A test of some distances:

chessboard_distance_2d:
,,,,,,,,,,,,,
,^^^^^^^^^^^,
,^*********^,
,^*xxxxxxx*^,
,^*xfffffx*^,
,^*xfHHHfx*^,
,^*xfH#Hfx*^,
,^*xfHHHfx*^,
,^*xfffffx*^,
,^*xxxxxxx*^,
,^*********^,
,^^^^^^^^^^^,
,,,,,,,,,,,,,

manhattan_distance_2d:
      ,      
     ,^,     
    ,^*^,    
   ,^*x*^,   
  ,^*xfx*^,  
 ,^*xfHfx*^, 
,^*xfH#Hfx*^,
 ,^*xfHfx*^, 
  ,^*xfx*^,  
   ,^*x*^,   
    ,^*^,    
     ,^,     
      ,      

euclidean_distance_2d:
   .:,,,:.   
  :-~^^^~-:  
 :-^+***+^-: 
.-^+oxxxo+^-.
:~+o8$f$8o+~:
,^*x$BHB$x*^,
,^*xfH#Hfx*^,
,^*x$BHB$x*^,
:~+o8$f$8o+~:
.-^+oxxxo+^-.
 :-^+***+^-: 
  :-~^^^~-:  
   .:,,,:.   

hexagonal_distance_2d:
   ,,,,,,,   
  .-^^^^^-.  
  ,^*****^,  
 .-=oxxxo=-. 
 ,^*xfffx*^, 
.-=o8kHk8o=-.
,^*xfH#Hfx*^,
.-=o8kHk8o=-.
 ,^*xfffx*^, 
 .-=oxxxo=-. 
  ,^*****^,  
  .-^^^^^-.  
   ,,,,,,,   

manhattan_distance_2d:
      ,      
     ,^,     
    ,^*^,    
   ,^*x*^,   
  ,^*xfx*^,  
 ,^*xfHfx*^, 
,^*xfH#Hfx*^,
 ,^*xfHfx*^, 
  ,^*xfx*^,  
   ,^*x*^,   
    ,^*^,    
     ,^,     
      ,      

octagonal_distance_2d:
   .,,,,,.   
  .,~^^^~,.  
 .,~=***=~,. 
.,~=*OxO*=~,.
,~=*O8f8O*=~,
,^*O8fBf8O*^,
,^*xfBWBfx*^,
,^*O8fBf8O*^,
,~=*O8f8O*=~,
.,~=*OxO*=~,.
 .,~=***=~,. 
  .,~^^^~,.  
   .,,,,,.   

sqr_euclidean_distance_2d:
             
             
             
             
     ^*^     
    ^fHf^    
    *H#H*    
    ^fHf^    
     ^*^     
             
             
             
             

minkowski_distance, p = 0.3
      ,      
      ^      
      *      
      x      
      f      
      H      
,^*xfH#Hfx*^,
      H      
      f      
      x      
      *      
      ^      
      ,      

minkowski_distance, p = 0.4
      ,      
      ^      
      *      
      x      
      f      
     -H-     
,^*xfH#Hfx*^,
     -H-     
      f      
      x      
      *      
      ^      
      ,      

minkowski_distance, p = 0.6
      ,      
      ^      
      *      
     ,x,     
    :=f=:    
   ,=OHO=,   
,^*xfH#Hfx*^,
   ,=OHO=,   
    :=f=:    
     ,x,     
      *      
      ^      
      ,      

minkowski_distance, p = 0.8
      ,      
     .^.     
     -*-     
    ,=x=,    
   ,=ofo=,   
 .-=o$H$o=-. 
,^*xfH#Hfx*^,
 .-=o$H$o=-. 
   ,=ofo=,   
    ,=x=,    
     -*-     
     .^.     
      ,      

