Une propriété de la parabole, animée !

En se dépaçant le long de sa directrice, la parabole est toujours vue sous un angle droit.

%@Auteur: Maxime Chupin
 
verbatimtex 
%&latex 
\documentclass{article} 
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{fourier}
\begin{document} 
\large
etex 
 
labeloffset := 8;
 
u=3cm;
v=3mm;
 
 
vardef proj(expr P,M,N) =
     save H; pair H;
     H = whatever [M,N]; H - P = whatever * (M - N) rotated 90; H
enddef;
 
 
vardef droite(expr a,b) =
    10[a,b]--10[b,a]
enddef;
 
path carre;
carre = (0,0)--(1,0)--(1,1)--(0,1)--cycle;
 
path parabole;
parabole = (3.466u,-2.30940u)..(3.1333u,-2.06559u)..(2.8u,-1.78885u)
            ..(2.1333u,-1.03279u)..{dir 90}(1.8u,0){dir 90}
            ..(2.1333u,1.03279u)..(2.4666u,1.46059u)..(2.8u,1.78885u)
            ..(3.1333u,2.06559u)..(3.466u,2.30940u);
 
pair O,I,M,M',K,K',F,H,H';
 
F  := (2.6u,0);
K  := (u,0);
K' := (u,3u);
 
for i=1 upto 60:
 beginfig(i);
 
    pickup pencircle scaled 1pt;
 
    % axe focal
    draw (0,0)--(5u,0);
    % directrice
    draw (u,-2u)--(u,2u);
 
    x := 2.6 + 0.4 * cosd(6i);
    M := ((x*u,0)--(x*u,3u)) intersectionpoint parabole;
 
    draw parabole withcolor red;
 
    % vecteurs i et j
    drawarrow (1.8u,0)--(2.4u,0);
    drawarrow (1.8u,0)--(1.8u,0.6u);
    pickup pencircle scaled 0.5pt;
 
    H := proj(M,K,K');
    dotlabel.lrt(btex $H$ etex, H);
 
    % droite pointille
    draw H--M dashed evenly;
    draw F--H dashed evenly;
 
    % milieu de HF
    I := 0.5[H,F];
 
    % M'
    draw droite(F,M);
    M' := droite(F,M) intersectionpoint parabole;
    dotlabel.urt(btex $M'$ etex, M');
 
    % projeté de M' sur la tangente issue de M
    O := proj(M',M,I);
    dotlabel.lft(btex $O$ etex,O);
 
    % carre
    draw carre scaled 10 rotated (angle(O-M)+90) shifted O
	dashed evenly withcolor 0.2white;
 
    % tangente en M
    pickup pencircle scaled 0.7pt;
    draw droite(I,M) withcolor blue;
    pickup pencircle scaled 0.5pt;
 
    % deuxieme tangente
    pickup pencircle scaled 0.7pt;
    draw droite(O,M') withcolor blue;
    pickup pencircle scaled 0.5pt;
 
    % projete de M' sur la directrice
    H':=proj(M',K,K');
    dotlabel.lrt(btex $H'$ etex, H');
    draw H'--M' dashed evenly;
 
 
    % K
    dotlabel.ulft(btex $K$ etex,(u,0));
    % F
    dotlabel.lrt(btex $F$ etex,F);
    dotlabel.lrt(btex $M$ etex,M);
 
    % S
    dotlabel.llft(btex $S$ etex,(1.8u,0));
    % D
    label.ulft(btex $\cal D$ etex,(u,-2u));
 
    % delta
    label.top(btex $\Delta$ etex,(4.8u,0));
 
    % i,j
    label.bot(btex $\vec i$ etex, (2.1u,0));
    label.lft(btex $\vec j$ etex, (1.8u,0.3u));
 
    clip currentpicture to 
	(0,-2u)--(0,2u)--(5u,2u)--(5u,-2u)--cycle;
 
  endfig;
endfor;
end