Re: ParametricPlot3D vs Reduce

• To: mathgroup at smc.vnet.net
• Subject: [mg124093] Re: ParametricPlot3D vs Reduce
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 8 Jan 2012 04:29:01 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <je96dj\$j22\$1@smc.vnet.net> <4F084325.1020903@gmail.com>

```Thanks a lot. As I mentioned in my second post on this subject, using RegionFunction also seems to deal with this problem:

ParametricPlot3D[rats, {b, -10, 10}, {d, -10, 10},
RegionFunction ->
Function[{x, y, z, u,
v}, -1 <= x <= 1 && -1 <= y <= 1 && -1 <= z <= 1],
AxesLabel -> {"a", "b", "c"}]

Andrzej

On 7 Jan 2012, at 14:05, Szabolcs Horv=E1t wrote:

> On 2012.01.07. 11:22, Andrzej Kozlowski wrote:
>> I just came across something somewhat baffling, though it could be the
>> result of an imperfect understanding of how 3D graphic functions work.
>> Consider the following three rational functions of two variables, which
>> we will think of as parameters of a point on a surface in 3D.
>>
>> rats = {(-b - 2*d - b^3*d^2)/(b*d), (2*b + d + b^4*d +
>>       2*b^3*d^2)/(b^2*d), (-1 - 2*b^3*d - b^2*d^2)/(b^2*d)};
>>
>> Now, note that:
>>
>> Reduce[Thread[-1<= rats<= 1], {b, d}]
>>
>> False
>>
>> in other words, there are no values of the parameters b and d for which
>> the point lies in the unit cube. However:
>>
>> ParametricPlot3D[rats, {b, -10, 10}, {d, -10, 10},
>>  PlotRange ->  {{-1, 1}, {-1, 1}, {-1, 1}},
>>  AxesLabel ->  {"a", "b", "c"}]
>>
>> There appear to be several polygons inside the unit cube that should not
>> be there?
>>
>
> Dear Andrzej,
>
> It appears that rats is discontinuous around 0:
>
> Plot[rats /. b -> 1.2 // Evaluate, {d, -1, 1},
> PlotStyle -> {{Red}, {Blue}, {Green}}, Axes -> False, Frame -> True]
>
> It seems that both Plot and ParametricPlot3D are not able to detect this discontinuity.  What you see in the unit box is the same thing as the vertical line at 0 in my Plot example above.
>
> We can fix this by specifying the Exclusions option manually:
>
> Plot[rats /. b -> 1.2 // Evaluate, {d, -1, 1},
> PlotStyle -> {{Red}, {Blue}, {Green}}, Axes -> False, Frame -> True,
> Exclusions -> {0}]
>
> For ParemetricPlot3D, this is done as follows:
>
> unitBox = {{-1, 1}, {-1, 1}, {-1, 1}}; (* avoid typing *)
>
> ParametricPlot3D[rats, {b, -1, 1}, {d, -1, 1},
> PlotRange -> 10 unitBox, MaxRecursion -> 2, PlotPoints -> 40,
> Exclusions -> {b == 0, d == 0}]
>
> If you change the plot range to 1 unitBox (instead of 10), you get an empty plot.
>
> Note that I needed to limit MaxRecursion and PlotPoints manually, otherwise my machine would run out of memory and freeze due to disk swapping...
>
> Alternatively we can chop up the parameter range into four pieces by hand, like this:
>
> With[{pr = 10 unitBox},
> Show[
>  ParametricPlot3D[rats, {b, 0, 1}, {d, 0, 1}, PlotRange -> pr],
>  ParametricPlot3D[rats, {b, -1, 0}, {d, 0, 1}, PlotRange -> pr],
>  ParametricPlot3D[rats, {b, 0, 1}, {d, -1, 0}, PlotRange -> pr],
>  ParametricPlot3D[rats, {b, -1, 0}, {d, -1, 0}, PlotRange -> pr]
>  ]
> ]
>
> --
> Szabolcs Horv=E1t
> Mma QA site proposal: http://area51.stackexchange.com/proposals/37304

```

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