Plotting a subset of the 4-sphere in the stereographic projection
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I'm looking for a program, library or function in a math language (SAGE will be fantastic) that allows me to plot in 3D a subset of $mathbb{S}^4$ through the canonical stereographic projection that sends it, minus a point, in $mathbb{R}^3$.
Someone know something that allows me to do it? Thank you in advance.
PS: Sorry for my English, it is not my mother language.
math-software stereographic-projections
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
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add a comment |
$begingroup$
I'm looking for a program, library or function in a math language (SAGE will be fantastic) that allows me to plot in 3D a subset of $mathbb{S}^4$ through the canonical stereographic projection that sends it, minus a point, in $mathbb{R}^3$.
Someone know something that allows me to do it? Thank you in advance.
PS: Sorry for my English, it is not my mother language.
math-software stereographic-projections
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
$endgroup$
add a comment |
$begingroup$
I'm looking for a program, library or function in a math language (SAGE will be fantastic) that allows me to plot in 3D a subset of $mathbb{S}^4$ through the canonical stereographic projection that sends it, minus a point, in $mathbb{R}^3$.
Someone know something that allows me to do it? Thank you in advance.
PS: Sorry for my English, it is not my mother language.
math-software stereographic-projections
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
$endgroup$
I'm looking for a program, library or function in a math language (SAGE will be fantastic) that allows me to plot in 3D a subset of $mathbb{S}^4$ through the canonical stereographic projection that sends it, minus a point, in $mathbb{R}^3$.
Someone know something that allows me to do it? Thank you in advance.
PS: Sorry for my English, it is not my mother language.
math-software stereographic-projections
math-software stereographic-projections
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
asked May 27 '18 at 12:39
Davide F.Davide F.
418213
418213
add a comment |
add a comment |
1 Answer
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Is your question connected to the so-called Hopf fibration as illustrated on the following figure ?
Fig. 1 : Foliation of the 4-sphere by an infinite set of torii, a few of them being represented by their family of Villarceau circles.
I have obtained this figure with the following Matlab program where the essential instruction is "u=a*v". I can provide more details ; you can find some in the following document : %http://mathhelpforum.com/differential-geometry/109193-hopf-fibration.html
clear all;close all;hold on;axis equal off;
set(gcf,'color','w');i=complex(0,1);
axis([-2,2,-2,2,-2,2]);
t=0:0.01:2*pi;
nc=30; % number of circles on each torus
nt=3; % number of torii
d=0.; % if d = 0 : torii ; if d > 0 : cyclides
cc=[0. 0.6 0.4
1 0. 0.5
0 0 0.]; % table of colors (must have nt rows)
for L=1:nt;
c=cc(L,:);
for K=1:nc;
a=d+L*exp(i*2*pi*K/nc);
R=1/sqrt(1+abs(a)^2);v=R*exp(i*t);
u=a*v;
ux=real(u);uy=imag(u);
vx=real(v);vy=imag(v);
D=1-vy;
x=ux./D;y=uy./D;z=vx./D; % stereographic projection
plot3(x,y,z,'color',c,'linewidth',1);
end;
end;
view([-44,6]);
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Is your question connected to the so-called Hopf fibration as illustrated on the following figure ?
Fig. 1 : Foliation of the 4-sphere by an infinite set of torii, a few of them being represented by their family of Villarceau circles.
I have obtained this figure with the following Matlab program where the essential instruction is "u=a*v". I can provide more details ; you can find some in the following document : %http://mathhelpforum.com/differential-geometry/109193-hopf-fibration.html
clear all;close all;hold on;axis equal off;
set(gcf,'color','w');i=complex(0,1);
axis([-2,2,-2,2,-2,2]);
t=0:0.01:2*pi;
nc=30; % number of circles on each torus
nt=3; % number of torii
d=0.; % if d = 0 : torii ; if d > 0 : cyclides
cc=[0. 0.6 0.4
1 0. 0.5
0 0 0.]; % table of colors (must have nt rows)
for L=1:nt;
c=cc(L,:);
for K=1:nc;
a=d+L*exp(i*2*pi*K/nc);
R=1/sqrt(1+abs(a)^2);v=R*exp(i*t);
u=a*v;
ux=real(u);uy=imag(u);
vx=real(v);vy=imag(v);
D=1-vy;
x=ux./D;y=uy./D;z=vx./D; % stereographic projection
plot3(x,y,z,'color',c,'linewidth',1);
end;
end;
view([-44,6]);
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
$endgroup$
add a comment |
$begingroup$
Is your question connected to the so-called Hopf fibration as illustrated on the following figure ?
