Animations

Cardioïde

On prend un cercle de diamètre 2a passant par l'origine. On trace une tangente quelconque à ce cercle en un point P. On projette ensuite O sur cette tangente, on obtient alors le point M. Si P parcourt le cercle alors M décrit une cardioïde.

%@AUTEUR: Maxime Chupin
%@DATE: 22 mars 2007
 
verbatimtex
%&latex
\documentclass{article}
\usepackage{amsmath}
\begin{document}
etex
 
u:=3cm;
a:= 2u;
 
vardef proj(expr P,M,N) =
    save H; pair H;
    H = whatever [M,N];
    H - P = whatever * (M - N) rotated 90;
    H
enddef;
 
path carre,cardioide;
carre = (0,0)--(1,0)--(1,1)--(0,1)--cycle;
 
 
for i:=0 upto 80:
    beginfig(i+1);
	path cercle,tangente,vec;
	pair P,M,O,P';
	pickup pencircle scaled 0.8;
	drawarrow (-a,0)--(2a,0);
        drawarrow (0,-a)--(0,a);
	O:=(0,0);
	pickup pencircle scaled 0.5;
	cercle := fullcircle scaled a shifted (a/2,0);
	draw cercle dashed evenly;
 
	P = point (4-i/10) of cercle;
 
	if i=0:
	    M = P;
	    cardioide = M;
	fi;
 
	if (i<>0) and (i<>40) and (i<>80):
	    P' = (xpart P +0.5*a, (xpart P +0.5*a)*((-xpart P)/(ypart P) + 
		a/(2*(ypart P)))+a*(xpart P)/(2*(ypart P)));
	    tangente = 5[P,P']--5[P',P];
	    M = proj(O,P,P');
	    vec = 20[O,M]--20[M,O];
	    cardioide := cardioide..M;
    	    pickup pencircle scaled 1;
	    drawarrow cardioide withcolor red;
    	    pickup pencircle scaled 0.5;
	    draw carre scaled 12 rotated (angle(M-P)+180) 
		shifted M dashed withdots scaled 0.3;
	    draw tangente withcolor blue dashed evenly;
	    draw vec withcolor blue dashed evenly;
	    dotlabel.urt(btex $M$ etex,M);
	    dotlabel.urt(btex $P$ etex,P);
	fi;
 
	if i=40 :
    	    pickup pencircle scaled 1;
	    M:=P;
	    cardioide := cardioide..M;
	    draw cardioide withcolor red;
	    dotlabel.urt(btex $P$ etex,P);
	fi;
 
	if i=80 :
    	    pickup pencircle scaled 1;
	    M:=P;
	    cardioide := cardioide..M;
	    draw cardioide withcolor red;
	fi;
	label.top(btex $a$ etex, (a/2,0));
	label.llft(btex $x$ etex, (0.99*2a,0));
	label.urt(btex $y$ etex, (0,0.9*a)); 
	dotlabel.urt(btex $O$ etex,O);
	label(btex $\boxed{r=a(1+\cos\theta)}$ etex,(3/2*a,a/2));
	clip currentpicture to (-a,-a)--(2*a,-a)--(2*a,a)--(-a,a)--cycle;
 
    endfig;
endfor;
end