The programs used

For the SOR method (which can be modified to GS and Jacobi):

#include<stdio.h>
#include<math.h>

int main(){

  //Set up the array 
 int i,j;
 int count=1;
 float ok=0.0001;
 float diff=0;
 float max=0.002;
 double v[100][100];
 double u[100][100];
 int flag=0;
 for(i=0; i<100; i++){
   for(j=0; j<100; j++){v[i][j]=0.0;u[i][j]=0.0;}}
 
 float w=1.93908;
 float alpha=0;
 
 //The SOR Method: Avoid boundaries
 //A grounded box surrounds a central
 //plate held at potential located at i=49
 //from j=25 to 75
 for(j=25;j<=75;j++){v[49][j]=1.0;u[49][j]=1.0;}
 
   
 //A
 while(max>ok){
   printf("%f\n",max);
   //(B-C)
   max=ok;
   count+=1;
   for(i=1; i<99; i++){  //B
     for(j=1; j<99; j++){  //C
       if(i==49)if(j>=25)if(j<=75){flag=100;}  \\check for boundry
       if(i==49)if(j<25){flag=0;}
       if(i==49)if(j>75){flag=0;}
       if(flag<1){alpha=0.25*(v[i+1][j]+u[i-1][j]+v[i][j+1]+u[i][j-1]);}
       if(flag<1){u[i][j]=v[i][j]+w*(alpha-v[i][j]);}
     diff=fabs(v[i][j]-u[i][j]);
     if(diff>max){max=diff;}
    
     }}  //~B ~C
   //~(B-C)
     for(i=1; i<99; i++){for(j=1; j<99; j++){v[i][j]=u[i][j];}}
  }//~A

 //Print to file
 FILE *fp;
 fp=fopen("so.txt","w");


 for(i=0; i<100; i++){for(j=0; j<100; j++){
   fprintf(fp,"%i %i %f \n", i,j,v[i][j]);}fprintf(fp,"\n");}
   

 printf("SOR took %i iterations.\n",count);

}

The above code plots a simple bar at potential V. We can modify it for our hyperbolic electordes as,

#include<stdio.h>
#include<math.h>

int main(){

  //Set up the array 
 int i,j;
 int count=1;
 float ok=0.0001;
 float diff=0;
 float max=0.002;
 double v[1000][1000];
 double u[1000][1000];
 int flag=0;
 for(i=0; i<1000; i++){
   for(j=0; j<1000; j++){v[i][j]=0.0;u[i][j]=0.0;}}
 
 float w=1.93908;
 float alpha=0;
 
 //Set hyperbolic conditions:
 
 // Upper
 for(j=250;j<=750;j++){
         i=10*(int)floor(50+sqrt(100+2.4*(j/10-50)*(j/10-50)));  
         v[i][j]=1.0; v[i+1][j]=1.0; v[i-1][j]=1.0; v[i][j+1]=1.0; 
                                                      \ v[i][j-1]=1.0;
         u[i][j]=1.0; u[i+1][j]=1.0; u[i-1][j]=1.0; u[i][j+1]=1.0; 
                                                     \ u[i][j-1]=1.0;}

 //Lower
 for(j=250;j<=750;j++){
         i=10*(int)floor(50-sqrt(100+2.4*(j/10-50)*(j/10-50)));  
         v[i][j]=1.0; v[i+1][j]=1.0; v[i-1][j]=1.0; v[i][j+1]=1.0; 
                                                       \ v[i][j-1]=1.0;
         u[i][j]=1.0; u[i+1][j]=1.0; u[i-1][j]=1.0; u[i][j+1]=1.0; 
                                                      \ u[i][j-1]=1.0;}

 
// Left
 for(i=250;i<=750;i++){
         j=10*(int)floor(50+sqrt(100+2.4*(i/10-50)*(i/10-50)));  
         v[i][j]=-1.0; v[i+1][j]=-1.0; v[i-1][j]=-1.0; v[i][j+1]=-1.0; 
                                                      \ v[i][j-1]=-1.0;
         u[i][j]=-1.0; u[i+1][j]=-1.0; u[i-1][j]=-1.0; u[i][j+1]=-1.0; 
                                                      \  u[i][j-1]=-1.0;}

 //Right
 for(i=250;i<=750;i++){
         j=10*(int)floor(50-sqrt(100+2.4*(i/10-50)*(i/10-50)));  
         v[i][j]=-1.0; v[i+1][j]=-1.0; v[i-1][j]=-1.0; v[i][j+1]=-1.0; 
                                                      \ v[i][j-1]=-1.0;
         u[i][j]=-1.0; u[i+1][j]=-1.0; u[i-1][j]=-1.0; u[i][j+1]=-1.0; 
                                                      \ u[i][j-1]=-1.0;}

 
   
 //A
 while(max>ok){
   printf("%f\n",max);
   //(B-C)
   max=ok;
   count+=1;
   for(i=1; i<999; i++){  //B
     for(j=1; j<999; j++){  //C
       // if(v[i][j]>1){v[i][j]=1.0;}  //Strays
       // if(v[i][j]<-1){v[i][j]=-1.0;}
       //if(v[i][j]<=-1.0){flag=100;}
       if(v[i][j]>=1){flag=100;}
       if(v[i][j]<=-1){flag=100;}
       if(v[i][j]<1 && v[i][j]>-1){flag=0;}
       //Problem:  what if more 
       if(flag<1){alpha=0.25*(v[i+1][j]+u[i-1][j]+v[i][j+1]+u[i][j-1]);}
       if(flag<1){
              u[i][j]=v[i][j]+w*(alpha-v[i][j]); 
              if(u[i][j]>1){u[i][j]=1;}  
              if(u[i][j]<-1){u[i][j]=-1;}}
     diff=fabs(v[i][j]-u[i][j]);
     if(diff>max){max=diff;}
    
     }}  //~B ~C
   //~(B-C)
     for(i=1; i<999; i++){for(j=1; j<999; j++){v[i][j]=u[i][j];}}
  }//~A

 //Print to file
 FILE *fp;
 fp=fopen("so2.txt","w");

 int num=0; int numy=0;
 for(i=0; i<1000; i++){for(j=0; j<1000; j++){
  fprintf(fp,"%i %i %f \n", i,j,v[i][j]);} fprintf(fp,"\n");
   numy++;
 }
   

 printf("SOR took %i iterations.\n",count);

}

We also used the following gnuplot script, where our previous program's output put a blank line following each j cycle to conform with gnuplot's contour method:

set data style lines
set terminal postscript
#set nokey
set cntrparam levels 20
set output 'hyper1.ps'
set title "Hyperbolic Electrodes via SOR, n=1000, Tolerate=0.0001"
splot 'so.txt'
set contour
set nosurface
set view 0,0
set output 'hyper2.ps'
splot 'so.txt'