%%% on January 19, 2026
%%%
\tikzmath{%
  function hyperplane(\varx, \vary, \varz) {%
    return \coefHx*\varx +\coefHy*\vary +\coefHz*\varz +\coefH;
  };
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%% General functions for the construction of the trace of a surface
%%%% of revolution in a half space determined by H>0
\tikzmath{%
  function PPNorm(\k, \j, \i) {%
    return norm3d(\PPx{\k,\j,\i}, \PPy{\k,\j,\i}, \PPz{\k,\j,\i});
  };
  function nFaceHx(\l1, \m1, \ru1, \l2, \m2, \ru2) {%
    return \PPy{\l1,\m1,\ru1}*\PPz{\l2,\m2,\ru2}
    -\PPz{\l1,\m1,\ru1}*\PPy{\l2,\m2,\ru2};
  };
  function nFaceHy(\l1, \m1, \ru1, \l2, \m2, \ru2) {%
    return \PPz{\l1,\m1,\ru1}*\PPx{\l2,\m2,\ru2}
    -\PPx{\l1,\m1,\ru1}*\PPz{\l2,\m2,\ru2};
  };
  function nFaceHz(\l1, \m1, \ru1, \l2, \m2, \ru2) {%
    return \PPx{\l1,\m1,\ru1}*\PPy{\l2,\m2,\ru2}
    -\PPy{\l1,\m1,\ru1}*\PPx{\l2,\m2,\ru2};
  };
}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Code elements
\tikzset{%
  points and H position/.code={%  used in |sor H|
    % Creates the list \positionH which takes values -4, 0, 1
    % depending on the position of the point \P with respect to H.
    \tikzmath{%
      integer \positionH;
      \coefHx = \ux*\aVar1 +\vx*\aVar2 +\wx*\aVar3;
      \coefHy = \uy*\aVar1 +\vy*\aVar2 +\wy*\aVar3;
      \coefHz = \uz*\aVar1 +\vz*\aVar2 +\wz*\aVar3;
      for \j in {0, ..., \NParallels}{%
        for \k in {0, ..., \NMeridians}{%
          \val = hyperplane(\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j});
          if \val >= 0 then {%
            if \val < \nearZero then {%
              \positionH{\k,\j} = 0;
            } else {%
              \positionH{\k,\j} = 1;
            };
          } else {%
            if \val > -\nearZero then {%
              \positionH{\k,\j} = 0;
            } else {%
              \positionH{\k,\j} = -4;
            };
          };
        };
      };
    }
  },
  new points on meridians and H/.code={% 
    % Points are costructed if there are two different proper vertices
    % with repect to H on a meridian.  Each time, the new point takes
    % the indices of the vertex that lies in the negative half-space
    % with a third position which equals either 2 or 4, depending on
    % the order of the two vertices along the meridian (2 if -4 goes
    % forward to become 0 and 4 if backward).
    %
    % Uses the list \positionH defined in |points and H position|.
    \tikzmath{% 
      integer \prevPosition, \currentPosition;
      % along every meridian
      for \k in {0, ..., \NMeridians}{%
        \prevPosition = \positionH{\k,0};
        for \j in {1, ..., \NParallels}{%
          \currentPosition = \positionH{\k,\j};
          if \prevPosition*\currentPosition <= 0 then {% <>0 and one =0
            if \prevPosition +\currentPosition < -1 then {% <0 and >=0
              \prevj = \j -1;
              \val0 = hyperplane(\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj});
              \val1 = hyperplane(\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j});
              \uTmp = (\Px{\k,\prevj}*\val1 -\Px{\k,\j}*\val0)/(\val1 -\val0);
              \vTmp = (\Py{\k,\prevj}*\val1 -\Py{\k,\j}*\val0)/(\val1 -\val0);
              \wTmp = (\Pz{\k,\prevj}*\val1 -\Pz{\k,\j}*\val0)/(\val1 -\val0);
              if \currentPosition == -4 then {%
                \Px{\k,\j,4} = \uTmp;
                \Py{\k,\j,4} = \vTmp;
                \Pz{\k,\j,4} = \wTmp;
              } else {
                \Px{\k,\prevj,2} = \uTmp;
                \Py{\k,\prevj,2} = \vTmp;
                \Pz{\k,\prevj,2} = \wTmp;
              };
            };
          };
          \prevPosition = \positionH{\k,\j};
        };
      };
    }
  },
  new points on parallels and H/.code={%
    % Points are costructed if there are two different proper vertices
    % with repect to H on a parallel.  Each time, the new point takes
    % the indices of the vertex that lies in the negative half-space
    % with a third position which equals either 1 or 3, depending on
    % the order of the two vertices along a parallel (1 if -4 goes
    % forward to its destiny...).
