ComplexPhasePortrait.jl

This package is a Julia implementation of the phase portrait ideas presented in Elias Wegert's book "Visual Complex Functions".

Installation

From the Julia command prompt:

] add ComplexPhasePortrait

Examples

There is so far one exported function, portrait, and here I will try to detail its use. First we need function data over a grid.

using ComplexPhasePortrait

nx = 1000
x = range(-1, stop=1, length=nx)
Z = x' .+ reverse(x)*im

f = z -> (z - 0.5im)^2 * (z + 0.5+0.5im)/z
fz = f.(Z)

Now a basic phase plot.

img = portrait(fz)

Now for a basic plot using NIST coloring.

img = portrait(fz, ctype="nist")

Lines of constant phase are given by

img = portrait(fz, PTstepphase)

Lines of constant modulus are given by

img = portrait(fz, PTstepmod)

Finally, a conformal grid is given by

img = portrait(fz, PTcgrid)

You can also use custom colormaps to create unique visualizations. Below are examples using the viridis colormap and a custom colormap.

using ComplexPhasePortrait, ColorSchemes

viridis_colormap = [RGB(c.r, c.g, c.b) for c in ColorSchemes.viridis.colors]

img = portrait(fz, PTcgrid, colormap=viridis_colormap)

using ComplexPhasePortrait, Colors
custom_colormap = [RGB(1.0, 0.0, 0.0), RGB(0.0, 1.0, 0.0), RGB(0.0, 0.0, 1.0)]
img = portrait(fz, PTcgrid, colormap=custom_colormap)

Plot recipes

ComplexPhasePortrait.jl has support for plotting recipes for Plots.jl and Makie.jl.

Plots.jl:

using Plots
using LaTeXStrings

using ComplexPhasePortrait
using IntervalSets

f = z -> (z - 0.5im)^2 * (z + 0.5+0.5im)/z

phaseplot(-1..1, -1..1, f, PTcgrid, :ctype=>"nist";
          xlabel=L"\Re\{z\}", ylabel=L"\Im\{z\}")

Makie.jl

Makie.jl is an optional dependency via package extensions, and the functionality to plot phase portraits becomes available if Makie.jl or one of its front-end packages is loaded before ComplexPhasePortrait.jl:

using GLMakie

using ComplexPhasePortrait
using IntervalSets

f = z -> (z - 0.5im)^2 * (z + 0.5+0.5im)/z

fig = Figure()
ax = Axis(fig[1, 1], aspect=1)
phase!(ax, -1..1, -1..1, f, portrait_type=PTcgrid, ctyle="nist")
display(fig)

Alternatively, one can use the function phase:

phase(x, y, f; kwargs...)

x and y can be vectors or ClosedIntervals; in the former case f can be a matrix of appropriate size or a Function, in the latter case, only a Function is possible.

Saving Images

To save the generated phase portraits as image file directly without use of Plots.jl or Makie.jl, you can use the save function from the Images.jl package. For example:

using Images
f = z -> (z - 0.5im)^2 * (z + 0.5+0.5im)/z
fz = f.(Z)
img = portrait(fz)
save("phase_portrait.png", img)