minkowski_distance, p = 1.0
      ,      
     ,^,     
    ,^*^,    
   ,^*x*^,   
  ,^*xfx*^,  
 ,^*xfHfx*^, 
,^*xfH#Hfx*^,
 ,^*xfHfx*^, 
  ,^*xfx*^,  
   ,^*x*^,   
    ,^*^,    
     ,^,     
      ,      

minkowski_distance, p = 1.2
     .,.     
    :-^-:    
   :~=*=~:   
  :~+oxo+~:  
 :~+o8f8o+~: 
.-=o8kHk8o=-.
,^*xfH#Hfx*^,
.-=o8kHk8o=-.
 :~+o8f8o+~: 
  :~+oxo+~:  
   :~=*=~:   
    :-^-:    
     .,.     

minkowski_distance, p = 1.4
     :,:     
   .,~^~,.   
  .-^+*+^-.  
 .-^*OxO*^-. 
 ,^*O8f8O*^, 
:~+O8kHk8O+~:
,^*xfH#Hfx*^,
:~+O8kHk8O+~:
 ,^*O8f8O*^, 
 .-^*OxO*^-. 
  .-^+*+^-.  
   .,~^~,.   
     :,:     

minkowski_distance, p = 1.6
    .:,:.    
   :-~^~-:   
  :~=+*+=~:  
 :~=*OxO*=~: 
.-=*x$f$x*=-.
:~+O$kHk$O+~:
,^*xfH#Hfx*^,
:~+O$kHk$O+~:
.-=*x$f$x*=-.
 :~=*OxO*=~: 
  :~=+*+=~:  
   :-~^~-:   
    .:,:.    

minkowski_distance, p = 1.8
    :,,,:    
  .,-^^^-,.  
 .,~=+*+=~,. 
 ,~+oOxOo+~, 
:-=ox$f$xo=-:
,^+O$BHB$O+^,
,^*xfH#Hfx*^,
,^+O$BHB$O+^,
:-=ox$f$xo=-:
 ,~+oOxOo+~, 
 .,~=+*+=~,. 
  .,-^^^-,.  
    :,,,:    

minkowski_distance, p = 2.0
   .:,,,:.   
  :-~^^^~-:  
 :-^+***+^-: 
.-^+oxxxo+^-.
:~+o8$f$8o+~:
,^*x$BHB$x*^,
,^*xfH#Hfx*^,
,^*x$BHB$x*^,
:~+o8$f$8o+~:
.-^+oxxxo+^-.
 :-^+***+^-: 
  :-~^^^~-:  
   .:,,,:.   

minkowski_distance, p = 2.2
   .:,,,:.   
  :-~^^^~-:  
 :~^+***+^~: 
.-^*oxxxo*^-.
:~+o8$f$8o+~:
,^*x$BHB$x*^,
,^*xfH#Hfx*^,
,^*x$BHB$x*^,
:~+o8$f$8o+~:
.-^*oxxxo*^-.
 :~^+***+^~: 
  :-~^^^~-:  
   .:,,,:.   

minkowski_distance, p = 2.4
   ::,,,::   
 .,-~^^^~-,. 
 ,~=+***+=~, 
:-=*OxxxO*=-:
:~+O8fff8O+~:
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
:~+O8fff8O+~:
:-=*OxxxO*=-:
 ,~=+***+=~, 
 .,-~^^^~-,. 
   ::,,,::   

minkowski_distance, p = 2.6
  .:,,,,,:.  
 .,~~^^^~~,. 
.,~=+***+=~,.
:~=*OxxxO*=~:
,~+O8fff8O+~,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,~+O8fff8O+~,
:~=*OxxxO*=~:
.,~=+***+=~,.
 .,~~^^^~~,. 
  .:,,,,,:.  

minkowski_distance, p = 2.8
  .:,,,,,:.  
 :-~^^^^^~-: 
.-^=+***+=^-.
:~=*OxxxO*=~:
,^+O8fff8O+^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^+O8fff8O+^,
:~=*OxxxO*=~:
.-^=+***+=^-.
 :-~^^^^^~-: 
  .:,,,,,:.  