    %
    % Uses the list \positionH defined in |points and H position|.
    \tikzmath{%
      integer \prevPosition, \currentPosition;
      % along every parallel
      for \j in {0, ..., \NParallels}{%
        \prevPosition = \positionH{0,\j};  % initialize \prevPosition
        for \k in {1, ..., \NMeridians}{%
          \currentPosition = \positionH{\k,\j};
          if \prevPosition *\currentPosition <= 0 then {% <>0 and one =0
            if \prevPosition +\currentPosition < -1 then {% <0 and >=0
              \prevk = int(Mod(\k -1, \NMeridians));
              \val0 = hyperplane(\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j});
              \val1 = hyperplane(\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j});
              \uTmp = (\Px{\prevk,\j}*\val1 -\Px{\k,\j}*\val0)/(\val1 -\val0);
              \vTmp = (\Py{\prevk,\j}*\val1 -\Py{\k,\j}*\val0)/(\val1 -\val0);
              \wTmp = (\Pz{\prevk,\j}*\val1 -\Pz{\k,\j}*\val0)/(\val1 -\val0);
              if \currentPosition == -4 then {% 
                \Px{\k,\j,3} = \uTmp;
                \Py{\k,\j,3} = \vTmp;
                \Pz{\k,\j,3} = \wTmp;
              } else {
                \Px{\prevk,\j,1} = \uTmp;
                \Py{\prevk,\j,1} = \vTmp;
                \Pz{\prevk,\j,1} = \wTmp;
              };
            };
          };
          \prevPosition = \positionH{\k,\j};
        };
      };
    }
  },
  drawing weights/.code={%
    % Construct drawning weights, \drawWeight, for each
    % quadrilateral.  Each point has a weight, \positionH, with
    % respect to $H$'s equation,  $1$, $0$, or $-4$.
    % The drawing weight is the sum of these weights.
    % Starts the construction of |\toBeDrawn|.
    \tikzmath{%
      integer \drawWeightTmp;
      for \j in {1, ..., \NParallels}{%
        \prevj = \j -1;
        for \k in {0, ..., \NMeridians}{%
          \prevk = int(Mod(\k -1, \NMeridians));
          \drawWeightTmp = \positionH{\prevk,\prevj} +\positionH{\k,\prevj}
          +\positionH{\k,\j} +\positionH{\prevk,\j};
          \drawWeight{\k,\j} = int(\drawWeightTmp);
        };
      };
    }
  },
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Main style and pic objects
\tikzset{%
  sor and H/.style args={name=#1,
    meridians=#2, parallels=#3,
    t_ini=#4, t_end=#5,
    rAxis=#6, cHyperplane=#7}{%
    evaluate={%
      real \s, \t, \ipFTo, \ipFTo;
      real \cax, \sax, \cay, \say, \caz, \saz;
      real \ux, \uy, \uz, \vx, \vy, \vz, \wx, \wy, \wz;
      real \coefHx, \coefHy, \coefHz, \coefH;
      integer \NMeridians, \NParallels;
      integer \j, \k, \i, \prevj, \prevk;
      \NMeridians = #2;
      \NParallels = #3;
      \initialT = #4;
      \finalT = #5;
      \i = 0; 
      for \tmp in {#6}{%  angles that characterise the axis of
                     %  rotation's position; in fact the new basis
                     %  obtained by the compositon R_\ay R_\az R_\ax.
        \i = \i +1;
        \aVar\i = \tmp;
      };
      \cax = cos(\aVar1);
      \sax = sin(\aVar1);
      \cay = cos(\aVar2);
      \say = sin(\aVar2);
      \caz = cos(\aVar3);
      \saz = sin(\aVar3);
      \i = 0; 
      for \tmp in {#7}{%  coefficients definig the hyperplane
        \i = \i +1;
        \aVar\i = \tmp;
      };
      \coefHx = \aVar1;
      \coefHy = \aVar2;
      \coefHz = \aVar3;
      \coefH = \aVar4;
    },
    new basis,
    the points={#1},
    points and H position,  % Transforms also the coefficients of H
                            % according to the new basis.
    drawing weights,
    new points on parallels and H,
    new points on meridians and H,
    face normal vector and ips
  },
  pics/rSurface and H/.style args={above=#1, below=#2, actions=#3}{% !!
