%%%%
%%%% y and z functions, i.e. the curve

%% sphere
%
\pgfkeys{/tikz/.cd,
  sphereRadius/.store in=\sphRadius,
  sphereRadius=1
}
\tikzmath{%
  function ySphere(\tvar) {% \tvar in [-90, 90]
    return \sphRadius*sin(\tvar);
  };
  function zSphere(\tvar) {%
    return \sphRadius*cos(\tvar);
  };
}

%% cone
%
\pgfkeys{/tikz/.cd,
  baseRadius/.store in=\baseRadius,
  baseRadius=1
}
\pgfkeys{/tikz/.cd,
  topRadius/.store in=\topRadius,
  topRadius=1
}
\pgfkeys{/tikz/.cd,
  height/.store in=\height,
  height=3
}
\tikzmath{%
   function yCone(\tvar) {%
    return \tvar*\height;
  };
  function zCone(\tvar) {%
    return \baseRadius +\tvar*(\topRadius -\baseRadius);
  };
}

%% hyperboloid1 and 2
%
\pgfkeys{/tikz/.cd,
  z half axis/.store in=\zHalfAxis,
  z half axis=1
}
\pgfkeys{/tikz/.cd,
  y half axis/.store in=\yHalfAxis,
  y half axis=1
}
\tikzmath{%
  function yHyperboloid1(\tvar) {% \tvar in [-y_max, y_max]
    return \tvar;
  };
  function zHyperboloid1(\tvar) {%
    return \zHalfAxis*pow(1 +pow(\tvar/\yHalfAxis, 2), .5);
  };
}
\tikzmath{%
  function yHyperboloid2(\tvar) {%
    return \yHalfAxis*pow(1 +pow(\tvar/\zHalfAxis, 2), .5);
  };
  function zHyperboloid2(\tvar) {% \tvar in [0, z_max]
    return \tvar;
  };
}

%
%% ellipsoid
\tikzmath{%
  function yEllipsoid(\tvar) {% \tvar in [-90, 90]
    return \yHalfAxis*sin(\tvar);
  };
  function zEllipsoid(\tvar) {%
    return \zHalfAxis*cos(\tvar);
  };
}

%
%% torus
\pgfkeys{/tikz/.cd,
  tMinorRadius/.store in=\tMinorRadius,
  tMinorRadius=1.6
}
\pgfkeys{/tikz/.cd,
  tMajorRadius/.store in=\tMajorRadius,
  tMajorRadius=3
}
\tikzmath{%
  function yTorus(\tvar) {% \tvar in [0, 1]
    return \tMinorRadius*sin(\tvar);
  };
  function zTorus(\tvar) {%
    return \tMajorRadius +\tMinorRadius*cos(\tvar);
  };
}



% %%%% Show coordinates pic object
% %% It needs layers (above and under <main>).
% \tikzset{%
%   pics/showCoordinates/.style n args={4}{%
%     %  #1 = color
%     %  #2 = length
%     %  #3 = S layer (foreground) for the arrow
%     %  #4 = U layer (background) for the arrow (behind main layer)
%     code={%
%       \draw[#1, thin] (0, 0) circle (.5);
%       \tikzmath{%
%         real \aConstant;
%         \aConstant = #2;
%       }
%       \begin{pgfonlayer}{main}
%         \draw[#1, ultra thick] (0, 0, 0) -- (\aConstant, 0, 0)
%         node[pos=1.25, scale=.7, text=K] {$x$};
%         \draw[#1, ultra thick] (0, 0, 0) -- (0, \aConstant, 0)
%         node[pos=1.25, scale=.7] {$y$};
%         \draw[#1, ultra thick] (0, 0, 0) -- (0, 0, \aConstant)
%         node[pos=1.25, scale=.7] {$z$};
%       \end{pgfonlayer}
%       % 
%       \begin{scope}[baseRadius=\aConstant/12, topRadius=0,
%         height=\aConstant/10,
%         rotating surface={yCurve=yCone, zCurve=zCone,
%           meridians=36, parallels=2, t_ini=0, t_end=1,
%           rAxis={90, 90}}]
%         \draw (\aConstant, 0, 0) pic[#1] {rSurface SU={#3}{#4}};
%       \end{scope}
%       \begin{scope}[baseRadius=\aConstant/12, topRadius=0,
%         height=\aConstant/10,
%         rotating surface={yCurve=yCone, zCurve=zCone,
%           meridians=36, parallels=2, t_ini=0, t_end=1,
%           rAxis={0, 0}}]
%         \draw (0, \aConstant, 0) pic[#1] {rSurface SU={#3}{#4}};
%       \end{scope}
%       \begin{scope}[baseRadius=\aConstant/12, topRadius=0,
%         height=\aConstant/10,
%         rotating surface={yCurve=yCone, zCurve=zCone,
%           meridians=36, parallels=2, t_ini=0, t_end=1,
%           rAxis={0, 90}}]
%         \draw (0, 0, \aConstant) pic[#1] {rSurface SU={#3}{#4}};
%       \end{scope}
%     }
%   }
% }
% \newcommand{\freeShowCoordinates}[6][]{%
%   % #1 = color (optional argument)
%   % #2 = length
%   % #3 = viewer's longitude
%   % #4 = viewer's latitude
%   % #5 = sun's longitude
%   % #6 = sun's latitude
%   \ifthenelse{\equal{#1}{}}{\colorlet{tmpRGB}{O}}{\colorlet{tmpRGB}{#1}}
%   \tikzexternalenable
%   \begin{tikzpicture}[view={#3}{#4}, sun={#5}{#6}{.5}]
%     \pgfdeclarelayer{-1}
%     \pgfdeclarelayer{1}
%     \pgfsetlayers{-1, main, 1}
%     \path (0, 0) pic {showCoordinates={tmpRGB}{#2}{1}{-1}};
%   \end{tikzpicture}
%   \tikzexternaldisable
% }