Join points or draw lines between them on a torus [duplicate]

Question

First I would have liked to map points and texture from a square map (box) in agent-based modellers to a torus.

StartShot = 
  ArrayPlot[NLGetPatches["covername"], 
    ColorRules ->  {"arable_land" -> Brown, "forests" -> Darker[Green]},
  Frame -> False, 
  DataRange -> {{0, 400}, {0, 400}}, 
  PlotRangePadding -> 0, 
  Method -> {"ShrinkWrap" -> True}]

startingpoints = {{348.488, 132.622}}
agents = 
  ListPlot[startingpoints, 
    PlotStyle -> Directive[PointSize[Medium], White], 
    AspectRatio -> 1,
    Axes -> None, 
    Frame -> False, 
    DataRange -> {{0, 400}, {0, 400}}];
Show[StartShot, agents, ImageSize -> 150]
paths = {{{348.488, 132.622}, {336.333, 63.6857}, {394.365, 24.5422},
          {39.3603, 78.1653}, {109.094, 84.2662}, {170.317, 50.3295},
          {195.403, 115.68}, {263.324, 132.615}, {316.947, 177.61},
          {381.382, 150.259}, {49.8526, 164.812}, {41.3217, 95.3342},
          {11.7384, 158.776}, {65.3616, 113.781}, {5.35985, 77.728},
          {18.7165, 9.01408}, {358.715, 372.961}, {394.767, 312.96},
          {340.367, 268.907}, {313.016, 333.343}, {269.92, 388.503}}};
arrows = {{{313.016, 333.343}, {269.92, 388.503}}}
lineplot = 
  ListLinePlot[paths,
    AspectRatio -> 1, 
    Axes -> None, 
    Frame -> False, 
    DataRange -> {{0, 400}, {0, 400}}, 
    PlotStyle -> White];
FlatTorus = 
  Show[StartShot, lineplot, agents,
    Epilog -> {White, Arrowheads[Small], Arrow /@ arrows}, 
    ImageSize -> 300]

Then I map the 'box' - flat coordinates:

    dataTorus = paths/400*2 π;
    r1 = 1; r2 = 0.3;
    f[{θ_, ϕ_}] := {(r1 + r2*Cos[ϕ])*Cos[θ], (r1 + r2*Cos[ϕ])*Sin[θ], r2*Sin[ϕ]}
    Show[ListPointPlot3D[Evaluate[f] /@ Flatten[dataTorus, 1], 
  PlotStyle -> Directive[PointSize[0.02]]], 
 ParametricPlot3D[
  Evaluate@f[{\[Theta], \[Phi]}], {\[Theta], 0, 2*\[Pi]}, {\[Phi], 0, 
   2*\[Pi]}, Mesh -> None, PlotStyle -> Directive[Texture[FlatTorus]],
   TextureCoordinateFunction -> ({#4, #5} &)], 
 Graphics3D[
  Line[f /@ 
      Table[{Interpolation[#[[All, 1]]][k], 
        Interpolation[#[[All, 2]]][k]}, {k, 1, Length@#, .01}]] &@
   Flatten[dataTorus, 1] (*after george2079*), Boxed -> False], PlotRange -> All, 
 Boxed -> False, Lighting -> "Neutral", Axes -> False]

Question:

I would like to connect my points with the shortest distances on the torus surface (in toroidal space) according to the order.

Problems:

1. The interpolation put curves (yellow line)
2. The long line (red) is not a shortest distance, it happened because the line is drawn in euclidean space between the points. The interpolation is done in the euclidean space, not in the toroidal space. The distances should seems to be equals, like the geodetics on the sphere surface.

Answer 1

You can draw the path through your points like this:

Graphics3D[Line[f /@ Table[{
       Interpolation[#[[All, 1]]][k],
       Interpolation[#[[All, 2]]][k]}, {k, 1, Length@#, .01}]] &@
         Flatten[dataTorus, 1] ]