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>