Plot on the surface of a Torus

Question

I'm trying to plot a system with a variable (or field) $\theta(x,t)\in[-\pi,\pi]$ which is an angle at every position $x$. The positional argument $x$ itself is a periodic coordinate $x\in[-\pi,\pi]$. I'm trying to show some topological properties of this field, and I felt that a natural way to do so would be to plot the curve $\theta$ versus $x$ for $\theta = \theta(x)$ at a fixed time $t$ on the surface of a torus ($S^1\times S^1$). This would clearly represent the periodic nature of $\theta$ and $x$. Is there a neat way to do this on Mathematica?

For example, my idea was to use the blue coordinate (refer image) as the position variable $x$ and the red coordinate at each $x$ to represent $\theta(x)$. In this way both $x$ and $\theta$ are periodic.

Edit: $t$ is time. I just put it there for completeness. I need to plot the variable at different time slices.

Answer 1

Ignoring $t$, which might seem irrelevant after some of the comments, one can plot $\theta = \theta(x)$ for $(x,\theta)\in S^1\times S^1$ with ParametricPlot3D. One issue is that some aspects of plotting functions do not automatically deal very well with discontinuities, such as MeshFunctions. To make x periodic over $[\pi, \pi]$, one might use Mod[#, 2 Pi, -Pi] &, but it creates discontinuities that cause spurious mesh lines along the discontinuities. Instead, consider the homeomorphism $S^1 \cong{\Bbb R}/(2\pi{\Bbb Z})$. If $x,\theta \in S^1$, then we can use the subgroup $2\pi{\Bbb Z}$ for the Mesh specification. However, a Mesh specification must be finite, so we need to know the bounds on $f(x)$ or otherwise specify a sufficiently broad range for Mesh.

With[{f = Function[x, 2 x]}, 
 ParametricPlot3D[{(2 + Cos[θ]) Cos[x], (2 + Cos[θ]) Sin[x], Sin[θ]},
  {x, -Pi, Pi}, {θ, -Pi, Pi},
  MeshFunctions -> {Function[{x0, y0, z0, x, θ}, θ - f[x]], #4 &, #5 &},
  Mesh -> {2 Pi*Range[-10, 10], {0}, {0}}, 
  MeshStyle -> {Directive[Thick, ColorData[97][1]], Thin, Thin}, 
  BoundaryStyle -> None]]

Answer 2

What you probably need is: https://reference.wolfram.com/language/ref/SliceDensityPlot3D.html

and the equation(s) of a Torus is(are) given by: http://mathworld.wolfram.com/Torus.html

Assuming that you want to work with (u as x) and (v as t), I guess you should try something like this:

\[Theta][u_, v_] := u + v;

c = 1; a = 4;
x = (c + a*Cos[v]) Cos[u];
y = (c + a*Cos[v]) Sin[u];
z = a*Sin[v];

SliceDensityPlot3D[\[Theta][u, 
  v], {(c - Sqrt[x^2 + y^2])^2 + z^2 == a^2}, {u, 0, 2 \[Pi]}, {v, 0, 
  2 \[Pi]}]