    % #1 = fg layer
    % #2 = bg layer
    % #3 = actions
    code={% 
      \tikzmath{%  
        integer \j, \k, \prevk, \prevj, \layerI;
      }
      %% faces
      \foreach \k [evaluate=\k as \prevk using int(\k -1)]
      in {1, ..., \NMeridians}{% 
        \foreach \j [evaluate=\j as \prevj using int(\j -1)]
        in {1, ..., \NParallels}{%
          \tikzmath{%
            \layerI = whatLayer(\ipFTo{\k,\j}, #1, #2);
          }
          \ifnum \drawWeight{\k,\j} > -1%  no negative vertex
            \begin{pgfonlayer}{\layerI}
              \path[., #3]
              (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
              -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
              -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
              -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
              -- cycle;
            \end{pgfonlayer}
          \else%  at least one negative vertex 
            \ifnum \drawWeight{\k,\j} > -5%  one negative vertex
              \oneNegativeVertex{#3}{0}%  1=in light
            \else%  at least two negative vertices
              \ifnum \drawWeight{\k,\j} > -9 %  two negative vertices
                \twoNegativeVertices{#3}{0}
              \else%  three negative vertices
                \ifnum \drawWeight{\k,\j} > -13%  three negative vertices
                  \threeNegativeVertices{#3}{0}
                \fi
              \fi
            \fi
          \fi
        }
      }
    }
  },
  pics/rSurface and HL/.style args={above=#1, below=#2, actions=#3}{%
    % #1 = fg layer
    % #2 = bg layer
    % #3 = actions (drawn must be provided if full opacity
    code={% 
      \setColorsForRSurface
      \tikzmath{%  
        integer \j, \k, \prevk, \prevj, \layerI, \rgbCst, \li;
      }
      %% faces
      \foreach \k [evaluate=\k as \prevk using int(\k -1)]
      in {1, ..., \NMeridians}{% 
        \foreach \j [evaluate=\j as \prevj using int(\j -1)]
        in {1, ..., \NParallels}{%
          \tikzmath{%
            \layerI = whatLayer(\ipFTo{\k,\j}, #1, #2);
            \li = rgbLeftIndex(\ipFTo{\k,\j}, \ipFSun{\k,\j});
            \rgbCst = rgbConstant(\ipFTo{\k,\j}, \ipFSun{\k,\j});
          }
          \ifnum \drawWeight{\k,\j} > -1%  no negative vertex
            \begin{pgfonlayer}{\layerI}
              \path[with L={1}, #3]
              (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
              -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
              -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
              -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
              -- cycle;
            \end{pgfonlayer}
          \else%  at least one negative vertex 
            \ifnum \drawWeight{\k,\j} > -5%  one negative vertex
              \oneNegativeVertex{#3}{1}%  1=in light
            \else%  at least two negative vertices
              \ifnum \drawWeight{\k,\j} > -9 %  two negative vertices
                \twoNegativeVertices{#3}{1}
              \else%  three negative vertices
                \ifnum \drawWeight{\k,\j} > -13%  three negative vertices
                  \threeNegativeVertices{#3}{1}
                \fi
              \fi
            \fi
          \fi
        }
      }
    }
  },
  pics/partial rSurface and H/.style args={above=#1, below=#2,
    actions=#3, i from=#4, i to=#5}{%
    % #1 = fg layer
    % #2 = bg layer
    % #3 = actions 
    % #4 = index of the first meridian
    % #5 = index of the last meridian 
    code={%
      \tikzmath{%  
        integer \j, \k, \prevk, \prevj, \layerI, \rgbCst, \kini, \kend;
        if #4 < #5 then {%
          \kini = int(#4 +1);
          \kend = #5;
        } else {%
          \kini = #4;
          \kend = int(#5 +1);          
        };
      }
      %% faces
      \foreach \k [evaluate=\k as \prevk using {int(Mod(\k -1, \NMeridians))}]
      in {\kini, ..., \kend}{% 
        \foreach \j [evaluate=\j as \prevj using int(\j -1)]
        in {1, ..., \NParallels}{%
          \tikzmath{%
            \layerI = whatLayer(\ipFTo{\k,\j}, #1, #2);
          }
