed T transformed S; draw z0--z2; draw z2--z3; pickup pencircle scaled 2pt; draw z0--z1; draw z1--z3; pickup pencircle scaled 0.5pt; label.urt(btex $P$ etex, z0+(0.1u,0u)); %droite dans le plan z4 = (1u,.5u); z5 = (4u,1u); z10= z4 shifted (0.5*(z5-z4)); z11= z4 shifted (0.9*(z5-z4)); label.bot(btex $\vect{u}$ etex, 0.5[z10,z11]); dotlabel.top(btex $A$ etex, z10); label.top(btex $B$ etex, z11); drawarrow z10--z11; draw z4--z5; z12= z4 shifted (0.1*(z5-z4)); dotlabel.top(btex $M$ etex, z12); endfig; end