The Doubly Periodic Scherk Surface

Matthias Weber
Indiana University
http://www.indiana.edu/~minimal

<< Own`Mesh`

Surface Data

ϵ = 10. ^-9;

G[z_] := -(Sqrt[Coth])[(z/2)];

n[z_] := StereographicProjection[G[z]]

ω1[z_] = 
        2  I ArcCot[(Sqrt[Coth])[(z/2)]];
ω2[z_] := 
        2 ArcTanh[(Sqrt[Coth])[(z/2)]];
ω3[z_] := I z;

f[z_] := Im[{ω1[z], ω2[z], ω3[z]}]

Mesh Setup

nr = 15;
nt = 15;
ϵ = 0.0000001;
r1 = 4;

dm = 
        RectangularDomain[NRange[0, r1, nr], 
          NRange[ϵ, Pi - ϵ, nt]];

Surface Plot

mp1 = 
        MeshPlot3D[{f[x + I y], n[x + I y]}, {x, y}, dm];

mp2 = 
        MeshJoin[mp1, 
          MeshRotate[mp1, 
            StraightLine[{1, -1, 0}, {0, 0, 0}]]];

mp3 = 
        MeshJoin[mp2, MeshReflect[mp2, Plane[{1, 0, 0}, 0]]];

mp4 = 
        MeshJoin[mp3, MeshReflect[mp3, Plane[{0, 1, 0}, 0]]];

gr = Mesh3DToGraphics3D[mp4];

Show[gr, Axes -> True];