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Re: ParametricPlot3D vs Reduce

  • To: mathgroup at smc.vnet.net
  • Subject: [mg124062] Re: ParametricPlot3D vs Reduce
  • From: Andrzej Kozlowski <akozlowski at gmail.com>
  • Date: Sat, 7 Jan 2012 05:23:58 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <09D35683-89B2-4778-8552-0B7F44CF5A82@gmail.com>

On 6 Jan 2012, at 23:09, 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?
>
> Andrzej Kozlowski
>
>

On the other hand this works fine:

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"}]

The surface becomes visible in a somewhat larger cube:

ParametricPlot3D[rats, {b, -10, 10}, {d, -10, 10},
 RegionFunction ->
  Function[{x, y, z, u,
    v}, -3 <= x <= 3 && -3 <= y <= 3 && -3 <= z <= 3],
 AxesLabel -> {"a", "b", "c"}]

So the polygons in the earlier picture (with PlotRange restricted to the 
unit cube) must be due to some artifact of the way ParametricPlot3D 
displays a surface. Possibly a bug?

Andrzej Kozlowski



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