Re: Ragged region boundary

• To: mathgroup at smc.vnet.net
• Subject: [mg123256] Re: Ragged region boundary
• From: Patrick Scheibe <pscheibe at trm.uni-leipzig.de>
• Date: Tue, 29 Nov 2011 07:05:24 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Hi,

first you could compile your torus-function and compare the runtimes

torus[c_, a_, u_, v_] := {(c + a Cos[v]) Cos[u], (c + a Cos[v]) Sin[u],
a Sin[v]};
torusCompiled =
Compile[{{c, _Real}, {a, _Real}, {u, _Real}, {v, _Real}},
{(c + a Cos[v]) Cos[u], (c + a Cos[v]) Sin[u], a Sin[v]},
CompilationTarget -> "C"];

In[38]:= First[
AbsoluteTiming[Do[#[1.3, 0.3, 2.3, 4.5], {1000000}]]] & /@ {torus,
torusCompiled}

Out[38]= {5.190318, 0.863473}

and then you could set MaxRecursion to a higher value. Since then the
runtime drops further, you could use ControlActive to use a very fast
version, when you use the sliders. See the code at the end.

Your "equator" is not the only unwanted boundary, but you don't see the
other one. These are the lines where your PlotRange ends. The doc to
BoundaryStyle says

"For 3D graphics, it is also used at the boundary of regions defined by
PlotRange."

So these lines are at 0 and 2Pi for both of your variables. I have no
idea how to turn them off.

Cheers
Patrick

Manipulate[
ParametricPlot3D[
torusCompiled[c, a, u, v], {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]},
MeshFunctions -> {{x, y, z, \[Theta], \[Phi]} \[Function]
z + Tan[tilt] x},
RegionFunction -> ({x, y, z, \[Theta], \[Phi]} \[Function]
z <= -Tan[tilt] x),
Mesh -> mesh,
MaxRecursion -> ControlActive[0, 5],
MeshStyle -> {Lighter[Yellow], Tube[tubeR]},
BoundaryStyle -> {Lighter[Red], Tube[tubeR]},
PlotStyle -> {Orange, Opacity[opac]},
PlotPoints -> ControlActive[10, plotPts],
Axes -> True,
AxesLabel -> {"x", "y", "z"},
PlotRange -> {{-4, 4}, {-4, 4}, {-3, 3}},
BoxRatios -> {8, 8, 6},
ViewPoint ->
{
viewR Cos[view\[Theta]] Sin[view\[Phi]],
viewR Sin[view\[Theta]] Sin[view\[Phi]],
viewR Cos[view\[Phi]]
},
PerformanceGoal -> "Quality"
],

{{c, 3}, 0, 4},
{{a, 1}, 0, 3},
{{sphereR, 0.1}, 0, 0.5},
{{tubeR, 0.05}, 0, 0.5},
{{opac, 0.5}, 0, 1},
{{mesh, 0}, 0, 16, 1},
{{tilt, N[ArcSin[a/c]]}, 0, \[Pi]/2, \[Pi]/36},
{{plotPts, 50}, 0, 200, 5},
{{viewR, 100}, 0, 100, 5},
{{view\[Theta], \[Pi]/2}, 0, 2 \[Pi], \[Pi]/36},
{{view\[Phi], \[Pi]/2}, 0, \[Pi], \[Pi]/36}
]

On Mon, 2011-11-28 at 05:54 -0500, Chris Young wrote:
> Would anyone have any tips on how to get a better region boundary here?
> Maybe via CountourPlot3D?
>
> Actually, if I use PlotPoints -> 200, it's OK, but it's extremely slow
> to plot then, of course.
>
> Also, I don't want the horizontal "boundary" line at the equator. Can't
> see why this is considered a boundary line anyway.
>
> http://home.comcast.net/~cy56/RegionBoundary.nb
>
> http://home.comcast.net/~cy56/RegionBoundary.png
>
>
> Manipulate[
>  ParametricPlot3D[
>   torus[c, a, u, v], {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]},
>   MeshFunctions -> {{x, y, z, \[Theta], \[Phi]} \[Function]
>      z + Tan[tilt] x},
>   RegionFunction -> ({x, y, z, \[Theta], \[Phi]} \[Function]
>      z <= -Tan[tilt] x),
>   Mesh -> mesh,
>   MeshStyle -> {Lighter[Yellow], Tube[tubeR]},
>   BoundaryStyle -> {Lighter[Red], Tube[tubeR]},
>   PlotStyle -> {Orange, Opacity[opac]},
>   PlotPoints -> plotPts,
>   Axes -> True,
>   AxesLabel -> {"x", "y", "z"},
>   PlotRange -> {{-4, 4}, {-4, 4}, {-3, 3}},
>   BoxRatios -> {8, 8, 6},
>   ViewPoint ->
>    {
>     viewR Cos[view\[Theta]] Sin[view\[Phi]],
>     viewR Sin[view\[Theta]] Sin[view\[Phi]],
>     viewR Cos[view\[Phi]]
>     },
>   PerformanceGoal -> "Quality"
>   ],
>
>  {{c, 3}, 0, 4},
>  {{a, 1}, 0, 3},
>  {{sphereR, 0.1}, 0, 0.5},
>  {{tubeR, 0.05}, 0, 0.5},
>  {{opac, 0.5}, 0, 1},
>  {{mesh, 1}, 0, 16, 1},
>  {{tilt, N[ArcSin[a/c]]}, 0, \[Pi]/2, \[Pi]/36},
>  {{plotPts, 50}, 0, 100, 5},
>  {{viewR, 100}, 0, 100, 5},
>  {{view\[Theta], \[Pi]/2}, 0, 2 \[Pi], \[Pi]/36},
>  {{view\[Phi], \[Pi]/2}, 0, \[Pi], \[Pi]/36}
>  ]
>
>

```

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