Mandelbrot set: Difference between revisions

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gif(smoothing, "Mandelbrot_smoothing.gif", fps=2)</syntaxhighlight>

===~~Normalized~~ ~~Counting~~, ~~Distance~~ ~~Estimation~~, Mercator Maps and Perturbation Theory ===

===Distance Estimation, Normal Maps, Mercator Maps and Perturbation Theory ===

This is a translation of the corresponding Python section. The Mandelbrot set is represented by distance estimation and normal maps using complex matrices (cf. Arnaud Chéritat: [https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/Mandelbrot_set#Boundary_detection_methods_via_distance_estimators ''Boundary detection methods via distance estimators''] and [https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/Mandelbrot_set#Normal_map_effect ''Normal map effect'']).

This is a translation of the corresponding Python section. The ''e^(-|z|)-smoothing'', ''normalized iteration count'' and ''exterior distance estimation'' algorithms are used.

<syntaxhighlight lang="julia">using Plots

gr(aspect_ratio=:equal, axis=true, ticks=true, legend=false, dpi=200)

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d, h = 800, 500 # pixel density (= image width) and image height

n, r = 200, 500 # number of iterations and escape radius (r > 2)

height, angle = 1.5, 45 # height factor of the incoming light and direction of this light

v = exp(angle / 180 * pi * im) # unit 2D vector in this direction

x = range(0, 2, length=d+1)

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C = (2.0 + 1.0im) .* (A' .+ B .* im) .- 0.5

Z, dZ = zero(C), zero(C)

Z, dZ, ddZ, U = zero(C), zero(C), zero(C), zero(C)

S, T~~, D~~ = ~~zeros(size(C)),~~ zeros(size(C)), zeros(size(C))

D, T = zeros(size(C)), zeros(size(C))

iteration(Z, dZ, ddZ, C) = Z .^ 2 .+ C, 2 .* Z .* dZ .+ 1, 2 .* (dZ .^ 2 .+ Z .* ddZ)

for k in 1:n

M = ~~abs~~.(Z) .< r

M = abs2.(Z) .< abs2(r)

S[M], T[M] ~~= S~~[M] .+ ~~exp.(.- abs.~~(Z[M])), T[M] .+ 1

Z[M], dZ[M], ddZ[M] = iteration(Z[M], dZ[M], ddZ[M], C[M])

Z[M], dZ[M] = Z[M] .^ 2 .+ C[M], 2 .* Z[M] .* dZ[M] .+ 1

end

N = abs.(Z) .> 2 # exterior distance estimation

⚫

heatmap(S .^ 0.1, c=:jet)

D[N] = log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])

⚫

heatmap(D .^ 0.1, c=:jet)

savefig("Mandelbrot_set_1.png")

N = abs.(Z) .> r # ~~normalized~~ ~~iteration~~ ~~count~~

N = abs.(Z) .> 2 # normal map (potential function)

T[N] = T[N] .- ~~log2.(log.(abs.(Z~~[N]~~)) ./ log(r))~~

U[N] = Z[N] ./ dZ[N]

T[N] = max.(((real.(U[N]) .* real.(v) .+ imag.(U[N]) .* imag.(v)) ./ abs.(U[N]) .+ height) ./ (1 .+ height), 0)

heatmap(T .^ 0.1, c=:~~jet~~)

heatmap(T .^ 1.0, c=:grays)

savefig("Mandelbrot_set_2.png")

N = abs.(Z) .> 2 # ~~exterior~~ distance estimation

N = abs.(Z) .> 2 # normal map (distance estimation)

D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ ~~abs~~.(dZ[N])

U[N] = Z[N] .* dZ[N] .* ((1 .+ log.(abs.(Z[N]))) .* conj.(dZ[N] .^ 2) .- log.(abs.(Z[N])) .* conj.(Z[N] .* ddZ[N]))

T[N] = max.(((real.(U[N]) .* real.(v) .+ imag.(U[N]) .* imag.(v)) ./ abs.(U[N]) .+ height) ./ (1 .+ height), 0)

heatmap(D .^ 0.1, c=:~~jet~~)

heatmap(T .^ 1.0, c=:grays)

savefig("Mandelbrot_set_3.png")</syntaxhighlight>

A small change in the code above creates Mercator maps and zoom images of the Mandelbrot set~~. Using the abs2 function and moving the iteration to a separate function speeds up the calculations inside the loop~~. See also the album [https://www.flickr.com/photos/arenamontanus/albums/72157615740829949 Mercator Mandelbrot Maps] by Anders Sandberg.

A small change in the code above creates Mercator maps and zoom images of the Mandelbrot set. See also the album [https://www.flickr.com/photos/arenamontanus/albums/72157615740829949 Mercator Mandelbrot Maps] by Anders Sandberg.

<syntaxhighlight lang="julia">using Plots

gr(aspect_ratio=:equal, axis=true, ticks=true, legend=false, dpi=200)

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N = abs.(Z) .> 2 # exterior distance estimation

D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])

D[N] = log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])

heatmap(D' .^ 0.1, c=:nipy_spectral)

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N = abs.(Z) .> 2 # exterior distance estimation

D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])

D[N] = log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])

heatmap(D .^ 0.3, c=:nipy_spectral)

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N = abs.(Z) .> 2 # exterior distance estimation

D[N] = 0.5 .* log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])

D[N] = log.(abs.(Z[N])) .* abs.(Z[N]) ./ abs.(dZ[N])

heatmap(D' .^ 0.03, c=:nipy_spectral)