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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;		 htrochsec11.jpg

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;		 htrochsec12.jpg

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;

htrochsec13.jpg

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;		 htrochsec21.jpg

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;		 htrochsec22.jpg

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;		 htrochsec23.jpg

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;		 htrochsec31.jpg

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;		 htrochsec32.jpg

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;		 htrochsec33.jpg

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;		 htrochsec41.jpg

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;		 htrochsec42.jpg

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;		 htrochsec43.jpg

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;		 htrochsec51.jpg

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;		 htrochsec52.jpg

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;		 htrochsec53.jpg

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;		 htrochsec61.jpg

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;		 htrochsec62.jpg

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;		 htrochsec63.jpg