Basic usage

Let's first initiate the module. We'll also initiate plotting with Plots.jl.

julia> using SchwarzChristoffel
julia> using Plots

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.GRBackend() 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.GRBackend() 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 + 0.1im1.2 + 0.1im
julia> m(ζ)1.0849884084568375 + 0.241083137360106im

or it could be evaluated inside the unit circle:

julia> ζ = 0.5 + 0.1im0.5 + 0.1im
julia> m(ζ;inside=true)-1.7897029858409896 - 1.2882910977565591im

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) .+ 0.1im10-element Vector{ComplexF64}:
 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> dz10-element Vector{ComplexF64}:
 0.8723555976319566 - 0.39447513501598724im
 0.9017237398517002 - 0.2893546703349404im
 0.9255615782055835 - 0.2179943990034074im
 0.9440588034755485 - 0.16826386737167404im
 0.9582547929005287 - 0.1326572835806059im
  0.969189657521706 - 0.106522445667103im
 0.9776947206989297 - 0.08691284233716035im
 0.9843890003381357 - 0.07191147884514731im
 0.9897232289776905 - 0.06023761983615078im
 0.9940246663281329 - 0.051014412241562im

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 Vector{Float64}:
 0.0
 0.39269908169872414
 0.7853981633974483
 1.1780972450961724
 1.5707963267948966
 1.9634954084936207
 2.356194490192345
 2.748893571891069
 3.141592653589793
 3.5342917352885173
 3.9269908169872414
 4.319689898685965
 4.71238898038469
 5.105088062083414
 5.497787143782138
 5.890486225480862
julia> w = (1 .+ 0.3cos.(n*θ)).*exp.(im*θ)16-element Vector{ComplexF64}:
                     1.3 + 0.0im
      0.6467156727579007 + 0.26787840265556284im
      0.9192388155425119 + 0.9192388155425117im
     0.26787840265556284 + 0.6467156727579007im
   7.960204194457797e-17 + 1.3im
     -0.2678784026555628 + 0.6467156727579007im
     -0.9192388155425117 + 0.9192388155425119im
     -0.6467156727579007 + 0.2678784026555629im
                    -1.3 + 1.5920408388915593e-16im
     -0.6467156727579008 - 0.26787840265556273im
      -0.919238815542512 - 0.9192388155425117im
    -0.26787840265556323 - 0.6467156727579005im
 -2.3880612583373386e-16 - 1.3im
       0.267878402655563 - 0.6467156727579007im
      0.9192388155425116 - 0.919238815542512im
      0.6467156727579005 - 0.26787840265556323im
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.GRBackend() 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.GRBackend() n=60}