minkowski_distance, p = 3.0
  .:,,,,,:.  
 :-~^^^^^~-: 
.-^+*****+^-.
:~+oOxxxOo+~:
,^*O8fff8O*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*O8fff8O*^,
:~+oOxxxOo+~:
.-^+*****+^-.
 :-~^^^^^~-: 
  .:,,,,,:.  

minkowski_distance, p = 3.2
  ::,,,,,::  
 :-~^^^^^~-: 
:-^+*****+^-:
:~+oOxxxOo+~:
,^*O$fff$O*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*O$fff$O*^,
:~+oOxxxOo+~:
:-^+*****+^-:
 :-~^^^^^~-: 
  ::,,,,,::  

minkowski_distance, p = 3.4
 .:,,,,,,,:. 
.,-~^^^^^~-,.
:-^+*****+^-:
,~+oOxxxOo+~,
,^*O$fff$O*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*O$fff$O*^,
,~+oOxxxOo+~,
:-^+*****+^-:
.,-~^^^^^~-,.
 .:,,,,,,,:. 

minkowski_distance, p = 3.6
 .:,,,,,,,:. 
.,-~^^^^^~-,.
:-^+*****+^-:
,~+oOxxxOo+~,
,^*O$fff$O*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*O$fff$O*^,
,~+oOxxxOo+~,
:-^+*****+^-:
.,-~^^^^^~-,.
 .:,,,,,,,:. 

minkowski_distance, p = 3.8
 .:,,,,,,,:. 
.,~~^^^^^~~,.
:~=+*****+=~:
,~+oxxxxxo+~,
,^*x$fff$x*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*x$fff$x*^,
,~+oxxxxxo+~,
:~=+*****+=~:
.,~~^^^^^~~,.
 .:,,,,,,,:. 

minkowski_distance, p = 4.0
 .:,,,,,,,:. 
.,~^^^^^^^~,.
:~=+*****+=~:
,^+oxxxxxo+^,
,^*x$fff$x*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*x$fff$x*^,
,^+oxxxxxo+^,
:~=+*****+=~:
.,~^^^^^^^~,.
 .:,,,,,,,:. 

minkowski_distance, p = 4.2
 .:,,,,,,,:. 
.,~^^^^^^^~,.
:~=+*****+=~:
,^+oxxxxxo+^,
,^*x$fff$x*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*x$fff$x*^,
,^+oxxxxxo+^,
:~=+*****+=~:
.,~^^^^^^^~,.
 .:,,,,,,,:. 

minkowski_distance, p = 4.4
 .:,,,,,,,:. 
.,~^^^^^^^~,.
:~=+*****+=~:
,^+oxxxxxo+^,
,^*x$fff$x*^,
,^*xfBHBfx*^,
,^*xfH#Hfx*^,
,^*xfBHBfx*^,
,^*x$fff$x*^,
,^+oxxxxxo+^,
:~=+*****+=~:
.,~^^^^^^^~,.
 .:,,,,,,,:. 

minkowski_distance, p = 4.6
 ::,,,,,,,:: 
:-~^^^^^^^~-:
:~=+*****+=~:
,^+OxxxxxO+^,
,^*x$fff$x*^,
,^*xfHHHfx*^,
,^*xfH#Hfx*^,
,^*xfHHHfx*^,
,^*x$fff$x*^,
,^+OxxxxxO+^,
:~=+*****+=~:
:-~^^^^^^^~-:
 ::,,,,,,,:: 

minkowski_distance, p = 4.8
 ::,,,,,,,:: 
:-~^^^^^^^~-:
:~=+*****+=~:
,^+OxxxxxO+^,
,^*x$fff$x*^,
,^*xfHHHfx*^,
,^*xfH#Hfx*^,
,^*xfHHHfx*^,
,^*x$fff$x*^,
,^+OxxxxxO+^,
:~=+*****+=~:
:-~^^^^^^^~-:
 ::,,,,,,,::