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Re: Plotting a hyperbolic paraboloid (saddle)

• To: mathgroup at smc.vnet.net
• Subject: [mg121404] Re: Plotting a hyperbolic paraboloid (saddle)
• From: "Christopher O. Young" <cy56 at comcast.net>
• Date: Wed, 14 Sep 2011 05:13:13 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <f74tck$95v$1@smc.vnet.net>

On 7/12/07 5:48 AM, in article f74tck$95v$1 at smc.vnet.net, "William S."
<wschacht47 at att.net> wrote:

> Does anyone know how to plot a hyperbolic paraboloid?


I thought it was time to get to understand 3D plotting basics in
Mathematica, so I tried three different ways of looking at the hyperbolic
paraboloid.

There's a picture at http://home.comcast.net/~cy56/SaddlePlots.png and a
Mathematica notebook at http://home.comcast.net/~cy56/SaddlePlots.nb

I think it's way too much of a struggle to get the axes to come out with the
same scales. I think this is something most students (and the rest of us)
would want to do most often. Couldn't there be a single option (maybe
"SameScaleAxes" or something similar?) to do this?

The contour plot version seems to be a little "wild" as I try to rotate it.
The size jumps around a lot.

I used "ColorFunctionScaling -> False" because I wanted to have custom
coloring running from red for negative values to green for positive values.

saddleContourPlot =
 ContourPlot3D[
   x * y == z,
   
   {x, -2.5, 2.5},
   {y, -2.5, 2.5},
   {z, -4, 4},
   
   PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}, {-4, 4}},
   AspectRatio -> 8/5,
   
   PlotPoints -> 50,
   Mesh -> 7,
   MeshFunctions -> {#3 &},
   ContourStyle -> Opacity[0.5],
   ColorFunctionScaling -> False,
   ColorFunction -> (Hue[0.35 (#3 + 4)/8 ] &)
   ] /. Line[pts_, opts___] :> {Gray, Tube[pts, 0.02, opts]}


The last line above just makes the contour lines into tubes. I got it from
the Help for Tubes. I wish there were a simple way to just have the contour
lines show up as tubes, maybe by having a "TubeRadius" option.

The plot below shows how the saddle surface in the form z = x * y gives us a
diagram of a multiplication table, with columns above each x * y point to
show us the value of the product.
  
saddleStepPlot =
  DiscretePlot3D[
   x * y,
   
   {x, -2, 2},
   {y, -2, 2},
   
   PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}, {-4, 4}},
   AspectRatio -> 2,
   
   (* SphericalRegion->True, *)
   
   ExtentSize -> Full,
   AxesLabel -> {"x", "y", "z"},
   PlotStyle -> Opacity[0.5],
   ColorFunctionScaling -> False,
   ColorFunction -> (Hue[0.35 (#3 + 4)/8 ] &)
   ];


This plot looks the same as the contour plot version. It sames to be more
stable when I try to rotate it.

saddleParamPlot = 
 ParametricPlot3D[
   {u, v, u*v},
   {u, -2.5, 2.5}, {v, -2.5, 2.5},
   
   RegionFunction ->
    Function[{x, y,
      z}, -2.5 <= x < 2.5 \[And] -2.5 <= y < 2.5 \[And] -4 <= z < 4],
   
   MeshFunctions -> {#3 &},
   Mesh -> 7,
   
   AxesLabel -> {"x", "y", "z"},
   
   PlotStyle -> Opacity[0.5],
   
   ColorFunctionScaling -> False,
   ColorFunction -> (Hue[0.35 (#3 + 4)/8 ] &),
   
   SphericalRegion -> True
   ] /. Line[pts_, opts___] :> {Gray, Tube[pts, 0.03, opts]}

Showing all the plots side by side:

GraphicsGrid[
 {
  {
   saddleContourPlot,
   saddleStepPlot,
   saddleParamPlot
   }
  }
 ]


-- Chris Young
cy56 at comcast.net
IntuMath.org