Basic usage

Let's first initiate the module. We'll also initiate plotting with Plots.jl, using PyPlot as the backend for the actual output.

julia> using SchwarzChristoffel

julia> using Plots

julia> pyplot()
Plots.PyPlotBackend()

Now, we create a polygon shape by specifying its vertices. Note that the vertices must be provided in counter-clockwise order.

julia> x = [-1.0,0.2,1.0,-1.0]; y = [-1.0,-1.0,0.5,1.0];

julia> p = Polygon(x,y)
Polygon with 4 vertices at
             (-1.0,-1.0) (0.2,-1.0) (1.0,0.5) (-1.0,1.0)
             interior angles/π = [0.5, 0.656, 0.422, 0.422]

Let's plot the polygon to make sure it matches what we wanted.

julia> plot(p)
Plot{Plots.PyPlotBackend() n=1}

Now, we create the map from the unit circle to the polygon.

julia> m = ExteriorMap(p)
Schwarz-Christoffel map of unit circle to exterior of polygon with 4 vertices

Let's visualize what we've constructed. Here, we will inspect the mapping from the exterior of the unit circle to the exterior of the polygon.

julia> plot(m)
Plot{Plots.PyPlotBackend() n=60}

We can now easily evaluate the map at any place we like. It could be evaluated outside the unit circle:

julia> ζ = 1.2 + 0.1im
1.2 + 0.1im

julia> m(ζ)
1.0849884084568373 + 0.2410831373600964im

or it could be evaluated inside the unit circle:

julia> ζ = 0.5 + 0.1im
0.5 + 0.1im

julia> m(ζ;inside=true)
-1.789702985840997 - 1.2882910977565625im

We can also evaluate the first and second derivative of the map at any place(s). Let's evaluate at a range of points outside the circle.

julia> dm = DerivativeMap(m)
d/dζ of Schwarz-Christoffel map of unit circle to exterior of polygon with 4 vertices

julia> ζ = collect(1.1:0.1:2.0) + 0.1im
10-element Array{Complex{Float64},1}:
 1.1+0.1im
 1.2+0.1im
 1.3+0.1im
 1.4+0.1im
 1.5+0.1im
 1.6+0.1im
 1.7+0.1im
 1.8+0.1im
 1.9+0.1im
 2.0+0.1im

julia> dz,ddz = dm(ζ);

julia> dz
10-element Array{Complex{Float64},1}:
 0.872356-0.394475im
 0.901724-0.289355im
 0.925562-0.217994im
 0.944059-0.168264im
 0.958255-0.132657im
  0.96919-0.106522im
 0.977695-0.0869128im
 0.984389-0.0719115im
 0.989723-0.0602376im
 0.994025-0.0510144im

Now let's try a more interesting shape. Here's a star-shaped body

julia> n = 8; dθ = 2π/(2n)
0.39269908169872414

julia> θ = collect(0:dθ:2π-dθ)
16-element Array{Float64,1}:
 0.0
 0.392699
 0.785398
 1.1781
 1.5708
 1.9635
 2.35619
 2.74889
 3.14159
 3.53429
 3.92699
 4.31969
 4.71239
 5.10509
 5.49779
 5.89049

julia> w = (1+0.3cos.(n*θ)).*exp.(im*θ)
16-element Array{Complex{Float64},1}:
          1.3+0.0im
     0.646716+0.267878im
     0.919239+0.919239im
     0.267878+0.646716im
   7.9602e-17+1.3im
    -0.267878+0.646716im
    -0.919239+0.919239im
    -0.646716+0.267878im
         -1.3+1.59204e-16im
    -0.646716-0.267878im
    -0.919239-0.919239im
    -0.267878-0.646716im
 -2.38806e-16-1.3im
     0.267878-0.646716im
     0.919239-0.919239im
     0.646716-0.267878im

julia> p = Polygon(w)
Polygon with 16 vertices at
             (1.3,0.0) (0.6467156727579007,0.26787840265556284) (0.9192388155425119,0.9192388155425117) (0.26787840265556284,0.6467156727579007) (7.960204194457797e-17,1.3) (-0.2678784026555628,0.6467156727579007) (-0.9192388155425117,0.9192388155425119) (-0.6467156727579007,0.2678784026555629) (-1.3,1.5920408388915593e-16) (-0.6467156727579008,-0.26787840265556273) (-0.919238815542512,-0.9192388155425117) (-0.26787840265556323,-0.6467156727579005) (-2.3880612583373386e-16,-1.3) (0.267878402655563,-0.6467156727579007) (0.9192388155425116,-0.919238815542512) (0.6467156727579005,-0.26787840265556323)
             interior angles/π = [0.248, 1.502, 0.248, 1.502, 0.248, 1.502, 0.248, 1.502, 0.248, 1.502, 0.248, 1.502, 0.248, 1.502, 0.248, 1.502]

julia> plot(p)
Plot{Plots.PyPlotBackend() n=1}

Construct the map and plot it

julia> m = ExteriorMap(p)
Schwarz-Christoffel map of unit circle to exterior of polygon with 16 vertices

julia> plot(m)
Plot{Plots.PyPlotBackend() n=60}