How to visualize a 4-dimensional parametric surface?

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What are some good ways, using ParametricPlot3D or otherwise, to visualize a surface (taking 2 real parameters) embedded in a 4-dimensional space?



Specifically, the question concerns the embedding of the Klein Bottle in $ mathbb R^4 $ given by 



 F[u_, v_] := x[u, v], y[u, v], z[u, v], w[u, v]


where



 x[u_, v_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
 y[u_, v_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
 z[u_, v_] := p Cos[u] (1 + ϵ Sin[v])
 w[u_, v_] := p Sin[u] (1 + ϵ Sin[v])


and the positive constants r, p, and ϵ will take convenient values.






plotting graphics3d visualization parametric-functions







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  • Klein Bottle: Eric Weisstein's notebook from MathWorld
    – kglr
    12 hours ago











  • The documentation of ParametricPlot3D contains some information about "Klein bottle".
    – Î‘λέξανδρος Ζεγγ
    10 hours ago














up vote
3
down vote

favorite












What are some good ways, using ParametricPlot3D or otherwise, to visualize a surface (taking 2 real parameters) embedded in a 4-dimensional space?



Specifically, the question concerns the embedding of the Klein Bottle in $ mathbb R^4 $ given by 



 F[u_, v_] := x[u, v], y[u, v], z[u, v], w[u, v]


where



 x[u_, v_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
 y[u_, v_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
 z[u_, v_] := p Cos[u] (1 + ϵ Sin[v])
 w[u_, v_] := p Sin[u] (1 + ϵ Sin[v])


and the positive constants r, p, and ϵ will take convenient values.






plotting graphics3d visualization parametric-functions







share|improve this question























  • Klein Bottle: Eric Weisstein's notebook from MathWorld
    – kglr
    12 hours ago











  • The documentation of ParametricPlot3D contains some information about "Klein bottle".
    – Î‘λέξανδρος Ζεγγ
    10 hours ago












up vote
3
down vote

favorite









up vote
3
down vote

favorite











What are some good ways, using ParametricPlot3D or otherwise, to visualize a surface (taking 2 real parameters) embedded in a 4-dimensional space?



Specifically, the question concerns the embedding of the Klein Bottle in $ mathbb R^4 $ given by 



 F[u_, v_] := x[u, v], y[u, v], z[u, v], w[u, v]


where



 x[u_, v_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
 y[u_, v_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
 z[u_, v_] := p Cos[u] (1 + ϵ Sin[v])
 w[u_, v_] := p Sin[u] (1 + ϵ Sin[v])


and the positive constants r, p, and ϵ will take convenient values.






plotting graphics3d visualization parametric-functions







share|improve this question















What are some good ways, using ParametricPlot3D or otherwise, to visualize a surface (taking 2 real parameters) embedded in a 4-dimensional space?



Specifically, the question concerns the embedding of the Klein Bottle in $ mathbb R^4 $ given by 



 F[u_, v_] := x[u, v], y[u, v], z[u, v], w[u, v]


where



 x[u_, v_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
 y[u_, v_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
 z[u_, v_] := p Cos[u] (1 + ϵ Sin[v])
 w[u_, v_] := p Sin[u] (1 + ϵ Sin[v])


and the positive constants r, p, and ϵ will take convenient values.






plotting graphics3d visualization parametric-functions




plotting graphics3d visualization parametric-functions






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 1 hour ago









J. M. is somewhat okay.♦

92.5k10286440




92.5k10286440










asked 13 hours ago









murray

5,8591634




5,8591634











  • Klein Bottle: Eric Weisstein's notebook from MathWorld
    – kglr
    12 hours ago











  • The documentation of ParametricPlot3D contains some information about "Klein bottle".
    – Î‘λέξανδρος Ζεγγ
    10 hours ago
















  • Klein Bottle: Eric Weisstein's notebook from MathWorld
    – kglr
    12 hours ago











  • The documentation of ParametricPlot3D contains some information about "Klein bottle".
    – Î‘λέξανδρος Ζεγγ
    10 hours ago















Klein Bottle: Eric Weisstein's notebook from MathWorld
– kglr
12 hours ago





Klein Bottle: Eric Weisstein's notebook from MathWorld
– kglr
12 hours ago













The documentation of ParametricPlot3D contains some information about "Klein bottle".
– Î‘λέξανδρος Ζεγγ
10 hours ago




The documentation of ParametricPlot3D contains some information about "Klein bottle".
– Î‘λέξανδρος Ζεγγ
10 hours ago










2 Answers
2






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up vote
4
down vote













A simple minded possibility is to use a perspective projection (similar to what I did here). Applied to the 4D Klein bottle, we have



With[p = 1/3, r = 1/2, ε = -1/3, (* Klein bottle parameters *)
 f = 3, d = 1, (* projection parameters *)
 k = 3 (* perspective over k-th coordinate *),
 ParametricPlot3D[Function[pt, f Delete[pt, k]/(d - Extract[pt, k])] @
 r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v]),
 r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v]),
 p Cos[u] (1 + ε Sin[v]), p Sin[u] (1 + ε Sin[v]),
 u, 0, 2 π, v, 0, 2 π,
 Mesh -> False, PlotPoints -> 55]]


projection of 4D Klein bottle



In addition to perspective projection, one might want to also apply a preliminary rotation (via e.g. RotationTransform) to the parametric equations before projecting over one of the axes, adding another element of flexibility.






