added Poincare half-plane and disk options

chakravala · May 4, 2019 · e2fdfd0 · e2fdfd0
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commit e2fdfd0
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Showing 2 changed files with 45 additions and 24 deletions.
diff --git a/src/Fatou.jl b/src/Fatou.jl
@@ -24,11 24,13 @@ export fatou, juliafill, mandelbrot, newton, basin, plot
  x0 = nothing, # orbit starting point
  orbit::Int = 0, # orbit cobweb depth
  depth::Int = 1, # depth of function composition
- cmap::String= "") # imshow color map
  cmap::String= "", # imshow color map
  plane::Bool = false, # convert input disk to half-plane
  disk::Bool = false) # convert output half-plane to disk
 
 `Define` the metadata for a `Fatou.FilledSet`.
 """
-struct Define{FT<:Function,QT<:Function,CT<:Function,M,N}
 struct Define{FT<:Function,QT<:Function,CT<:Function,M,N,P,D}
  E::Any # input expression
  F::FT # primary map
  Q::QT # escape criterion
@@ -47,6 49,8 @@ struct Define{FT<:Function,QT<:Function,CT<:Function,M,N}
  orbit::Int # orbit cobweb depth
  depth::Int # depth of function composition
  cmap::String # imshow color map
  plane::Bool # convert input disk to half-plane
  disk::Bool # convert output half-plane to disk
  function Define(E;
  Q=:(abs2(z)),
  C=:((angle(z)/(2π))*n^p),
@@ -63,14 67,16 @@ struct Define{FT<:Function,QT<:Function,CT<:Function,M,N}
  x0=nothing,
  orbit::Int=0,
  depth::Int=1,
- cmap::String="")
  cmap::String="",
  plane::Bool=false,
  disk::Bool=false)
  !(typeof(∂) <: Array) && (∂ = [-float(∂),∂,-∂,∂])
  length(∂) == 2 && (∂ = [∂[1],∂[2],∂[1],∂[2]])
  !newt ? (f = genfun(E,[:z,:c]); q = genfun(Q,[:z,:c])) :
  (f = genfun(newton_raphson(E,m),[:z,:c]); q = genfun(Expr(:call,:abs,E),[:z,:c]))
  c = genfun(C,[:z,:n,:p])
  e = typeof(E) == String ? parse(E) : E
- return new{typeof(f),typeof(q),typeof(c),mandel,newt}(e,f,q,c,convert(Array{Float64,1},∂),UInt16(n),UInt8(N),float(ϵ),iter,float(p),newt,m,mandel,seed,x0,orbit,depth,cmap)
  return new{typeof(f),typeof(q),typeof(c),mandel,newt,plane,disk}(e,f,q,c,convert(Array{Float64,1},∂),UInt16(n),UInt8(N),float(ϵ),iter,float(p),newt,m,mandel,seed,x0,orbit,depth,cmap,plane,disk)
  end
 end
 
@@ -79,14 85,14 @@ end
 
 Compute the `Fatou.FilledSet` set using `Fatou.Define`.
 """
-struct FilledSet{FT,QT,CT,M,N}
- meta::Define{FT,QT,CT,M,N}
 struct FilledSet{FT,QT,CT,M,N,P,D}
  meta::Define{FT,QT,CT,M,N,P,D}
  set::Matrix{Complex{Float64}}
  iter::Matrix{UInt8}
  mix::Matrix{Float64}
- function FilledSet{FT,QT,CT,M,N}(K::Define{FT,QT,CT,M,N}) where {FT,QT,CT,M,N}
  function FilledSet{FT,QT,CT,M,N,P,D}(K::Define{FT,QT,CT,M,N,P,D}) where {FT,QT,CT,M,N,P,D}
  (i,s) = Compute(K)
- return new{FT,QT,CT,M,N}(K,s,i,broadcast(K.C,s,broadcast(float,i./K.N),K.p))
  return new{FT,QT,CT,M,N,P,D}(K,s,i,broadcast(K.C,s,broadcast(float,i./K.N),K.p))
  end
 end
 
