documentclass[border=0pt,10pt,dvipsnames]{standalone}
usepackage{amsmath}
usepackage{pstricks}
usepackage{dashrule}
usepackage{pst-plot}%, pst-math}
usepackage{pst-func}
usepackage{pst-ode}
usepackage{pst-3dplot}
usepackage{pst-solides3d}
usepackage[off]{auto-pst-pdf}
usepackage[nomessages]{fp}
psset{algebraic,unit=2}
usepackage[active,tightpage]{preview}
PreviewBorder=12pt
PreviewEnvironment{pspicture}




makeatletter
deflistplotIIID{defpst@par{}pst@object{listplotIIID}}
deflistplotIIID@i#1{%
@nameuse{beginplot@psplotstyle}%
addto@pscode{%
   /viewpointXYZ {pst@solides@viewpoint} def
   /Decran pst@solides@Decranspace def % distance de l'ecran
    viewpointXYZ /ZpointVue ED /YpointVue ED /XpointVue ED
  /THETA {YpointVue XpointVue atan} bind def
  /PHI   {ZpointVue XpointVue dup mul YpointVue dup mul add sqrt atan} bind def
  /Dobs  {XpointVue dup mul YpointVue dup mul add ZpointVue dup mul add sqrt} bind def
  XpointVue YpointVue ZpointVue /viewpoint defpoint3d
    /XYZ [#1] def
  /@tabXYZ [
    0 3 XYZ length 3 sub {/i exch def
     XYZ i get
     XYZ i 1 add get
     XYZ i 2 add get
     3dto2d
     pst@numberpsunit mul exch
     pst@numberpsunit mul exch
    }  for
    ] bind def
 [ @tabXYZ aload pop
    }%
@nameuse{endplot@psplotstyle}}%
makeatother


defr(#1){0.25*sqrt((9-#1*#1)*(#1*#1+1))}
defrs(#1){-(1/4)*#1*(#1^2-4)/sqrt(-(#1^2-9)*(#1^2+1))}
defrss(#1){-(1/4)*(#1^6-12*#1^4-27*#1^2+36)/((#1^2-9)*(#1^2+1)*sqrt(-(#1^2-9)*(#1^2+1))) }


begin{document}
begin{pspicture}(-2.1,-1.1)(2.5,1.2)
psset[pst-solides3d]{viewpoint=20 5 20,Decran=20,lightsrc=viewpoint}
defFunction[algebraic]{rot}(t,v){r(t)*sin(v)} {t} {r(t)*cos(v)}
psSolid[object=surfaceparametree,base=-3 3 0 6.28,
inhue=0.01 0.11, hue=0.1 0,
function=rot,linewidth=0.5pt,grid=false,opacity=0.8,ngrid=200 70,
intersectiontype=0,
%% Intersections with planes [a b c d] = ax + by + cz + d = 0
intersectionplan={[0 1 0 0] [0 1 0 -2.828] [0 0 1 0]},
intersectioncolor=(red) (red)  (blue),
intersectionlinewidth=1 1]

axesIIID(1,3,1)(2,3.5,2)

pstODEsolve[algebraic,algebraicOutputFormat]{test}{r(x[2])*sin(x[0])
| x[2] | r(x[2])*cos(x[0])}{0}{14}{2000}{1.57 1 2.8284 0}{ x[1] |
 -2*rs(x[2])*x[1]*x[3]/(r(x[2]))| x[3] |
 -rss(x[2])*rs(x[2])*x[3]*x[3]/(rs(x[2])*rs(x[2])+1)
+rs(x[2])*r(x[2])*x[1]*x[1]/(rs(x[2])*rs(x[2])+1)}

listplotIIID[linecolor=black,linewidth=1pt]{test}

pstODEsolve[algebraic,algebraicOutputFormat]{test}{r(x[2])*sin(x[0]) |
 x[2] | r(x[2])*cos(x[0])}{0}{2}{500}{1.57 -1 2.8284 0}{ x[1] |
 -2*rs(x[2])*x[1]*x[3]/(r(x[2]))| x[3] |
 -rss(x[2])*rs(x[2])*x[3]*x[3]/(rs(x[2])*rs(x[2])+1)
+rs(x[2])*r(x[2])*x[1]*x[1]/(rs(x[2])*rs(x[2])+1)}
listplotIIID[linecolor=black,linewidth=1pt]{test}
psSolid[object=point,args=0.75 2.828 0]
end{pspicture}
end{document}

documentclass[border=0pt,10pt,dvipsnames,pstricks]{standalone}
usepackage{amsmath}
usepackage{dashrule}
usepackage{pst-plot}%, pst-math}
usepackage{pst-func}
usepackage{pst-ode}
usepackage{pst-3dplot}