share|improve this answer





























    up vote
    3
    down vote













    There are an infinite number of projections of multidimensional figures on 3D. I will show one variant that is suitable for this case



    F[u_, v_] := x[u, v], y[u, v], z[u, v], t[u, v]
    x[u_, v_, r_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
    y[u_, v_, r_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
    z[u_, v_, p_, [Epsilon]_] := p Cos[u] (1 + [Epsilon] Sin[v])
    t[u_, v_, p_, [Epsilon]_] := p Sin[u] (1 + [Epsilon] Sin[v])

    KleinBottle4D[p_, r_, [Epsilon]_, [Alpha]_] := 
     ParametricPlot3D[x[u, v, r], y[u, v, r], 
     Cos[[Alpha]]*z[u, v, p, [Epsilon]] + 
     Sin[[Alpha]]*t[u, v, p, [Epsilon]], u, 0, 2*Pi, v, 0, 2*Pi,
     ColorFunction -> Hue, PlotRange -> All, Mesh -> None]
    KleinBottle4D[1/3, 1/2, -1/3, Pi/4]


    fig1






    share|improve this answer




















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      2 Answers
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      active

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      2 Answers
      2






      active

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      active

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      up vote
      4
      down vote













      A simple minded possibility is to use a perspective projection (similar to what I did here). Applied to the 4D Klein bottle, we have



      With[p = 1/3, r = 1/2, ε = -1/3, (* Klein bottle parameters *)
       f = 3, d = 1, (* projection parameters *)
       k = 3 (* perspective over k-th coordinate *),
       ParametricPlot3D[Function[pt, f Delete[pt, k]/(d - Extract[pt, k])] @
       r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v]),
       r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v]),
       p Cos[u] (1 + ε Sin[v]), p Sin[u] (1 + ε Sin[v]),
       u, 0, 2 π, v, 0, 2 π,
       Mesh -> False, PlotPoints -> 55]]


      projection of 4D Klein bottle



      In addition to perspective projection, one might want to also apply a preliminary rotation (via e.g. RotationTransform) to the parametric equations before projecting over one of the axes, adding another element of flexibility.






      share|improve this answer


























        up vote
        4
        down vote













        A simple minded possibility is to use a perspective projection (similar to what I did here). Applied to the 4D Klein bottle, we have



        With[p = 1/3, r = 1/2, ε = -1/3, (* Klein bottle parameters *)
         f = 3, d = 1, (* projection parameters *)
         k = 3 (* perspective over k-th coordinate *),
         ParametricPlot3D[Function[pt, f Delete[pt, k]/(d - Extract[pt, k])] @
         r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v]),
         r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v]),
         p Cos[u] (1 + ε Sin[v]), p Sin[u] (1 + ε Sin[v]),
         u, 0, 2 π, v, 0, 2 π,
         Mesh -> False, PlotPoints -> 55]]


        projection of 4D Klein bottle



        In addition to perspective projection, one might want to also apply a preliminary rotation (via e.g. RotationTransform) to the parametric equations before projecting over one of the axes, adding another element of flexibility.






        share|improve this answer
























          up vote
          4
          down vote










          up vote
          4
          down vote









          A simple minded possibility is to use a perspective projection (similar to what I did here). Applied to the 4D Klein bottle, we have



          With[p = 1/3, r = 1/2, ε = -1/3, (* Klein bottle parameters *)
           f = 3, d = 1, (* projection parameters *)
           k = 3 (* perspective over k-th coordinate *),
           ParametricPlot3D[Function[pt, f Delete[pt, k]/(d - Extract[pt, k])] @
           r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v]),
           r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v]),
           p Cos[u] (1 + ε Sin[v]), p Sin[u] (1 + ε Sin[v]),
           u, 0, 2 π, v, 0, 2 π,
           Mesh -> False, PlotPoints -> 55]]


          projection of 4D Klein bottle



          In addition to perspective projection, one might want to also apply a preliminary rotation (via e.g. RotationTransform) to the parametric equations before projecting over one of the axes, adding another element of flexibility.






          share|improve this answer














          A simple minded possibility is to use a perspective projection (similar to what I did here). Applied to the 4D Klein bottle, we have



          With[p = 1/3, r = 1/2, ε = -1/3, (* Klein bottle parameters *)
           f = 3, d = 1, (* projection parameters *)
           k = 3 (* perspective over k-th coordinate *),
           ParametricPlot3D[Function[pt, f Delete[pt, k]/(d - Extract[pt, k])] @
           r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v]),
           r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v]),
           p Cos[u] (1 + ε Sin[v]), p Sin[u] (1 + ε Sin[v]),
           u, 0, 2 π, v, 0, 2 π,
           Mesh -> False, PlotPoints -> 55]]


          projection of 4D Klein bottle



          In addition to perspective projection, one might want to also apply a preliminary rotation (via e.g. RotationTransform) to the parametric equations before projecting over one of the axes, adding another element of flexibility.