          \ifnum \drawWeight{\k,\j} > -1%  no negative vertex
            \begin{pgfonlayer}{\layerI}
              \path[., #3]
              (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
              -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
              -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
              -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
              -- cycle;
            \end{pgfonlayer}
          \else%  at least one negative vertex 
            \ifnum \drawWeight{\k,\j} > -5%  one negative vertex
              \oneNegativeVertex{#3}{0}%  1=in light
            \else%  at least two negative vertices
              \ifnum \drawWeight{\k,\j} > -9 %  two negative vertices
                \twoNegativeVertices{#3}{0}
              \else%  three negative vertices
                \ifnum \drawWeight{\k,\j} > -13%  three negative vertices
                  \threeNegativeVertices{#3}{0}
                \fi
              \fi
            \fi
          \fi
        }
      }  
    }
  },
  pics/partial rSurface and HL/.style args={above=#1, below=#2,
    actions=#3, i from=#4, i to=#5}{%
    % #1 = fg layer
    % #2 = bg layer
    % #3 = actions 
    % #4 = index of the first meridian
    % #5 = index of the last meridian 
    code={%
      \setColorsForRSurface
      \tikzmath{%  
        integer \j, \k, \prevk, \prevj, \layerI, \rgbCst, \kini, \kend;
        if #4 < #5 then {%
          \kini = int(#4 +1);
          \kend = #5;
        } else {%
          \kini = #4;
          \kend = int(#5 +1);          
        };
      }
      %% faces
      \foreach \k [evaluate=\k as \prevk using {int(Mod(\k -1, \NMeridians))}]
      in {\kini, ..., \kend}{%
        %\ifnum\k<\NMeridians+1
        %\tikzmath{%
        %  \k = int(Mod(\k, \NMeridians));
        %}
        \foreach \j [evaluate=\j as \prevj using int(\j -1)]
        in {1, ..., \NParallels}{%
          \tikzmath{%
            \k = int(Mod(\k, \NMeridians));
            if \k==0 then {%
              \k = \NMeridians;
            };
            \layerI = whatLayer(\ipFTo{\k,\j}, #1, #2);
            \li = rgbLeftIndex(\ipFTo{\k,\j}, \ipFSun{\k,\j});
            \rgbCst = rgbConstant(\ipFTo{\k,\j}, \ipFSun{\k,\j});
          }
          \ifnum \drawWeight{\k,\j} > -1%  no negative vertex
            \begin{pgfonlayer}{\layerI}
              \path[with L=1, #3]
              (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
              -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
              -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
              -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
              -- cycle;
            \end{pgfonlayer}
          \else%  at least one negative vertex 
            \ifnum \drawWeight{\k,\j} > -5%  one negative vertex
              \oneNegativeVertex{#3}{1}%  1=in light
            \else%  at least two negative vertices
              \ifnum \drawWeight{\k,\j} > -9 %  two negative vertices
                \twoNegativeVertices{#3}{1}
              \else%  three negative vertices
                \ifnum \drawWeight{\k,\j} > -13%  three negative vertices
                  \threeNegativeVertices{#3}{1}
                \fi
              \fi
            \fi
          \fi
        }
        %\fi
      }  
    }
  },
  with L/.code={% #1 = 0 no light, #1 = 1 light is on, i.e. for HL
    \ifnum#1=0
      \tikzset{.}
    \else
      \tikzset{fill, rgbLeft{\li}!\rgbCst!rgbRight, line width=.01pt}
    \fi    
  }
}

\newcommand{\oneNegativeVertex}[2]{% 
  \ifnum \positionH{\prevk,\prevj}<0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\prevk,\prevj,1}, \Py{\prevk,\prevj,1}, \Pz{\prevk,\prevj,1})
      -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
      -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
      -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
      -- (\Px{\prevk,\prevj,2}, \Py{\prevk,\prevj,2}, \Pz{\prevk,\prevj,2})
      -- cycle;
    \end{pgfonlayer}
  \fi
  \ifnum \positionH{\k,\prevj}<0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
