Cycles on a torus

Question

I can draw a torus using a ParametricPlot3D:

torus = ParametricPlot3D[
         {Cos[θ] (1 + .3 Cos[φ]), Sin[θ] (1 + .3 Cos[φ]), .3 Sin[φ]}, 
         {θ, 0, 2 π}, {φ, 0, 2 π}, 
         PlotStyle -> Opacity[0.3]]

And now I want to label its trivial "cycles", as in this picture:

Except without $c$. I've seen ways of overlaying parametric plots on toruses but I'm sure there is an easier way. I just can't seem to find it.

The hardest part is to plot all invisible lines as dashed. I’d gladly see the solution to that as well. — Vsevolod A., Jun 11 '18 at 17:22
Can you plot any of the cycles, independently of the formatting? Have you tried anything? — MarcoB, Jun 11 '18 at 17:35

Michael E2 · Accepted Answer · 2018-06-11T18:41:56.870

tFN = Function[{θ, ϕ},
       {Cos[θ] (1 + 3/10 Cos[ϕ]), Sin[θ] (1 + 3/10 Cos[ϕ]), 3/10 Sin[ϕ]}]

torus = ParametricPlot3D[
  tFN[θ, ϕ], {θ, 0, 2 π}, {ϕ, 0, 2 π},
   PlotStyle -> Opacity[0.3], Mesh -> {{3 Pi/2}, {Pi/2}}]

Mathematica graphics

Show[
 Normal[torus] /. Line :> Arrow@*Reverse,
 Graphics3D[{Text["a", tFN[0, Pi/2], {-2, 0}], 
   Text["b", tFN[3 Pi/2, 0], {2, 0}]}]
 ]

Mathematica graphics

Update: Dashed lines

Extending Silvia's DashedGraphics3D function to handle Arrow, we can get a fixed image (from any preassigned ViewPoint).

Clear[DashedGraphics3D]
DashedGraphics3D::optx = 
  "Invalid options for Graphics3D are omitted: `1`.";
Off[OptionValue::nodef];
Options[DashedGraphics3D] = {ViewAngle -> 0.4, 
   ViewPoint -> {3, -1, 0.5}, ViewVertical -> {0, 0, 1}, 
   ImageSize -> 800};
DashedGraphics3D[basegraph_, effectFunction_: Identity, 
   opts : OptionsPattern[]] /; ! 
   MatchQ[Flatten[{effectFunction}], {(Rule | RuleDelayed)[__] ..}] :=
  Module[{basegraphClean = basegraph /. (Lighting -> _) :> Sequence[],
    exceptopts, fullopts, frontlayer, dashedlayer, borderlayer, 
   face3DPrimitives = {Cuboid, Cone, Cylinder, Sphere, Tube, 
     BSplineSurface}}, 
  exceptopts = FilterRules[{opts}, Except[Options[Graphics3D]]];
  If[exceptopts =!= {}, Message[DashedGraphics3D::optx, exceptopts]];
  fullopts = 
   Join[FilterRules[Options[DashedGraphics3D], Except[#]], #] &@
    FilterRules[{opts}, Options[Graphics3D]];
  frontlayer = 
   Show[basegraphClean /.
       {Line[pts__] :> {Thick, Line[pts]}, 
        Arrow[pts__] :> {Thick, Arrow[pts]}} /.

      h_[pts___] /; 
        MemberQ[face3DPrimitives, h] :> {EdgeForm[{Thick}], h[pts]}, 
     fullopts, Lighting -> {{"Ambient", White}}] // Rasterize;
  dashedlayer = 
   Show[basegraphClean /.
       {Polygon[__] :> {}, 
        Line[pts__] :> {Dashed, Line[pts]}, 
        Arrow[pts__] :> {Dashed, Arrow[pts]}} /.

      h_[pts___] /; MemberQ[face3DPrimitives, h] :> {FaceForm[], 
        EdgeForm[{Dashed}], h[pts]}, fullopts] // Rasterize;
  borderlayer = 
   Show[basegraphClean /. RGBColor[__] :> Black, 
       ViewAngle -> (1 - .001) OptionValue[ViewAngle], 
       Lighting -> {{"Ambient", Black}}, fullopts, Axes -> False, 
       Boxed -> False] // Rasterize // GradientFilter[#, 1] & // 
    ImageAdjust;
  ImageSubtract[frontlayer, dashedlayer] // effectFunction // 
      ImageAdd[frontlayer // ColorNegate, #] & // 
     ImageAdd[#, borderlayer] & // ColorNegate // ImageCrop]

torus = ParametricPlot3D[
   tFN[\[Theta], \[Phi]], {\[Theta], 0, 2 \[Pi]}, {\[Phi], 0, 
    2 \[Pi]}, PlotStyle -> White, Mesh -> {{3 Pi/2}, {Pi/2}}, 
   MeshStyle -> Thick, PlotPoints -> {50, 25}];
annotated = Show[
   Normal[torus] /. Line :> Arrow@*Reverse,
   Graphics3D[{Text[Style["a", Large], tFN[0, Pi/2], {-3, 0}], 
     Text[Style["b", Large], tFN[3 Pi/2, 0], {3, -1}]}],
   SphericalRegion -> True, Boxed -> False, Axes -> False
   ];
DashedGraphics3D[%, ViewPoint -> {1.3, -2.4, 2.}]

Mathematica graphics

Cycles on a torus

1 Answers1