Combining two parameters in animation

Question

I need to make the following animations on the torus :

t=ParametricPlot3D[{ (3 + Cos[v]) Sin[u], (3+ Cos[v]) Cos[u], Sin[v]-12}, {u, 0, 2 Pi}, {v, 0, 2 Pi},PlotStyle -> Opacity[.4],MeshStyle->None]

t1=ParametricPlot3D[{Cos[Sqrt[2] t](3 + Cos[t]), Sin[Sqrt[2] t] ( 3+Cos[t]),Sin[t]-12}, {t, 0, 40},PlotStyle -> {Blue,Opacity[7]}]

t2=ParametricPlot3D[{Cos[Sqrt[3] t](3 + Cos[t]), Sin[Sqrt[3] t] ( 3+Cos[t]),Sin[t]-12}, {t, 0, 30},PlotStyle -> {Red,Opacity[7]}]

First, the blue line "t1" wrapping the torus "t". Then, the red line "t2" wrapping the torus "t".

Answer 1

t1t2[tmax_] := ParametricPlot3D[{{Cos[Sqrt[2]  40 t/2/Pi] (3 + Cos[40 t/2/Pi]), 
       Sin[Sqrt[2] 40 t/2/Pi] (3 + Cos[t]), 
       Sin[40 t/2/Pi] - 12}, 
      {Cos[Sqrt[3]  30 t/2/Pi] (3 + Cos[30 t/2/Pi]),
       Sin[Sqrt[3] 30 t/2/Pi] (3 + Cos[30 t/2/Pi]), 
       Sin[30 t/2/Pi] - 12}}, 
  {t, 0, tmax}, 
  PlotStyle -> {Blue, Red}]

Manipulate[Show[t, t1t2[tmax], Boxed -> False, Axes -> False], 
 {tmax, Pi/6, 2 Pi}]

If you like replace Manipulate with Animate.