The Singly Periodic Scherk Surfaces

<<Own`Mesh`

We use a level curve parametrization for the singly periodic Scherk surface

Weierstrass Data

dh[x_]:=1;
G[x_]:=(Sqrt[Cot[x]]);

ω1[z_]:= 1/2 (1/G[z]-G[z])dh[z];
ω2[z_]:=1/2 I (1/G[z]+G[z])dh[z];
ω3[z_]:= dh[z];

Explicit Integral

f[z_]:=Re[{(1/(2 (Sqrt[2])))(-I π+Log[-1+(Sqrt[2]) (Sqrt[Cot[z]])-Cot[z]]-Log[1+(Sqrt[2]) (Sqrt[Cot[z]])+Cot[z]]),(1/(Sqrt[2]))(I (π+ArcTan[1-(Sqrt[2]) (Sqrt[Cot[z]])]-ArcTan[1+(Sqrt[2]) (Sqrt[Cot[z]])])),z}];

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

y1=3;
nr1=5;
nt=5;

ϵ=0.000001;

domain1=RectangularDomain[NRange[ϵ,Pi/2,nr1],NRange[0,y1,nt]];

fr1=MeshPlot3D[{f[x+I y],n[x+I y]},{x,y},domain1];

fr2=MeshReflect[fr1,Plane[{0,1,0},0]];
fr3=MeshJoin[fr1,fr2];
fr4=MeshReflect[fr3,Plane[{1,0,0},0]];
fr5=MeshJoin[fr3,fr4];
fr6=MeshRotate[fr5,StraightLine[{0,0,0},{1,1,0}]];
fr7=MeshJoin[fr5,fr6];

fr8=MeshTranslate[fr7,{0,0,Pi}];
fr9=MeshTranslate[fr7,{0,0,-Pi}];
fr=MeshJoin[fr7,fr8,fr9];

Show[gr=Mesh3DToGraphics3D[fr]]