input geoelem.mp ;

verbatimtex 
%&latex 
\documentclass{article} 
\usepackage{palatino} 
\newcommand{\vect}[1]{\overrightarrow{#1}}
\newcommand{\Vect}[1]{\overrightarrow{\strut #1}}
\begin{document} 
etex 
                
beginfig(1);
numeric u;
u=1.5cm ;
draw fullcircle scaled 3cm;

dotlabel.lft(btex $0$ etex, (0,0));

drawarrow (cosd(30)*1.4*u,sind(30)*1.4*u)..(cosd(45)*1.4*u,sind(45)*1.4*u)..
(cosd(60)*1.4*u,sind(60)*1.4*u);
drawarrow (cosd(30)*1.4*u,-sind(30)*1.4*u)..(cosd(45)*1.4*u,-sind(45)*1.4*u)..
(cosd(60)*1.4*u,-sind(60)*1.4*u);
label.rt (btex $+$ etex,(cosd(45)*1.6*u,sind(45)*1.6*u));
label.rt (btex $-$ etex,(cosd(45)*1.6*u,-sind(45)*1.6*u));

draw (0,0)--(cosd(30)*u,sind(30)*u);
draw (0,0)--(cosd(30)*u,-sind(30)*u);
dotlabel.lrt(btex $A$ etex, (cosd(30)*u,-sind(30)*u));
dotlabel.urt(btex $B$ etex, (cosd(30)*u,sind(30)*u));
z1=(cosd(30)*u,sind(30)*u);z2=(0,0);z3=(cosd(30)*u,-sind(30)*u);

mark_angle(z3,z2,z1,2); %ca tourne dans le sens trigo 
label.top(btex $\alpha$ etex, (0.4*u,-0.15*u));

label.top(btex $1$ etex, (0.4*u,0.2*u));
label.rt(btex $l$ etex, (1.1*u,-0.15*u));


endfig;


beginfig(2);
numeric u;
u=1.5cm ;
draw fullcircle scaled 3cm;

dotlabel.lft(btex $0$ etex, (0,0));

drawarrow (cosd(30)*1.4*u,sind(30)*1.4*u)..(cosd(45)*1.4*u,sind(45)*1.4*u)..
(cosd(60)*1.4*u,sind(60)*1.4*u);
drawarrow (cosd(30)*1.4*u,-sind(30)*1.4*u)..(cosd(45)*1.4*u,-sind(45)*1.4*u)..
(cosd(60)*1.4*u,-sind(60)*1.4*u);
label.rt (btex $+$ etex,(cosd(45)*1.6*u,sind(45)*1.6*u));
label.rt (btex $-$ etex,(cosd(45)*1.6*u,-sind(45)*1.6*u));

draw (0,0)--(cosd(30)*u,sind(30)*u);
draw (0,0)--(cosd(30)*u,-sind(30)*u);
dotlabel.lrt(btex $A$ etex, (cosd(30)*u,-sind(30)*u));
dotlabel.urt(btex $B$ etex, (cosd(30)*u,sind(30)*u));
z1=(cosd(30)*u,sind(30)*u);z2=(0,0);z3=(cosd(30)*u,-sind(30)*u);

label.rt(btex $l$ etex, (1.1*u,-0.15*u));
label.lft(btex $2\pi -l$ etex, (-1.1*u,-0.15*u));


endfig;



beginfig(3);
numeric u;
u=2.5cm ;
draw fullcircle scaled 5cm;

drawarrow (0,0)--(1.8*u,0);
drawarrow (0,0)--(u,0);
drawarrow (0,0)--(-cosd(45)*1.4*u,sind(45)*1.4*u);
drawarrow (0,0)--(-cosd(45)*u,sind(45)*u);

drawarrow (0.3*u,0){up}..(0,0.3*u)..(-0.3*cosd(45)*u,0.3*sind(45)*u);

dotlabel.llft(btex $0$ etex, (0,0));
dotlabel.lrt(btex $A$ etex, (u,0));
dotlabel.lft(btex $B$ etex, ((-cosd(45))*u,sind(45)*u));

label.bot(btex $M$ etex,(1.8*u,0));
label.top(btex $N$ etex,(-cosd(45)*1.4*u,sind(45)*1.4*u));
label.top(btex $l$ etex,(0.4*u,1*u));