      -- (\Px{\k,\prevj,3}, \Py{\k,\prevj,3}, \Pz{\k,\prevj,3})
      -- (\Px{\k,\prevj,2}, \Py{\k,\prevj,2}, \Pz{\k,\prevj,2})
      -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
      -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
      -- cycle;
    \end{pgfonlayer}
  \fi
  \ifnum \positionH{\k,\j}<0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
      -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
      -- (\Px{\k,\j,4}, \Py{\k,\j,4}, \Pz{\k,\j,4})
      -- (\Px{\k,\j,3}, \Py{\k,\j,3}, \Pz{\k,\j,3})
      -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
      -- cycle;
    \end{pgfonlayer}
  \fi
  \ifnum \positionH{\prevk,\j}<0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
      -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
      -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
      -- (\Px{\prevk,\j,1}, \Py{\prevk,\j,1}, \Pz{\prevk,\j,1})
      -- (\Px{\prevk,\j,4}, \Py{\prevk,\j,4}, \Pz{\prevk,\j,4})
      -- cycle;
    \end{pgfonlayer}
  \fi
}
\newcommand{\twoNegativeVertices}[2]{%
  \ifnum \positionH{\prevk,\prevj}<0%
    \ifnum \positionH{\k,\prevj}<0%
      \begin{pgfonlayer}{\layerI}
        \path[with L=#2, #1]
        (\Px{\prevk,\prevj,2}, \Py{\prevk,\prevj,2}, \Pz{\prevk,\prevj,2})
        -- (\Px{\k,\prevj,2}, \Py{\k,\prevj,2}, \Pz{\k,\prevj,2})
        -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
        -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
        -- cycle;
      \end{pgfonlayer}
    \fi
  \fi
  \ifnum \positionH{\k,\prevj}<0%
    \ifnum \positionH{\k,\j}<0%
      \begin{pgfonlayer}{\layerI}
        \path[with L=#2, #1]
        (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
        -- (\Px{\k,\prevj,3}, \Py{\k,\prevj,3}, \Pz{\k,\prevj,3})
        -- (\Px{\k,\j,3}, \Py{\k,\j,3}, \Pz{\k,\j,3})
        -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
        -- cycle;
      \end{pgfonlayer}
    \fi
  \fi
  \ifnum \positionH{\k,\j}<0%
    \ifnum \positionH{\prevk,\j}<0%
      \begin{pgfonlayer}{\layerI}
        \path[with L=#2, #1]
        (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
        -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
        -- (\Px{\k,\j,4}, \Py{\k,\j,4}, \Pz{\k,\j,4})
        -- (\Px{\prevk,\j,4}, \Py{\prevk,\j,4}, \Pz{\prevk,\j,4})
        -- cycle;
      \end{pgfonlayer}
    \fi
  \fi
  \ifnum \positionH{\prevk,\j}<0%
    \ifnum \positionH{\prevk,\prevj}<0%
      \begin{pgfonlayer}{\layerI}
        \path[with L=#2, #1]
        (\Px{\prevk,\prevj,1}, \Py{\prevk,\prevj,1}, \Pz{\prevk,\prevj,1})
        -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
        -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
        -- (\Px{\prevk,\j,1}, \Py{\prevk,\j,1}, \Pz{\prevk,\j,1})
        -- cycle;
      \end{pgfonlayer}
    \fi
  \fi
}
\newcommand{\threeNegativeVertices}[2]{%
  \ifnum \positionH{\prevk,\prevj}>0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\prevk,\prevj}, \Py{\prevk,\prevj}, \Pz{\prevk,\prevj})
      -- (\Px{\k,\prevj,3}, \Py{\k,\prevj,3}, \Pz{\k,\prevj,3})
      -- (\Px{\prevk,\j,4}, \Py{\prevk,\j,4}, \Pz{\prevk,\j,4})
      -- cycle;
    \end{pgfonlayer}
  \fi
  \ifnum \positionH{\k,\prevj}>0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\prevk,\prevj,1}, \Py{\prevk,\prevj,1}, \Pz{\prevk,\prevj,1})
      -- (\Px{\k,\prevj}, \Py{\k,\prevj}, \Pz{\k,\prevj})
      -- (\Px{\k,\j,4}, \Py{\k,\j,4}, \Pz{\k,\j,4})
      -- cycle;
    \end{pgfonlayer}
  \fi
  \ifnum \positionH{\k,\j}>0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\k,\prevj,2}, \Py{\k,\prevj,2}, \Pz{\k,\prevj,2})
      -- (\Px{\k,\j}, \Py{\k,\j}, \Pz{\k,\j})
      -- (\Px{\prevk,\j,1}, \Py{\prevk,\j,1}, \Pz{\prevk,\j,1})
      -- cycle;
    \end{pgfonlayer}
  \fi
  \ifnum \positionH{\prevk,\j}>0%
    \begin{pgfonlayer}{\layerI}
      \path[with L=#2, #1]
      (\Px{\prevk,\prevj,2}, \Py{\prevk,\prevj,2}, \Pz{\prevk,\prevj,2})
      -- (\Px{\k,\j,3}, \Py{\k,\j,3}, \Pz{\k,\j,3})
      -- (\Px{\prevk,\j}, \Py{\prevk,\j}, \Pz{\prevk,\j})
      -- cycle;
    \end{pgfonlayer}
  \fi
}