@@ -100,7 106,7 @@ Compute the `Fatou.FilledSet` set using `Fatou.Define`.
 julia> fatou(K)
 ```
 """
-fatou(K::Define{FT,QT,CT,M,N}) where {FT,QT,CT,M,N} = FilledSet{FT,QT,CT,M,N}(K)
 fatou(K::Define{FT,QT,CT,M,N,P,D}) where {FT,QT,CT,M,N,P,D} = FilledSet{FT,QT,CT,M,N,P,D}(K)
 
 """
  juliafill(::Expr; # primary map, (z, c) -> F
@@ -115,7 121,9 @@ fatou(K::Define{FT,QT,CT,M,N}) where {FT,QT,CT,M,N} = FilledSet{FT,QT,CT,M,N}(K)
  x0 = nothing, # orbit starting point
  orbit::Int = 0, # orbit cobweb depth
  depth::Int = 1, # depth of function composition
- cmap::String= "") # imshow color map
  cmap::String= "", # imshow color map
  plane::Bool = false, # convert input disk to half-plane
  disk::Bool = false) # convert output half-plane to disk
 
 `Define` filled Julia basin in `Fatou`
 
@@ -138,8 146,10 @@ function juliafill(E;
  x0=nothing,
  orbit::Int=0,
  depth::Int=1,
- cmap::String="")
- return Define(E,Q=Q,C=C,∂=∂,n=n,N=N,ϵ=ϵ,iter=iter,p=p,newt=newt,m=m,x0=x0,orbit=orbit,depth=depth,cmap=cmap)
  cmap::String="",
  plane::Bool=false,
  disk::Bool=false)
  return Define(E,Q=Q,C=C,∂=∂,n=n,N=N,ϵ=ϵ,iter=iter,p=p,newt=newt,m=m,x0=x0,orbit=orbit,depth=depth,cmap=cmap,plane=plane,disk=disk)
 end
 
 """
@@ -157,7 167,9 @@ end
  x0 = nothing, # orbit starting point
  orbit::Int = 0, # orbit cobweb depth
  depth::Int = 1, # depth of function composition
- cmap::String= "") # imshow color map
  cmap::String= "", # imshow color map
  plane::Bool = false, # convert input disk to half-plane
  disk::Bool = false) # convert output half-plane to disk
 
 `Define` Mandelbrot basin in `Fatou`
 
@@ -181,9 193,11 @@ function mandelbrot(E;
  x0=nothing,
  orbit::Int=0,
  depth::Int=1,
- cmap::String="")
  cmap::String="",
  plane::Bool=false,
  disk::Bool=false)
  m ≠ 0 && (newt = true)
- return Define(E,Q=Q,C=C,∂=∂,n=n,N=N,ϵ=ϵ,iter=iter,p=p,newt=newt,m=m,mandel=true,seed=seed,x0=x0,orbit=orbit,depth=depth,cmap=cmap)
  return Define(E,Q=Q,C=C,∂=∂,n=n,N=N,ϵ=ϵ,iter=iter,p=p,newt=newt,m=m,mandel=true,seed=seed,x0=x0,orbit=orbit,depth=depth,cmap=cmap,plane=plane,disk=disk)
 end
 
 """
@@ -201,7 215,9 @@ end
  x0 = nothing, # orbit starting point
  orbit::Int = 0, # orbit cobweb depth
  depth::Int = 1, # depth of function composition
- cmap::String= "") # imshow color map
  cmap::String= "", # imshow color map
  plane::Bool = false, # convert input disk to half-plane
  disk::Bool = false) # convert output half-plane to disk
 
 `Define` Newton basin in `Fatou`
 
@@ -224,8 240,10 @@ function newton(E;