label.bot(btex $\vect{u}$ etex, (0.5*u,0));
label.lft(btex $\vect{v}$ etex, (-cosd(45)*0.5*u-0.1*u,sind(45)*0.5*u));
endfig;


beginfig(4);
numeric u;
u=2.5cm ;
draw fullcircle scaled 5cm;

drawarrow (0,0)--(1.8*u,0);
drawarrow (0,0)--(u,0);
drawarrow (0,0)--(-cosd(45)*1.4*u,sind(45)*1.4*u);
drawarrow (0,0)--(-cosd(45)*u,sind(45)*u);

drawarrow (-0.3*cosd(45)*u,0.3*sind(45)*u)..(-0.3*u,0)..(0,-0.3*u)
..(0.3*u,0);

dotlabel.llft(btex $0$ etex, (0,0));
dotlabel.lrt(btex $B$ etex, (u,0));
dotlabel.lft(btex $A$ etex, ((-cosd(45))*u,sind(45)*u));

label.bot(btex $N$ etex,(1.8*u,0));
label.top(btex $M$ etex,(-cosd(45)*1.4*u,sind(45)*1.4*u));

label.bot(btex $\vect{u}$ etex, (0.5*u,0));
label.lft(btex $\vect{v}$ etex, (-cosd(45)*0.5*u-0.1*u,sind(45)*0.5*u));
label.top(btex $2\pi -l$ etex,(0.4*u,1*u));
endfig;


beginfig(5);
numeric u;
u=1cm ;

drawarrow (0,0)--(4*u,0);
drawarrow (0,0.1*u)--(2.4*u,0.1*u);

label.bot(btex $\vect{u}$ etex, (2*u,0));
label.top(btex $\vect{v}$ etex, (1.6*u,0.1*u));
endfig;

beginfig(6);
numeric u;
u=1cm ;
dotlabel.lft(btex $$ etex, (0,0));
drawarrow (0,0)--(4*u,0);
drawarrow (0,0)--(-2.4*u,0);

label.bot(btex $\vect{u}$ etex, (2*u,0));
label.top(btex $\vect{v}$ etex, (-1.2*u,0u));

label.top(btex $\pi$ etex, (0,0.5*u));
drawarrow (0.5*u,0){up}..(0,0.5*u)..(-0.5*u,0){down};
endfig;



beginfig(7);
numeric u;
u=1cm ;
dotlabel.lft(btex $$ etex, (0,0));
drawarrow (0,0)--(1.5*u,0);
drawarrow (0,0)--(-cosd(45)*u,sind(45)*u);
drawarrow (0,0)--(-cosd(60)*1.2*u,-sind(60)*1.2*u);

label.bot(btex $\vect{u}$ etex, (1*u,0));
label.lft(btex $\vect{v}$ etex, (-cosd(45)*u-0.1*u,sind(45)*u));
label.ulft(btex $\vect{w}$ etex, (-cosd(60)*1.2*u-0.1*u,-sind(60)*1.2*u));

drawarrow (0.3*u,0){up}..(-cosd(45)*0.3*u,sind(45)*0.3*u);
drawarrow (-cosd(45)*0.4*u,sind(45)*0.4*u)..(-0.4*u,0)..(-cosd(60)*0.4*u,-sind(60)*0.4*u);
drawarrow 
(0.7*u,0){up}..(-cosd(45)*0.7*u,sind(45)*0.7*u)..(-cosd(60)*0.7*u,-sind(60)*0.7*u);

endfig;


beginfig(8);
numeric u;
u=1cm ;
dotlabel.lft(btex $$ etex, (0,0));
drawarrow (0,0)--(1.5*u,0);
drawarrow (0,0)--(-cosd(45)*u,sind(45)*u);

label.bot(btex $\vect{u}$ etex, (1*u,0));
label.lft(btex $\vect{v}$ etex, (-cosd(45)*u-0.1*u,sind(45)*u));

drawarrow (0.3*u,0){up}..(-cosd(45)*0.3*u,sind(45)*0.3*u);
drawarrow reverse((0.6*u,0){up}..(-cosd(45)*0.6*u,sind(45)*0.6*u));

endfig;

beginfig(9);
numeric u;
u=1cm ;
dotlabel.lft(btex $$ etex, (0,0));
drawarrow (0,0)--(1.5*u,0);
drawarrow (0,0)--(cosd(45)*u,sind(45)*u);
drawarrow (0,0)--(-cosd(45)*u,-sind(45)*u);

