Plotting a subset of the 4-sphere in the stereographic projection

Question

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.

Answer 1

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]);