H -Surface of H.A. Schwarz

<<Own`Mesh`

Weierstrass Data

Choose lattice parameter

τ=2.I;

dh[z_]:=1
ρ=1.;
a0=1/4.;
G2[z_]:=ρ θ11[z-a0,τ]/θ11[z+a0,τ]

G[z_]:=G2[z]^(2/3)

phi1[z_]:=G[z ] dh[z];
phi2[z_]:=1/G[z]dh[z];

ω1[z_] :=( phi2[z]-phi1[z] )/2;
ω2[z_] :=I( phi2[z]+phi1[z])/2;
ω3[z_]:=dh[z]

cp=.25+τ/4.;

f0[w_]:=Re[{
		NIntegrate[ω1[z],{z,cp,w}],NIntegrate[ω2[z],{z,cp,w}],
	w}]

δ0=f0[.5]

f[w_]:=Re[{
		NIntegrate[ω1[z],{z,cp,w}],NIntegrate[ω2[z],{z,cp,w}],
	w}]-δ0

n[w_]:=StereographicProjection[G[w]]

Parametrization

ϵ=10^-7;

y1=Im[τ]/2.;
nx=4;
ny=5;

domain=RectangularDomain[Union[NRange[0,a0,nx],NRange[a0,.5,nx],SameTest->AlmostEqual],NRange[0,y1,ny]];

Surface Patch Creation

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

c0=f[τ/2].{Cos[30. Degree],Sin[-30 Degree],0};

len=2(f[τ/2]-f[a0]).{Cos[30. Degree],Sin[-30 Degree],0}

fr2=MeshJoin[fr1,MeshReflect[fr1,Plane[{0,1,0},0]]];

fr3=MeshJoin[fr2,MeshReflect[fr2,Plane[{0,0,1},2f[a0][[3]]]]];

fr4=MeshJoin[fr3,MeshReflect[fr3,Plane[{Cos[30. Degree],Sin[30 Degree],0},c0]]];

fr5=MeshJoin[fr4,MeshReflect[fr4,Plane[{Cos[30 Degree],Sin[-30 Degree],0},c0]]];

fr6=MeshJoin[fr5,MeshTranslate[fr5,len{Cos[30. Degree],Sin[30 Degree],0}],MeshTranslate[fr5,len{Cos[30. Degree],Sin[-30 Degree],0}]];

fr7=MeshJoin[fr6,MeshTranslate[fr6,{0,0,1}],MeshTranslate[fr6,{0,0,-1}]];

Show[gr=Mesh3DToGraphics3D[fr7],Axes->True]