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

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

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}
]

• Can all the $n$ curves be combined into only one, closed(head to tail), space curve? Commented Apr 30, 2018 at 9:03
• @user6043040 see the edit and play with it
– Kuba
Commented Apr 30, 2018 at 9:08
• Thank you very much @Kuba This answers all my questions! The CoordinateTransform is so much powerful! It seems, in order to obtain denser mesh, the least common multiple of m and n has to be as larger as possible. Commented Apr 30, 2018 at 9:21