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Des kiéroïdes...

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Contruction d'une kiéroïde.

verbatimtex
%&latex
\documentclass{minimal}
\usepackage[charter]{mathdesign}
\usepackage[utf8]{inputenc}
\begin{document}
etex
 
% l'unité graphique
u:=1cm;
 
% paramètres
largeurDroite:=12u;
 
% fonction(s)
def droite(expr a,b)=
  10[a,b]--10[b,a]
enddef;
 
def demiedroite(expr a,b)=
  0.01[a,b]--10[a,b] 
enddef;
 
compt:=0;
for i=0 step 0.05 until 3.5: 
  beginfig(compt);
    a:=-3u+i*u;
    b:=-4u;
    c:=abs(a-b); % rayon du cercle
    % les ensembles de construction
    pair O;
    O:=(0,0);
    path supportCercle,  supportSegment, cercle,kieroide;
 
    supportCercle:=(-largeurDroite,a)--(+largeurDroite,a);
    supportSegment:=(-largeurDroite,b)--(+largeurDroite,b);
    cercle:=fullcircle scaled (2*c); 
 
 
    pair P,N,M;
    path  demieNO, cercleP;
 
    for t:=0 step 0.01 until 1:
    % le point P parcours la droite supportCercle
      P:=point t of supportCercle;
    % cercle centré en P
      cercleP:=cercle shifted P;
    % M comme intersection
      N:=(xpart P,b);
      demieNO:=demiedroite(N,O);
      M:=demieNO intersectionpoint cercleP;
    %on définie la courbe
      if t=0:
	kieroide:=M;
      else:
	kieroide:=kieroide--M;
      fi; 
    endfor;
 
    % on redéfinit les points pour le tracé d,un configuration particulière
    P:=point 0.4 of supportCercle;
    cercleP:=cercle shifted P;
    % M comme intersection
    N:=(xpart P,b);
    demieNO:=demiedroite(N,O);
    M:=demieNO intersectionpoint cercleP;
    % on dessine
    draw supportCercle withcolor blue dashed evenly withpen pencircle scaled 1pt;
    draw cercleP dashed evenly  withcolor (0.7,0.7,0.7)  withpen pencircle scaled 1pt;
    draw supportSegment dashed evenly  withcolor (0.7,0.7,0.7)  withpen pencircle scaled 1pt;
    draw N--O withcolor (0.7,0.7,0.7) withpen pencircle scaled 1pt;
    draw N--P withcolor (0.7,0.7,0.7) withpen pencircle scaled 1pt;
    draw M--P withcolor (0.7,0.7,0.7) withpen pencircle scaled 1pt;
    draw kieroide withcolor red withpen pencircle scaled 1pt;
    % les labels
    label(btex \itshape La Kiéroïde etex, (5u, u));
    dotlabel.bot(btex $P$ etex, P);
    dotlabel.bot(btex $M$ etex, M);
    dotlabel.bot(btex $N$ etex, N);
    dotlabel.bot(btex $O$ etex, O);
    label.ulft(btex $\mathcal C$ etex, P+(cosd(125)*c,sind(125)*c));
    label.bot(btex $\Delta$ etex, (8u,a));
    clip currentpicture to (10u,-5u)--(-10u,-5u)--(-10u,5u)--(10u,5u)--cycle;
  endfig;
  compt:=compt+1;
endfor;
end.