# Geodesics on a torus

Is it possible to draw geodesics between the points in a path on a torus - toroidal surface?

geodesics: generalization of the notion of a "straight line" to "curved spaces"

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}}}; The plot has some problem because periodic boundary conditions (PBCs).

• Possible duplicate: Morphing a “sheet of paper” into a torus – shrx Jan 8 '16 at 11:55
• – shrx Jan 8 '16 at 11:56
• That are not about geodesics. Please consider the picture description as well to see the problem in full scale. – pnz Jan 8 '16 at 12:23
• Related Q&A about geodesics on a torus: 115435 – shrx May 19 '16 at 10:55

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 == v1, v' == 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]
] • Thank you! It seems good, I will try to implement it. – pnz Jan 8 '16 at 16:51
• Plotted geodesics on circular tori numerically before using constant meridional curvature condition and the Clairaut.s Law. Depending on ratio of major/minor radii of torus and starting angle to meridian all trajectories can be obtained. The trajectories are also available in closed form using elliptic integrals. – Narasimham Jan 9 '16 at 19:00

With code from here, one can plot geodesics on general discretized surfaces.

R = 2;
r = 1;
M = RegionBoundary@BoundaryDiscretizeRegion[
ImplicitRegion[
(R - Sqrt[x^2 + y^2])^2 + z^2 - r^2 <= 0,
{{x, -4, 4}, {y, -4, 4}, {z, -4, 4}}
],
MaxCellMeasure -> 0.01
];
data = GeodesicData[M];

SeedRandom;
p0 = RegionNearest[M, RandomPoint[M]];
u0 = RandomReal[{10, 1000}] RandomPoint[Sphere[]];
result = ShootGeodesic[M, p0, u0, "GeodesicData" -> data];
Show[M, Graphics3D[{Specularity[White, 30], Sphere[p0, 0.1], Gray,  Tube[result[["Trajectory"]], 0.01]}]]
` • After three years revisioning my question, I think none of the answers is exactly what I was looking for. I wanted to plot a random walk created in 2D with periodic conditions ('2D torus') to a 3D torus. This means wrapping or transform the 2D domain values to a torus and calculate geodesics between all the points of the list. Here you can find some additional detail to understand the question well: library.wolfram.com/infocenter/MathSource/9281 – pnz May 22 at 11:41