label.bot(btex $\vect{u}$ etex, (1*u,0));
label.lft(btex $\vect{v}$ etex, (cosd(45)*u-0.1*u,sind(45)*u));
label.rt(btex $-\vect{v}$ etex, (-cosd(45)*u,-sind(45)*u));

drawarrow (0.3*u,0){up}..(cosd(45)*0.3*u,sind(45)*0.3*u);
drawarrow (0.6*u,0){up}..(cosd(45)*0.6*u,sind(45)*0.6*u)..(-0.6*u,0)..(-cosd(45)*0.6*u,-sind(45)*0.6*u);

endfig;



beginfig(10);
numeric u;
u=1cm ;
dotlabel.lft(btex $$ etex, (0,0));
drawarrow (0,0)--(1.5*u,0);
drawarrow (0,0)--(-1.5*u,0);
drawarrow (0,0)--(cosd(45)*u,sind(45)*u);


label.bot(btex $\vect{u}$ etex, (1*u,0));
label.bot(btex $-\vect{u}$ etex, (-1*u,0));
label.lft(btex $\vect{v}$ etex, (cosd(45)*u-0.1*u,sind(45)*u));

drawarrow (0.6*u,0){up}..(cosd(45)*0.6*u,sind(45)*0.6*u);
drawarrow (-0.3*u,0){down}..(0,-0.3*u)..(0.3*u,0)..(cosd(45)*0.3*u,sind(45)*0.3*u);

endfig;

beginfig(11);
numeric u;
u=1cm ;
dotlabel.lft(btex $$ etex, (0,0));
drawarrow (0,0)--(1.5*u,0);
drawarrow (0,0)--(-1.5*u,0);
drawarrow (0,0)--(cosd(45)*u,sind(45)*u);
drawarrow (0,0)--(-cosd(45)*u,-sind(45)*u);

label.bot(btex $\vect{u}$ etex, (1*u,0));
label.bot(btex $-\vect{u}$ etex, (-1*u,0));
label.lft(btex $\vect{v}$ etex, (cosd(45)*u-0.1*u,sind(45)*u));
label.rt(btex $-\vect{v}$ etex, (-cosd(45)*u,-sind(45)*u));

drawarrow (0.5*u,0){up}..(cosd(45)*0.5*u,sind(45)*0.5*u);
drawarrow (-0.5*u,0){down}..(-cosd(45)*0.5*u,-sind(45)*0.5*u);

endfig;


beginfig(12);
numeric u;
u=1cm ;
transform S,T;
T = identity rotated -135 ;
S = identity rotated  60 ;

z0=(0,0) ; z1=(2*u,0); z2=(1*u,0);
z3-z1=(z2-z1) transformed T;

draw z0--z1; draw z1--z3;

dotlabel.bot(btex $A$ etex, z0);
dotlabel.bot(btex $B$ etex, z1);
dotlabel.bot(btex $C$ etex, z3);

z4 = z3+(z3-z1);
draw z3--z4 dashed evenly;
z5-z3 = (z4-z3) transformed S;
draw z3--z5;
dotlabel.lft(btex $D$ etex, z5);

z6=0.5[z3,z5];
z7=0.5[z3,z4];
drawarrow z7{up}..z6;
z8=0.5[z1,z3];
drawarrow z2{up}..z8;

label.top(btex $-\frac{3 \pi}{4}$ etex, (1*u,0.5*u));
label.top(btex $\frac{\pi}{3}$ etex, (3*u,1.2*u));

endfig;

beginfig(13);
numeric u;
u=1cm ;
dotlabel.bot(btex $O$ etex, (0,0));
draw (0,0)--(3*u,0) withcolor blue;
drawarrow (0,0)--(2*u,0);
draw (0,0)--(4*u,3*u) withcolor blue;
drawarrow (0,0)--(2*u,1.5*u);

dotlabel.bot(btex $A$ etex , (2*u,0));
dotlabel.top(btex $B$ etex , (2*u,1.5*u));
label.bot(btex $\vect{u}$ etex, (1*u,0));
label.top(btex $\vect{v}$ etex, (1*u,0.75*u));



endfig;




beginfig(14);
%coordonnees polaires
numeric u;
pair i,j;
u=1cm; 

