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Re: ParametricPlot3D vs Reduce

• To: mathgroup at smc.vnet.net
• Subject: [mg124076] Re: ParametricPlot3D vs Reduce
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Sun, 8 Jan 2012 04:23:07 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201201071020.FAA19453@smc.vnet.net>

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)};

ParametricPlot3D does not plot the intersection of {fx, fy, fz}. For
the intersection use

Plot3D[1, {b, -10, 10}, {d, -10, 10},
 RegionFunction ->
  Function[{b, d, z}, And @@ Thread[-1 <= rats <= 1]]]

RegionPlot[And @@ Thread[-1 <= rats <= 1],
 {b, -10, 10}, {d, -10, 10}]

As expected, the results are empty. However, applying only two of the conditions

Plot3D[1, {b, -10, 10}, {d, -10, 10},
   RegionFunction ->

    Function[{b, d, z}, And @@ Thread[-1 <= Drop[rats, #] <= 1]]] & /@
 Range[3]

RegionPlot[And @@ Thread[-1 <= Drop[rats, #] <= 1],
   {b, -10, 10}, {d, -10, 10}] & /@ Range[3]


Bob Hanlon

On Sat, Jan 7, 2012 at 5:20 AM, Andrzej Kozlowski <akozlowski at gmail.com> 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?
>
> Andrzej Kozlowski
>
>
>



--
Bob Hanlon

• References:
  • ParametricPlot3D vs Reduce
    • From: Andrzej Kozlowski <akozlowski@gmail.com>