Mandelbrot set: Difference between revisions

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===Normal Map Effect, Mercator Projection and Perturbation Theory===

This is a translation of the corresponding Python section: see there for more explanations. 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#Normal_map_effect ''Normal map effect'']).

This is a translation of the corresponding Python section: see there for more explanations. 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#Normal_map_effect ''Normal map effect'']). See also [https://www.shadertoy.com/view/wtscDX Julia Stripes] on Shadertoy.

<syntaxhighlight lang="julia">using Plots

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

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

stripes, damping = 5.0, 2.0 # stripe density and damping parameter

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

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

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

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

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

for k in 1:n

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

S[M], T[M] = S[M] .+ sin.(stripes .* angle.(Z[M])), T[M] .+ 1

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

end

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N = abs.(Z) .> 2 # normal map effect 1 (potential function)

P, Q = S[N] ./ T[N], (S[N] .+ sin.(stripes .* angle.(Z[N]))) ./ (T[N] .+ 1)

F = 1 .- log2.(log.(abs.(Z[N])) ./ log(r))

H = P .+ (Q .- P) .* F .* F .* (3 .- 2 .* F) # hermite interpolation

U = Z[N] ./ dZ[N] # normal vectors to the equipotential lines

U, S = U ./ abs.(U), 1 .+ ~~sin.(100~~ .* ~~angle.(U))~~ ./ 10 # unit normal vectors and ~~stripes~~

U, v = U ./ abs.(U), exp(direction / 180 * pi * im) # unit normal vectors and unit 2D vector

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

D[N] = max.((real.(U) .* real(v) .+ imag.(U) .* imag(v) .+ (1 .+ H ./ damping) .* height) ./ (1 + height), 0)

heatmap(T .^ 1.0, c=:bone_1)

heatmap(D .^ 1.0, c=:bone_1)

savefig("Mandelbrot_normal_map_1.png")

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

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

U = U ./ abs.(U) # unit normal vectors to ~~the~~ ~~equidistant~~ ~~lines~~

U, v = U ./ abs.(U), exp(direction / 180 * pi * im) # unit normal vectors and unit 2D vector

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

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

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

heatmap(D .^ 1.0, c=:afmhot)

savefig("Mandelbrot_normal_map_2.png")</syntaxhighlight>