# Une propriété de la parabole, animée !

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

parabole.mp
%@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