  x0=nothing,
  orbit::Int=0,
  depth::Int=1,
- cmap::String="")
- return Define(E,C=C,∂=∂,n=n,N=N,ϵ=ϵ,iter=iter,p=p,newt=true,m=m,mandel=mandel,seed=seed,x0=x0,orbit=orbit,depth=depth,cmap=cmap)
  cmap::String="",
  plane::Bool=false,
  disk::Bool=false)
  return Define(E,C=C,∂=∂,n=n,N=N,ϵ=ϵ,iter=iter,p=p,newt=true,m=m,mandel=mandel,seed=seed,x0=x0,orbit=orbit,depth=depth,cmap=cmap,plane=plane,disk=disk)
 end
 
 # load additional functionality
@@ -245,24 263,27 @@ L"\$\\displaystyle D_2(\\epsilon) = \\left\\{z\\in\\mathbb{C}:\\left|\\,z - \\fr
 """
 basin(K::Define,j) = K.newt ? nrset(K.E,K.m,j) : jset(K.E,j)
 
 plane(z::Complex) = (2z.re/(z.re^2 (1-z.im)^2)) im*(1-z.re^2-z.im^2)/(z.re^2 (1-z.im)^2)
 disk(z::Complex) = (2z.re/(z.re^2 (1 z.im)^2)) im*(z.re^2 z.im^2-1)/(z.re^2 (1 z.im)^2)
 
 # define function for computing orbit of a z0 input
-function orbit(K::Define{FT,QT,CT,M,N},z0::Complex{Float64}) where {FT,QT,CT,M,N}
- M ? (z = K.seed) : (z = z0)
 function orbit(K::Define{FT,QT,CT,M,N,P,D},z0::Complex{Float64}) where {FT,QT,CT,M,N,P,D}
  M ? (z = K.seed) : (z = P ? plane(z0) : z0)
  zn = 0x00
  while (N ? (K.Q(z,z0)::Float64>K.ϵ)::Bool : (K.Q(z,z0)::Float64<K.ϵ))::Bool && K.N>zn
  z = K.F(z,z0)::Complex{Float64}
  zn =0x01
  end; #end
  # return the normalized argument of z or iteration count
- return (zn::UInt8,z::Complex{Float64})
  return (zn::UInt8,(D ? disk(z) : z)::Complex{Float64})
 end
 
 """
  Compute(::Fatou.Define)::Union{Matrix{UInt8},Matrix{Float64}}
 
 `Compute` the `Array` for `Fatou.FilledSet` as specefied by `Fatou.Define`.
 """
-function Compute(K::Define{FT,QT,CT,M,N})::Tuple{Matrix{UInt8},Matrix{Complex{Float64}}} where {FT,QT,CT,M,N}
 function Compute(K::Define{FT,QT,CT,M,N,D})::Tuple{Matrix{UInt8},Matrix{Complex{Float64}}} where {FT,QT,CT,M,N,D}
  # generate coordinate grid
  Kyn = round(UInt16,(K.∂[4]-K.∂[3])/(K.∂[2]-K.∂[1])*K.n)
  x = range(K.∂[1] 0.0001,stop=K.∂[2],length=K.n)

diff --git a/src/pyplot.jl b/src/pyplot.jl
@@ -18,11 18,11 @@ function plot(K::FilledSet;c::String="",bare::Bool=false)
  # annotate title using LaTeX
  ttext = "f:z\\mapsto $(rdpm(Algebra.latex(K.meta.E))),\\,"
  if K.meta.newt
- title(latexstring("$ttext m = $(K.meta.m), ")*t)
  PyPlot.title(latexstring("$ttext m = $(K.meta.m), ")*t)
  # annotate y-axis with Newton's method
  ylabel(L"Fatou\,set:\,"*L"z\,↦\,z-m\,×\,f(z)\,/\,f\,'(z)")
  else
- title(latexstring(ttext)*t)
  PyPlot.title(latexstring(ttext)*t)
  end
  tight_layout()
  colorbar()