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Spatial Accuracy in visualizing 3D model sections


Villarceau circles are generated as intersections between a common torus and plane tangent to meridional section circles. Attempted to rotate and cut a torus to visualize such circular section lines in Mathematica.

The graphics show resulting section lines needing a small correction in cutting plane placement to produce a more precisely defined Villarceau circle pair. 

Is there another 3D Graphics directive for improved visualization accuracy? Or do I miss something here?

c = 5; a = 3; (* Programmed: Narasimham GL ,6 Oct 2013, Graphics accuracy *);
r[u_] = c + a Cos[u]; z[u_] = a Sin[u];
Torus = ParametricPlot3D[{r[t] Cos[v], r[t] Sin[v], z[t]}, {t, 0, 
   2 Pi}, {v, 0, 2 Pi}, Mesh -> {30, 40}]
al = ArcSin[a/c]; csal = Cos[al]; snal = Sin[al];
TiltTorus = 
  ParametricPlot3D[{r[t] Cos[v] csal - z[t] snal, r[t] Sin[v], 
    r[t] Cos[v] snal + z[t] csal}, {t, 0, 2 Pi}, {v, 0, 2 Pi}, 
   Mesh -> {30, 40}];
TiltTorusNeg = 
 ParametricPlot3D[{r[t] Cos[v] csal - z[t] snal, 
   r[t] Sin[v], -(r[t] Cos[v] snal + z[t] csal)}, {t, 0, 2 Pi}, {v, 0,
    2 Pi}, Mesh -> {30, 40}]
error = -.122
Show[{TiltTorusNeg, TiltTorus}, 
 PlotRange -> {{-8, 8}, {-8, 8}, {error, -6}}]
Show[{TiltTorusNeg, TiltTorus}, 
 PlotRange -> {{-8, 8}, {-8, 8}, {0, -6}}, 
 PlotLabel -> "Graphics InAccuracy"]
Show[TiltTorus, PlotRange -> {{-8, 8}, {-8, 8}, {0, -.06}}]
Section1 = 
 Show[TiltTorus, PlotRange -> {{-8, 8}, {-8, 8}, {error, -6}}]
Section2 = 
 Show[TiltTorus, PlotRange -> {{-8, 8}, {-8, 8}, {0, -6}}, 
  PlotLabel -> "Graphics InAccuracy"]

Regards
Narasimham



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