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 49 mins ago

























          answered 1 hour ago









          J. M. is somewhat okay.♦

          92.5k10286440




          92.5k10286440




















              up vote
              3
              down vote













              There are an infinite number of projections of multidimensional figures on 3D. I will show one variant that is suitable for this case



              F[u_, v_] := x[u, v], y[u, v], z[u, v], t[u, v]
              x[u_, v_, r_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
              y[u_, v_, r_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
              z[u_, v_, p_, [Epsilon]_] := p Cos[u] (1 + [Epsilon] Sin[v])
              t[u_, v_, p_, [Epsilon]_] := p Sin[u] (1 + [Epsilon] Sin[v])

              KleinBottle4D[p_, r_, [Epsilon]_, [Alpha]_] := 
               ParametricPlot3D[x[u, v, r], y[u, v, r], 
               Cos[[Alpha]]*z[u, v, p, [Epsilon]] + 
               Sin[[Alpha]]*t[u, v, p, [Epsilon]], u, 0, 2*Pi, v, 0, 2*Pi,
               ColorFunction -> Hue, PlotRange -> All, Mesh -> None]
              KleinBottle4D[1/3, 1/2, -1/3, Pi/4]


              fig1






              share|improve this answer
























                up vote
                3
                down vote













                There are an infinite number of projections of multidimensional figures on 3D. I will show one variant that is suitable for this case



                F[u_, v_] := x[u, v], y[u, v], z[u, v], t[u, v]
                x[u_, v_, r_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
                y[u_, v_, r_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
                z[u_, v_, p_, [Epsilon]_] := p Cos[u] (1 + [Epsilon] Sin[v])
                t[u_, v_, p_, [Epsilon]_] := p Sin[u] (1 + [Epsilon] Sin[v])

                KleinBottle4D[p_, r_, [Epsilon]_, [Alpha]_] := 
                 ParametricPlot3D[x[u, v, r], y[u, v, r], 
                 Cos[[Alpha]]*z[u, v, p, [Epsilon]] + 
                 Sin[[Alpha]]*t[u, v, p, [Epsilon]], u, 0, 2*Pi, v, 0, 2*Pi,
                 ColorFunction -> Hue, PlotRange -> All, Mesh -> None]
                KleinBottle4D[1/3, 1/2, -1/3, Pi/4]


                fig1






                share|improve this answer






















                  up vote
                  3
                  down vote










                  up vote
                  3
                  down vote









                  There are an infinite number of projections of multidimensional figures on 3D. I will show one variant that is suitable for this case



                  F[u_, v_] := x[u, v], y[u, v], z[u, v], t[u, v]
                  x[u_, v_, r_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
                  y[u_, v_, r_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
                  z[u_, v_, p_, [Epsilon]_] := p Cos[u] (1 + [Epsilon] Sin[v])
                  t[u_, v_, p_, [Epsilon]_] := p Sin[u] (1 + [Epsilon] Sin[v])

                  KleinBottle4D[p_, r_, [Epsilon]_, [Alpha]_] := 
                   ParametricPlot3D[x[u, v, r], y[u, v, r], 
                   Cos[[Alpha]]*z[u, v, p, [Epsilon]] + 
                   Sin[[Alpha]]*t[u, v, p, [Epsilon]], u, 0, 2*Pi, v, 0, 2*Pi,
                   ColorFunction -> Hue, PlotRange -> All, Mesh -> None]
                  KleinBottle4D[1/3, 1/2, -1/3, Pi/4]


                  fig1






                  share|improve this answer












                  There are an infinite number of projections of multidimensional figures on 3D. I will show one variant that is suitable for this case



                  F[u_, v_] := x[u, v], y[u, v], z[u, v], t[u, v]
                  x[u_, v_, r_] := r (Cos[u/2] Cos[v] - Sin[u/2] Sin[2 v])
                  y[u_, v_, r_] := r (Sin[u/2] Cos[v] + Cos[u/2] Sin[2 v])
                  z[u_, v_, p_, [Epsilon]_] := p Cos[u] (1 + [Epsilon] Sin[v])
                  t[u_, v_, p_, [Epsilon]_] := p Sin[u] (1 + [Epsilon] Sin[v])

                  KleinBottle4D[p_, r_, [Epsilon]_, [Alpha]_] := 
                   ParametricPlot3D[x[u, v, r], y[u, v, r], 
                   Cos[[Alpha]]*z[u, v, p, [Epsilon]] + 
                   Sin[[Alpha]]*t[u, v, p, [Epsilon]], u, 0, 2*Pi, v, 0, 2*Pi,
                   ColorFunction -> Hue, PlotRange -> All, Mesh -> None]
                  KleinBottle4D[1/3, 1/2, -1/3, Pi/4]


                  fig1







                  share|improve this answer












                  share|improve this answer



                  share|improve this answer










                  answered 8 hours ago









                  Alex Trounev

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