usepackage{pst-solides3d}
usepackage[nomessages]{fp}
psset{algebraic,unit=2}

makeatletter
define@key[psset]{}{valuerange}[1-]{expandafterpst@getrange#1@nil}
defpst@getrange#1-#2@nil{%
  ifxrelax#1relax defpst@startvalue{1 }elsedefpst@startvalue{#1 }fi%
  ifxrelax#2relax defpst@endvalue{1e32 }elsedefpst@endvalue{#2 }fi%
}
psset{valuerange=1-}

deflistplotIIID{defpst@par{}pst@object{listplotIIID}}
deflistplotIIID@i#1{%
@nameuse{beginplot@psplotstyle}%
addto@pscode{%
   /viewpointXYZ {pst@solides@viewpoint} def
   /Decran pst@solides@Decranspace def % distance de l'ecran
    viewpointXYZ /ZpointVue ED /YpointVue ED /XpointVue ED
  /THETA {YpointVue XpointVue atan} bind def
  /PHI   {ZpointVue XpointVue dup mul YpointVue dup mul add sqrt atan} bind def
  /Dobs  {XpointVue dup mul YpointVue dup mul add ZpointVue dup mul add sqrt} bind def
  XpointVue YpointVue ZpointVue /viewpoint defpoint3d
    /XYZ [#1] def
  /@tabXYZ [
    0 3 XYZ length 3 sub {
     /i exch def
     XYZ i get
     XYZ i 1 add get
     XYZ i 2 add get
     i 3 div dup /ii ED
       pst@startvalue ge {
         ii pst@endvalue le {
         3dto2d
         pst@numberpsunit mul exch
         pst@numberpsunit mul exch
       } { pop pop pop } ifelse
     }{ pop pop pop } ifelse
    }  for
    ] bind def
 [ @tabXYZ aload pop
    }%
@nameuse{endplot@psplotstyle}}%
makeatother

defr(#1){0.25*sqrt((9-#1*#1)*(#1*#1+1))}
defrs(#1){-(1/4)*#1*(#1^2-4)/sqrt(-(#1^2-9)*(#1^2+1))}
defrss(#1){-(1/4)*(#1^6-12*#1^4-27*#1^2+36)/((#1^2-9)*(#1^2+1)*sqrt(-(#1^2-9)*(#1^2+1))) }


begin{document}
begin{pspicture}(-2.1,-1.1)(2.5,1.2)
psset[pst-solides3d]{viewpoint=20 5 20,Decran=20,lightsrc=viewpoint}
defFunction[algebraic]{rot}(t,v){r(t)*sin(v)} {t} {r(t)*cos(v)}
%psSolid[object=surfaceparametree,base=-3 3 0 6.28,inhue=0.01 0.11, hue=0.1 0,
%   function=rot,linewidth=0.5pt,grid=false,opacity=0.8,ngrid=200 70,
%   intersectiontype=0,
%   %% Intersections with planes [a b c d] = ax + by + cz + d = 0
%   intersectionplan={[0 1 0 0] [0 1 0 -2.828] [0 0 1 0]},
%   intersectioncolor=(red) (red)  (blue),
%   intersectionlinewidth=1 1]

axesIIID(1,3,1)(2,3.5,2)

pstODEsolve[algebraic,algebraicOutputFormat,%saveData
  ]{test}{r(x[2])*sin(x[0])
| x[2] | r(x[2])*cos(x[0])}{0}{14}{2000}{1.57 1 2.8284 0}{ x[1] |
 -2*rs(x[2])*x[1]*x[3]/(r(x[2]))| x[3] |
 -rss(x[2])*rs(x[2])*x[3]*x[3]/(rs(x[2])*rs(x[2])+1)
+rs(x[2])*r(x[2])*x[1]*x[1]/(rs(x[2])*rs(x[2])+1)}

listplotIIID[linewidth=1pt,linecolor=red,valuerange={-50}]{test}
listplotIIID[linewidth=1pt,linecolor=green,valuerange={50-100}]{test}
listplotIIID[linewidth=1pt,linecolor=blue,valuerange={100-}]{test}

pstODEsolve[algebraic,algebraicOutputFormat]{test}{
  r(x[2])*sin(x[0]) |
  x[2]               |
  r(x[2])*cos(x[0])}{0}{2}{500}{1.57 -1 2.8284 0}{ x[1] |
 -2*rs(x[2])*x[1]*x[3]/(r(x[2]))| x[3] |
 -rss(x[2])*rs(x[2])*x[3]*x[3]/(rs(x[2])*rs(x[2])+1)
    +rs(x[2])*r(x[2])*x[1]*x[1]/(rs(x[2])*rs(x[2])+1)}

listplotIIID[linewidth=1pt,linecolor=cyan,valuerange={-50}]{test}
listplotIIID[linewidth=1pt,linecolor=magenta,valuerange={50-100}]{test}
listplotIIID[linewidth=1pt,linecolor=gray,valuerange={100-}]{test}

%psSolid[object=point,args=0.75 2.828 0]

end{pspicture}

end{document}