Revisions to How to draw Fractal images of iteration functions on the Riemann sphere?

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edited Apr 13, 2017 at 12:55

1

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the north pole corresponds to the point at infinity) and using a technique similar to the one in this answer this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the north pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the north pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

Rollback to Revision 1

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edited Nov 22, 2012 at 13:50

J. M.'s missing motivation ♦

125.5k
11
407
581

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the southnorth pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the south pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the north pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

typo

Source Link

edited Nov 22, 2012 at 13:41

J. M.'s missing motivation ♦

125.5k
11
407
581

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the northsouth pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the north pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

Here is my modest attempt, based on the formulae for stereographic projection in this Wikipedia entry (where the south pole corresponds to the point at infinity) and using a technique similar to the one in this answer:

newtonRaphson = Compile[{{n, _Integer}, {c, _Complex}},
                        Arg[FixedPoint[(# - (#^n - 1)/(n #^(n - 1))) &, c, 30]]]

tex = Image[DensityPlot[
                        newtonRaphson[3, Cot[ϕ/2] Exp[I θ]], {θ, -π, π}, {ϕ, 0, π}, 
                        AspectRatio -> Automatic, 
                        ColorFunction -> (Which[# < .3, Red, # > .7, Yellow, True, Blue] &), 
                        Frame -> False, ImagePadding -> None, PlotPoints -> 400, 
                        PlotRange -> All, PlotRangePadding -> None],
            ImageResolution -> 256];

(* yes, I know that I could have used SphericalPlot3D[]... *)
ParametricPlot3D[{Sin[ϕ] Cos[θ], Sin[ϕ] Sin[θ], Cos[ϕ]}, {θ, -π, π}, {ϕ, 0, π}, 
                 Axes -> None, Boxed -> False, Lighting -> "Neutral", Mesh -> None,
                 PlotStyle -> Texture[tex], TextureCoordinateFunction -> ({#4, #5} &)]

$stereographic projection of Newton-Raphson fractal$

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created Nov 22, 2012 at 13:29

J. M.'s missing motivation ♦

125.5k
11
407
581

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