How to create such 3D curve embeded in a certain hyperboloid

Question

The following hyperboloid surface is created by:

ContourPlot3D[
 x^2 + y^2 - z^2 == 1, {x, -3, 3}, {y, -3, 3}, {z, -3/2, 3/2}, 
 RegionFunction -> Function[{x, y, z}, x^2 + y^2 <= 4], 
 PlotTheme -> "Classic", Boxed -> False, Axes -> False, 
 MeshFunctions -> {(Cos@Sin@#1)^2 &, (Sin@Sin@#2/2)^2 &}]

How can I obtain a closed, sine/cosine like space curve embedded in the surface? Below the curve model is used for the purpose of depiction

Answer 1

ParametricPlot3D[
 Evaluate@CoordinateTransform["Cylindrical" -> "Cartesian", Table[
    {Sqrt[1 + Sin[ th + d ]^2], th, Sin[ th + d ]}, {d, 0, 2 Pi, 
     2 Pi/16}
    ]],
 {th, 0, 2 Pi}
 ]

Manipulate[
 Graphics3D[
  GeometricTransformation[
   First@ParametricPlot3D[
     Evaluate@
      CoordinateTransform[
       "Cylindrical" -> "Cartesian", {Sqrt[1 + Sin[ m th ]^2], th, 
        Sin[m th ]}],
     {th, 0, 2 Pi}
     ],
   Dynamic@Table[RotationTransform[t, {0, 0, 1}], {t, 0, 2 Pi, 2 Pi/n}]
   ]
  ],
 {{n, 10}, 1, 25,1},
 {{m, 1}, 1, 25,1}
 ]

Can all the n curves be combined into only one, closed(head to tail), space curve? – user6043040

Manipulate[
 ParametricPlot3D[
  Evaluate@
   CoordinateTransform[
    "Cylindrical" -> "Cartesian", {Sqrt[1 + Sin[ m /n th ]^2], th, 
     Sin[m /n th ]}],
  {th, 0, 2 n Pi}
  ],
 {{n, 1}, 1, 25, 1},
 {{m, 1}, 1, 25, 1}
 ]