Geodesics on a torus

I don't know if there's a simple way to find geodesics on a torus, but I can give you a general way to find geodesics on any curved surface.

First, I define the torus:

r = 3;
torus[{u_, v_}] := {Cos[u]*(Sin[v] + r), Sin[u]*(Sin[v] + r), Cos[v]}

My initial attempt was then to use variational methods to derive a formula for geodesics:

Needs["VariationalMethods`"]
eq = EulerEquations[Sqrt[Total[D[torus[{u, v[u]}], u]^2]], v[u], u]; 

And use ParametricNDSolve & FindRoot to find the right parameters that connect the start and end point on the torus:

geodesic[{{u1_, v1_}, {u2_, v2_}}] := Module[{start, g, sol},
  If[u2 < u1, Return[geodesic[{{u2, v2}, {u1, v1}}]]];
  sol = ParametricNDSolve[Flatten[{
      eq, v[0] == v1, v'[0] == a
      }], v, {u, 0, u2 - u1}, {a}];
  start = a /. FindRoot[Evaluate[(v[a][u2 - u1] - v2 /. sol)], {a, 0}];
  g = v[start] /. sol;
  Function[t, {u1 + t*(u2 - u1), g[t*(u2 - u1)]}]
  ]

So given two points, geodesic will return a function that maps a number $0\leq t\leq 1$ to torus coordinates of the right geodesic:

LocatorPane[
 Dynamic[pts],
 Dynamic[ParametricPlot[Evaluate[geodesic[pts][t]], {t, 0, 1}, 
   PlotRange -> {{-π, π}, {-π, π}}, Axes -> True, 
   AspectRatio -> 1/r]]]

Show[
 ParametricPlot3D[
  torus[{u, v}], {u, -π, π}, {v, -π, π}, 
  PlotStyle -> White, ImageSize -> 500],
 ParametricPlot3D[Evaluate[torus[geodesic[pts][t]]], {t, 0, 1}, 
  PlotStyle -> Red]
 ]

Unfortunately, for some points, FindRoot becomes very slow or doesn't even find the right solution. (In that case, geodesic still returns a proper geodesic, it just doesn't end where you want it to end.)

So my second attempt uses unconstrained minimization, i.e. I optimize N "control points" along a path to get the shortest path, then interpolate between the control points:

Clear[geodesicFindMin]
geodesicFindMin[{p1_, p2_}, nPts_: 25] := 
 Module[{approximatePts, optimizeOffset, optimizeOffsets, direction, 
   normal, pathLength, optimalPath, interpolations, len, solution},
  direction = p2 - p1;
  normal = {{0, 1}, {-1, 0}}.direction;

  approximatePts = Join[
    {p1},
    Table[
     p1 + i*direction/(nPts + 1) + optimizeOffset[i]*normal, {i, 
      nPts}],
    {p2}];

  pathLength = Total[Norm /@ Differences[torus /@ approximatePts]];

  {len, solution} = 
   Quiet[FindMinimum[pathLength, 
     Table[{optimizeOffset[i], 0}, {i, nPts}]]];
  optimalPath = approximatePts /. solution;

  interpolations = 
   ListInterpolation[#, {{0, 1}}] & /@ Transpose[optimalPath];

  Function[t, #[t] & /@ interpolations]
  ]

Usage is the same as before, only this version works much smoother:

LocatorPane[
 Dynamic[pts],
 Dynamic[ParametricPlot[Evaluate[geodesicFindMin[pts][t]], {t, 0, 1}, 
   PlotRange -> {{-π, π}, {-2 π, 2 π}}, Axes -> True, 
   AspectRatio -> 2/r]]]

Show[
 ParametricPlot3D[
  torus[{u, v}], {u, -π, π}, {v, -π, π}, 
  PlotStyle -> Directive[White], ImageSize -> 500],
 ParametricPlot3D[Evaluate[torus[geodesicFindMin[pts][t]]], {t, 0, 1},
   PlotStyle -> Red]
 ]