L'éditeur de formules de Carrara Studio -
Page 30
Hypotrochoïdes en "plaques"
Paramétrisation d'une hypotrochoïde
On appelle hypotrochoïde la courbe plane décrite par un point
P lié à un cercle mobile (de rayon b) roulant sans glisser sur un cercle fixe
(de rayon a), le cercle mobile se touvant à l'intérieur du cercle fixe.
Soit h la distance du point P au centre du cercle mobile. On
pose m = a/b (donc m>1) et h = k*b.
Une paramétrisation de l'hypotrochoïde est alors :
x= m*(b-1)*cos(p)+k*b*cos((m-1)*p));
y= m*(b-1)*sin(p)-k*b*sin((m-1)*p));
le paramètre p variant de 0 à 2*PI.
Paramétrisation de la surface
On introduit un deuxième paramètre (q) de la façon suivante
:
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
1) m = 3
k = 0.5
Objet
Texture
Rendu
m=3;
k=0.5;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=3;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 1
Objet
Texture
Rendu
m=3;
k=1;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=3;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 2
Objet
Texture
Rendu
m=3;
k=2;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=3;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
2) m = 4
k = 0.5
Objet
Texture
Rendu
m=4;
k=0.5;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=4;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 1
Objet
Texture
Rendu
m=4;
k=1;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=4;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 2
Objet
Texture
Rendu
m=4;
k=2;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=4;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
3) m = 5
k = 0.5
Objet
Texture
Rendu
m=5;
k=0.5;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=5;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 1
Objet
Texture
Rendu
m=5;
k=1;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=5;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 2
Objet
Texture
Rendu
m=5;
k=2;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=5;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
4) m = 6
k = 0.5
Objet
Texture
Rendu
m=6;
k=0.5;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=6;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 1
Objet
Texture
Rendu
m=6;
k=1;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=6;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 2
Objet
Texture
Rendu
m=6;
k=2;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=6;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
5) m = 7
k = 0.5
Objet
Texture
Rendu
m=7;
k=0.5;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=7;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 1
Objet
Texture
Rendu
m=7;
k=1;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=7;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 2
Objet
Texture
Rendu
m=7;
k=2;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=7;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
6) m = 8
k = 0.5
Objet
Texture
Rendu
m=8;
k=0.5;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=8;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 1
Objet
Texture
Rendu
m=8;
k=1;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=8;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;
k = 2
Objet
Texture
Rendu
m=8;
k=2;
c=0.2;b=2;
p=2*PI*u;
q=PI*v;
x=sin(q)*(m*(b-1)*cos(p)+k*b*cos((m-1)*p) );
y=sin(q)*(m*(b-1)*sin(p)-k*b*sin((m-1)*p) );
z=c*cos(q);
mt=8;
nt=15;
p=2*PI*u;
q=PI*v;
red=abs(cos(mt*p));
blue=abs(cos(nt*q));
green=1-red*blue;