Spatial Accuracy in visualizing 3D model sections
- To: mathgroup at smc.vnet.net
- Subject: [mg131799] Spatial Accuracy in visualizing 3D model sections
- From: Narasimham <mathma18 at gmail.com>
- Date: Mon, 7 Oct 2013 08:24:41 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- Delivered-to: l-mathgroup@wolfram.com
- Delivered-to: mathgroup-outx@smc.vnet.net
- Delivered-to: mathgroup-newsendx@smc.vnet.net
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