i=(1u,0u);
j=(0u,1u);
transform T,R;
T= identity shifted i;
R= identity shifted j;
z0=(0,0);z1= z0 transformed T;
z2=z0 transformed R;

drawarrow z0--z1 ; 
drawarrow z0--z2 ;
label.llft(btex $O$ etex, z0);
label.bot(btex $\vect{i}$ etex, 0.5[z0,z1]);
label.lft(btex $\vect{j}$ etex, 0.5[z0,z2]);
z3= z0 shifted 3i;z4= z0 shifted -i;
z5= z0 shifted 2j; z6= z0 shifted -0.5j;
draw z4--z3+i withcolor blue; draw z6--z5+j withcolor blue;

z7 = z0 shifted 3(i+j) ; 
draw z0--z7 ;

z8 = 0.8[z0,z7];
z9 = 0.4[z0,z7];
draw z0--z8 withcolor red;

label.top(btex $r$ etex , 0.5[z9,z8]);

drawarrow 1.5i{up}..z9;% withcolor green;
drawarrow 3.2i{up}..z8;% withcolor green;

label.rt(btex $\theta$ etex , 0.5[z9,1.5i]);

dotlabel.top(btex $M$ etex , z8);
%dotlabel.top(btex $A$ etex , z9);
endfig;


beginfig(15); %rotation du plan orienté
numeric u;
u=1cm;
z0=(1u,1u); z1=(3u,1u); 
drawarrow z0..z1 ;
z2-z0 =(z1-z0) rotated 60; 
drawarrow z0..z2;
draw z1{up}..z2 dashed evenly;
z3=0.3[z0,z1];
z4=0.3[z0,z2];
drawarrow z3{up}..z4;
%mark_angle(z1,z0,z2,0);
draw_marked(z0--z1,2);
draw_marked(z0--z2,2);
dotlabel.bot(btex $I$ etex, z0); 
label.bot(btex $M$ etex, z1);
label.top(btex $M'$ etex, z2);
z5=(1.8u,1.3u);
label.top(btex $\alpha$ etex, z5);
z6=(2u,0.5u);
label.bot(btex $\alpha=\frac{\pi}{3}$ etex, z6);
endfig;




beginfig(16);
numeric u;
u=2.5cm ;
draw fullcircle scaled 5cm;
draw fullcircle scaled 8cm;

draw (0,0)--(1.8*u,0);
draw (0,0)--(-1.8*u,0);
draw (0,0)--(0,1.8*u);
draw (0,0)--(0,-1.8*u);

drawarrow (0,0)--(u,0);
drawarrow (0,0)--(0,u);

drawarrow (0,0)--(cosd(120)*1.6*u,sind(120)*1.6*u);
draw (cosd(120)*1.6*u,sind(120)*1.6*u)--(cosd(120)*1.6*u,0) dashed evenly;
draw (cosd(120)*1.6*u,sind(120)*1.6*u)--(0,sind(120)*1.6*u) dashed evenly;
draw (cosd(120)*u,sind(120)*u)--(cosd(120)*u,0) dashed evenly;
draw (cosd(120)*u,sind(120)*u)--(0,sind(120)*u) dashed evenly;
drawarrow (0,0)--(cosd(120)*u,sind(120)*u);

drawarrow (0.3*u,0){up}..(0,0.3*u)..(0.3*cosd(120)*u,0.3*sind(120)*u);

dotlabel.llft(btex $0$ etex, (0,0));
dotlabel.lft(btex $N$ etex, ((cosd(120))*u,sind(120)*u));
dotlabel.top(btex $M$ etex,(cosd(120)*1.6*u,sind(120)*1.6*u));
label(btex $\theta$ etex, (0.3*u,0.3*u));

label.bot(btex $\vect{\imath}$ etex, (0.5*u,0));
label.rt(btex $\vect{\jmath}$ etex, (0,0.5*u));
%label.lft(btex $\vect{u}$ etex, ((cosd(120))*u*0.5,sind(120)*u*0.5));
label.bot(btex $x$ etex, (cosd(120)*1.6*u,0)) ;
label.rt(btex $y$ etex, (0,sind(120)*1.6*u)) ;
label.bot(btex $\cos \theta$ etex, (cosd(120)*u,0)) ;
label.rt(btex $\sin \theta$ etex, (0,sind(120)*u)) ;

endfig;