Le cercle de Monge

La courbe orthoptique — lieu des points à partir desquels une courbe est vue sous un angle droit — de l'ellipse est un cercle, le cercle de Monge; il a pour diamètre la diagonale du rectangle qui contient l'ellipse.

%@Auteur: Maxime Chupin
 
verbatimtex 
%&latex 
\documentclass{article} 
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{fourier}
\begin{document} 
etex 
 
input geometrie2d;
u=3cm;
v=3;
 
path r;
path carre;
carre = (0,0)--(1,0)--(1,1)--(0,1)--cycle;
 
for i=0 upto 72:
    beginfig(i+1);
 
    % excentricité : 0.65
 
    numeric a,b,c,e,ya,xa,yb,xb,w,m,xm,ym;
    e := 0.65;
    a := 1.3u;
    b := a*sqrt(1-e**2);
    c := a*e;
    pair M,T,F,H,A,B,M';
    path ellipse,tangente,monge,tang,arc,vraitang;
    ellipse = fullcircle xscaled 2a yscaled 2b;
 
    % axes
    pickup pencircle scaled 0.6pt;
    draw(-5u,0)--(5u,0);
    draw(0,-5u)--(0,5u);
    pickup pencircle scaled 1pt;
    draw ellipse withcolor green;
    drawarrow (0,0)--(0.6u,0);
    drawarrow (0,0)--(0,0.6u);
    pickup pencircle scaled 0.5pt;
 
    % directrice
    z3=(a/e,0);
    z4=(a/e,2u);
    draw 5[z3,z4]--5[z4,z3];
    z6=(-a/e,0);
    z7=(-a/e,2u);
    draw 5[z6,z7]--5[z7,z6];
 
    % M
    xm:=a*cosd(30+5*i);
    ym:=b*sind(30+5*i);
    M = (xm,ym);
 
    %tangente en M
	path tangente;
 
 
    if (xm<>0) and (ym<>0):
	  ya=0;
	  xa=a*a/xm;
 
	  yb=b*b/ym;
	  xb=0;
    fi;
 
    if (ym=0) and (xm<>0):
	  ya=0;
	  xa=(a*a)/xm;
 
	  yb=2u;
	  xb=(a*a)/xm;
    fi;
 
    if (xm=0) and (ym<>0):
	  xa=0;
	  ya=(b*b)/ym;
 
	  xb=2u;
	  yb=(b*b)/ym;
    fi;
 
    %définition de daux points de coordonnés xa,ya et xb,yb
    A = (xa,ya);
    B = (xb,yb);
 
    % cercle de monge
    monge := fullcircle scaled 2sqrt(a*a+b*b);
 
 
 
    pickup pencircle scaled 0.8pt;
    % tracé de la droite
    draw 10[A,B]--10[B,A] withcolor blue;
    tangente := 10[A,B]--10[B,A];
 
 
    % point d'intersection
    arc := halfcircle scaled 2sqrt(a*a+b*b) rotated (15+5*i) ;
    T = arc intersectionpoint tangente;
 
    % tracé du cercle
 
    if i=0:
	r := T;
    else:
	r := r--T;
	draw r withcolor red;
    fi;
 
 
    % tangente en M'
    F=(0,0);
    if (xm<>0) and (ym<>0):	
	w=1*u;
	m=(a*a)/(b*b)*ym/xm*u;
    fi;
 
    if (xm=0) and (ym<>0):
	w=0;
	m=2u;
    fi;
 
    if (ym=0) and (xm<>0):
	w=2u;
	m=0;
    fi;
 
    z2=(w,m);
    H=z2;
    tang = 5[F,H]--5[H,F];
    vraitang := tang shifted T;
    draw tang shifted T withcolor blue;
    % z5= vraitang intersectionpoint ellipse;
 
 
    %carré
    pickup pencircle scaled 0.7pt;
    draw carre scaled 10 rotated (angle(T-M)+90) shifted T
	dashed evenly withcolor 0.2white;
 
 
    % labels
 
    % i,j
    label.bot(btex $\vec \imath$ etex, (0.3u,0));
    label.lft(btex $\vec \jmath$ etex, (-0.01u,0.3u));
 
    % points
    % label.lft(btex $M'$ etex, z5);
    dotlabel.urt(btex $T$ etex, T);
    dotlabel.urt(btex $M$ etex, M);
    dotlabel.urt(btex $O$ etex, (0,0));
    % droites
    label.lft(btex $\delta$ etex,(0,-1.8u));
    label.lft(btex $\cal D$ etex,(-2u,-1.8u));
    label.rt(btex ${\cal D}'$ etex,(2u,-1.8u));
 
    % titre
    label.urt(btex \begin{LARGE}\textit{Le cercle de Monge}\end{LARGE} etex, (-1.5*a,1.3*a)); 
    label.urt(btex \begin{LARGE}$x^2+y^2=a^2+b^2$\end{LARGE} etex, (0.3*a,1.3*a)); 
 
    clip currentpicture to 
	(-2.5u,-2u)--(-2.5u,2u)--(2.5u,2u)--(2.5u,-2u)--cycle;
    endfig;
endfor;
 
end