Fig. 1 : Foliation of the 4-sphere by an infinite set of torii, a few of them being represented by their family of Villarceau circles.
I have obtained this figure with the following Matlab program where the essential instruction is "u=a*v". I can provide more details ; you can find some in the following document : %http://mathhelpforum.com/differential-geometry/109193-hopf-fibration.html
clear all;close all;hold on;axis equal off;
set(gcf,'color','w');i=complex(0,1);
axis([-2,2,-2,2,-2,2]);
t=0:0.01:2*pi;
nc=30; % number of circles on each torus
nt=3; % number of torii
d=0.; % if d = 0 : torii ; if d > 0 : cyclides
cc=[0. 0.6 0.4
1 0. 0.5
0 0 0.]; % table of colors (must have nt rows)
for L=1:nt;
c=cc(L,:);
for K=1:nc;
a=d+L*exp(i*2*pi*K/nc);
R=1/sqrt(1+abs(a)^2);v=R*exp(i*t);
u=a*v;
ux=real(u);uy=imag(u);
vx=real(v);vy=imag(v);
D=1-vy;
x=ux./D;y=uy./D;z=vx./D; % stereographic projection
plot3(x,y,z,'color',c,'linewidth',1);
end;
end;
view([-44,6]);
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
$endgroup$
add a comment |
$begingroup$
Is your question connected to the so-called Hopf fibration as illustrated on the following figure ?
Fig. 1 : Foliation of the 4-sphere by an infinite set of torii, a few of them being represented by their family of Villarceau circles.
I have obtained this figure with the following Matlab program where the essential instruction is "u=a*v". I can provide more details ; you can find some in the following document : %http://mathhelpforum.com/differential-geometry/109193-hopf-fibration.html
clear all;close all;hold on;axis equal off;
set(gcf,'color','w');i=complex(0,1);
axis([-2,2,-2,2,-2,2]);
t=0:0.01:2*pi;
nc=30; % number of circles on each torus
nt=3; % number of torii
d=0.; % if d = 0 : torii ; if d > 0 : cyclides
cc=[0. 0.6 0.4
1 0. 0.5
0 0 0.]; % table of colors (must have nt rows)
for L=1:nt;
c=cc(L,:);
for K=1:nc;
a=d+L*exp(i*2*pi*K/nc);
R=1/sqrt(1+abs(a)^2);v=R*exp(i*t);
u=a*v;
ux=real(u);uy=imag(u);
vx=real(v);vy=imag(v);
D=1-vy;
x=ux./D;y=uy./D;z=vx./D; % stereographic projection
plot3(x,y,z,'color',c,'linewidth',1);
end;
end;
view([-44,6]);
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
$endgroup$
Is your question connected to the so-called Hopf fibration as illustrated on the following figure ?
Fig. 1 : Foliation of the 4-sphere by an infinite set of torii, a few of them being represented by their family of Villarceau circles.
I have obtained this figure with the following Matlab program where the essential instruction is "u=a*v". I can provide more details ; you can find some in the following document : %http://mathhelpforum.com/differential-geometry/109193-hopf-fibration.html
clear all;close all;hold on;axis equal off;
set(gcf,'color','w');i=complex(0,1);
axis([-2,2,-2,2,-2,2]);
t=0:0.01:2*pi;
nc=30; % number of circles on each torus
nt=3; % number of torii
d=0.; % if d = 0 : torii ; if d > 0 : cyclides
cc=[0. 0.6 0.4
1 0. 0.5
0 0 0.]; % table of colors (must have nt rows)
for L=1:nt;
c=cc(L,:);
for K=1:nc;
a=d+L*exp(i*2*pi*K/nc);
R=1/sqrt(1+abs(a)^2);v=R*exp(i*t);
u=a*v;
ux=real(u);uy=imag(u);
vx=real(v);vy=imag(v);
D=1-vy;
x=ux./D;y=uy./D;z=vx./D; % stereographic projection
plot3(x,y,z,'color',c,'linewidth',1);
end;
end;
view([-44,6]);
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
edited Dec 15 '18 at 22:54
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
answered Dec 11 '18 at 17:03
Jean MarieJean Marie
29.7k42051
29.7k42051
add a comment |
add a comment |
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