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Mandelbrot set: Difference between revisions

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Mandelbrot set (view source)

Revision as of 14:00, 16 July 2023

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→‎Normal Map Effect, Mercator Projection and Perturbation Theory: Much nicer stripes
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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'']). 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, 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, Sv = U ./ abs.(U), 1exp(direction .+/ sin.(100180 .* angle.(U))pi ./* 10im) # unit normal vectors and stripesunit 2D vector
TD[N] = max.((real.(U) .* real(v) .+ imag.(U) .* imag(v) .+ S(1 .+ H ./ damping) .* height) ./ (1 + height), 0)
 
heatmap(TD .^ 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, v = U ./ abs.(U), exp(direction / 180 * pi * im) # unit normal vectors toand theunit equidistant2D linesvector
TD[N] = max.((real.(U) .* real(v) .+ imag.(U) .* imag(v) .+ height) ./ (1 + height), 0)
 
heatmap(TD .^ 1.0, c=:afmhot)
savefig("Mandelbrot_normal_map_2.png")</syntaxhighlight>
 
Majow
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