Re: Plotter for complex polynomials (complex coefficients)

• To: mathgroup at smc.vnet.net
• Subject: [mg124896] Re: Plotter for complex polynomials (complex coefficients)
• From: Chris Young <cy56 at comcast.net>
• Date: Sat, 11 Feb 2012 06:36:33 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jgauk3\$eu3\$1@smc.vnet.net> <jgdmtm\$qo5\$1@smc.vnet.net>

```On 2012-02-02 10:01:26 +0000, DrMajorBob said:

> This seems a little faster:
>
> Manipulate[
>   Module[{f, z},(*Convert 2D point to complex point*)
>    f[z_] = a.{1, I} z^3 + b.{1, I} z^2 + c.{1, I} z + d.{1, I};
>    Plot3D[Abs@f[x + y I], {x, -6, 6}, {y, -6, 6}, PlotPoints -> 100,
>     MaxRecursion -> 2, Mesh -> 11,
>     MeshStyle -> Directive[Gray, AbsoluteThickness[0.01]],
>     MeshFunctions -> ({x,
>         y} \[Function] (\[Pi] - Abs@Arg[f[x + I y]])/\[Pi]),
>     ColorFunctionScaling -> False,
>     ColorFunction -> ({x, y} \[Function]
>        Hue[0.425 \[LeftFloor]12 (\[Pi] -
>                Abs@Arg@f[x + I y])/\[Pi]\[RightFloor]/12, sat, bri]),
>     PlotStyle -> Opacity[opac],
>     AxesLabel -> {"x", "i y",
>       "|f(x + iy)|"}]],(*Item["The complex coefficients"],*){a, {-2, \
> -2}, {2, 2}}, {b, {-2, -2}, {2, 2}}, {c, {-2, -2}, {2,
>     2}}, {d, {-2, -2}, {2, 2}}, {{opac, 0.75, "Opacity"}, 0,
>    1}, {{sat, 0.75, "Saturation"}, 0, 1}, {{bri, 1, "Brightness"}, 0,
>    1}, ControlPlacement -> {Left, Left, Left, Left, Bottom, Bottom,
>     Bottom}]
>
> Bobby

Thanks, I incorporated this into the latest plotter.

http://home.comcast.net/~cy56/Mma/ComplexCoeffPlotter.nb

http://home.comcast.net/~cy56/Mma/ComplexCoeffPlotterPic.png

Took me a long time to figure out that the two options

PlotRange -> {{-2, 2}, {-2, 2}, {6, 0}},
PlotRangePadding -> None

have to be in the same "option set" in order to really cut out the
padding, so that the grid will go right through the dots.

Chris
cy56 at comcast.net

Manipulate[
Module[
{\[ScriptCapitalC], \[ScriptCapitalH],
f, z,
arg},

\[ScriptCapitalC] =  ( \[ScriptCapitalP] \[Function] \
\[ScriptCapitalP] . {1, I});                 (*
Convert 2D vector to complex point *)
\[ScriptCapitalH] = ({\
\[ScriptCapitalP], h} \[Function] Append[\[ScriptCapitalP], h]);   (*
Convert 2D point \[ScriptCapitalP] to 3D point with height h *)

(* The complex polynomial, with complex coefficients: *)

f[z_] = (\[ScriptCapitalC] @ a) z^3 + (\[ScriptCapitalC] @
b) z^2 + (\[ScriptCapitalC] @ c) z + (\[ScriptCapitalC] @ d);

(* The "complex argument" (i.e., angle),
modified to run from 0 at -\[Pi] to 0.4 at 0 and back again,
in 12 steps: *)

arg[z_] = 1/12 (0.4) Round[12  (\[Pi] - Abs[ Arg[z]])/\[Pi]];

Show[
Graphics3D[
{
Red,        Sphere[\[ScriptCapitalH][a, 0], dotR],
Yellow, Sphere[\[ScriptCapitalH][b, 0], dotR],
Green,    Sphere[\[ScriptCapitalH][c, 0], dotR],
Blue,      Sphere[\[ScriptCapitalH][d, 0], dotR]
}
],

Plot3D[
Abs[f[x + y I]], {x, -3, 3}, {y, -3, 3},

PlotPoints -> plotPts,
MaxRecursion -> maxRecurs,
Mesh -> {12, 5},
MeshStyle -> Directive[Gray, AbsoluteThickness[0.01]],

ColorFunctionScaling -> False,

MeshFunctions -> {
({x, y} \[Function] arg[f[x + y I]]),
({x, y, z} \[Function] z)
},
ColorFunction -> ({x, y} \[Function] Hue[arg[f[x + y I]], sat, bri]),
PlotStyle -> Opacity[opac],
ClippingStyle -> None
],

Lighting -> "Neutral",
Axes -> True,
PlotRange -> {{-2, 2}, {-2, 2}, {6, 0}},
PlotRangePadding -> None,
BoxRatios -> {4, 4, 6},
AxesLabel -> {"x", "y \[ImaginaryI]", "|f(x + iy)|"},
FaceGrids -> {{0, 0, 1}, {0, 0, -1}}
]
],

{{a, {1, 0}}, {-2, -2}, {2, 2}, 0.5},
{{b, {0, 1}}, {-2, -2}, {2, 2}, 0.5},
{{c, {-1, 0}}, {-2, -2}, {2, 2}, 0.5},
{{d, {0, -1}}, {-2, -2}, {2, 2}, 0.5},

{{opac, 0.95, "Opacity"}, 0, 1, 0.125},
{{sat, 0.75, "Saturation"}, 0, 1, 0.125},
{{bri, 1, "Brightness"}, 0, 1, 0.125},
{{plotPts, 100}, 10, 200, 10},
{{maxRecurs, 2}, 1, 4, 1},

{{dotR, 0.05}, 0.01, 0.2, 0.01},

ControlPlacement -> {Left, Left, Left, Left,
Bottom, Bottom, Bottom, Bottom, Bottom, Bottom}
]

```

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