spherical coordinates in tikz 3d

Question

Can we specify direction in spherical coordinates?

I know we can do polar (angle:radius) but what is we are using tikz-3d and want to specify (r, theta, phi) where theta is the azimuthal angle?

Answer 1

TikZ provides with the apparently undocumented library 3d a xyz spherical coordinate system.

It accepts the keys radius (now fixed), angle = longitude, latitude and with my help rho and theta.

The first TikZ picture shows my example, the second a PGF picture example of the TikZ/PGF manual.

(I removed the 3d library again and implemented the xyz spherical similar to how it is done in tikzlibrary3d.code.tex. It simply uses the \pgfpointspherical macro, which does all the calculations and uses the appropriate vectors.)

Code

\documentclass[tikz,convert=false]{standalone}
%\usetikzlibrary{3d}
\makeatletter
\pgfqkeys{/tikz/cs}{
  latitude/.store in=\tikz@cs@latitude,% not needed with '3d' library
  longitude/.style={angle={#1}},% not needed with '3d' library
  theta/.style={latitude={#1}},
  rho/.style={angle={#1}}
}
\tikzdeclarecoordinatesystem{xyz spherical}{% needed even with '3d' library!
  \pgfqkeys{/tikz/cs}{angle=0,radius=0,latitude=0,#1}%
  \pgfpointspherical{\tikz@cs@angle}{\tikz@cs@latitude}{\tikz@cs@xradius}% fix \tikz@cs@radius to \tikz@cs@xradius
}
\makeatother

\tikzset{my color/.code=\pgfmathparse{(#1+90)/180*100}\pgfkeysalso{every path/.style={color=red!\pgfmathresult!blue}}}
\begin{document}
\begin{tikzpicture}[radius=+0.4pt]% (this is the radius of little dots on the lines)
\foreach \lat in {-90,-80,...,90} {
 \tikzset{my color=\lat}
 \foreach \lon in {0,10,...,359} {
  \filldraw (xyz spherical cs: radius=1, angle=\lon,    latitude=\lat) circle[]
         -- (xyz spherical cs: radius=1, angle=\lon+10, latitude=\lat);
 }}
\end{tikzpicture}

\begin{tikzpicture}
\foreach \lat in {-90,-75,...,30}
  \filldraw[line join=round, fill=lightgray]
    \foreach \lon in {0,20,...,359} {
         (xyz spherical cs: radius=1, rho=\lon,    theta=\lat   )
      -- (xyz spherical cs: radius=1, rho=\lon+20, theta=\lat   )
      -- (xyz spherical cs: radius=1, rho=\lon+20, theta=\lat+15)
      -- (xyz spherical cs: radius=1, rho=\lon,    theta=\lat+15)
      -- cycle
    };
\end{tikzpicture}
\end{document}

Output

Answer 2

I believe this is an interesting question.

First, I want to point out that the notation for physicist is not the same as for mathematicians. What physicist call theta (ϴ), mathematicians call phi (ϕ) and vice-versa. I refer the reader to the Wikipedia website for the conventions used. The macro is simple and I include it next with the convention asked here.

\newcommand{\sphToCart}[3]
        {
          \def\rpar{#1}
          \def\thetapar{#2}
          \def\phipar{#3}

          \pgfmathsetmacro{\x}{\rpar*sin(\phipar)*cos(\thetapar)}
          \pgfmathsetmacro{\y}{\rpar*sin(\phipar)*sin(\thetapar)}
          \pgfmathsetmacro{\z}{\rpar*cos(\phipar)}
        }

Here is complete example where we use this macro several times to create a spherical triangle.

\documentclass[12pt]{article}
\usepackage{pgfplots}
\usepackage{tikz}
\usepackage{tikz-qtree}
\usepackage{tkz-berge}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc,3d,decorations.markings, backgrounds, positioning,intersections,shapes}


 \newcommand{\sphToCart}[3]
        {
          \def\rpar{#1}
          \def\thetapar{#2}
          \def\phipar{#3}

          \pgfmathsetmacro{\x}{\rpar*sin(\phipar)*cos(\thetapar)}
          \pgfmathsetmacro{\y}{\rpar*sin(\phipar)*sin(\thetapar)}
          \pgfmathsetmacro{\z}{\rpar*cos(\phipar)}
        }


\begin{document}

\begin{tikzpicture}[scale=1.3]
  \coordinate (O) at (0,0,0);

  \tdplotsetmaincoords{60}{135}
  \pgfmathsetmacro\R{sqrt(3)} 
  \fill[ball color=white!10, opacity=0.2, name path global=C] (O) 
      circle (\R); % 3D lighting effect
  \begin{scope}[tdplot_main_coords, shift={(0,0)}]
    \pgfmathsetmacro\R{sqrt(3)} 
    \pgfmathsetmacro{\thetavec}{0};
    \pgfmathsetmacro{\phivec}{0};
    \pgfmathsetmacro{\gammav}{0};
    \tdplotsetrotatedcoords{\phivec}{\thetavec}{\gammav};


    % draw point with azimuth -20 degrees, polar angle 90
    \def\thetaA{-20}
    \def\phiA{90}
    \sphToCart{\R}{\thetaA}{\phiA}
    \coordinate (A) at (\x,\y,\z);

    % save legend location
    \pgfmathsetmacro{\dx}{\x+1.2};
    \pgfmathsetmacro{\dy}{\y+0.9};
    \pgfmathsetmacro{\dz}{\z-1.0};


    \node[] at (\dx,\dy,\dz) {Point $A:( r=\R, \theta=\thetaA, \phi=\phiA)$};
    \node[yshift=-5mm, xshift=6mm] at (\dx,\dy,\dz) 
        { $( x=\x, y=\y, z=\z)$};


    \def\thetaA{110}
    \def\phiA{90}
    \sphToCart{\R}{\thetaA}{\phiA}
    \coordinate (B) at (\x,\y,\z);

    % save legend location  (relative to this point)
    \pgfmathsetmacro{\dx}{\x-1.2};
    \pgfmathsetmacro{\dy}{\y+2.5};
    \pgfmathsetmacro{\dz}{\z-1.0};

    \node[] at (\dx,\dy,\dz) {Point $B:( r=\R, \theta=\thetaA, \phi=\phiA)$};
    \node[yshift=-5mm, xshift=6mm] at (\dx,\dy,\dz) 
        { $( x=\x, y=\y, z=\z)$};


    \def\thetaA{70}
    \def\phiA{-20}
    \sphToCart{\R}{\thetaA}{\phiA}
    \coordinate (C) at (\x,\y,\z);


    % save legend location  (relative to this point)
    \pgfmathsetmacro{\dx}{\x-2};
    \pgfmathsetmacro{\dy}{\y+3};
    \pgfmathsetmacro{\dz}{\z+1.0};

    \node[] at (\dx,\dy,\dz) {Point $C:( r=\R, \theta=\thetaA, \phi=\phiA)$};
    \node[yshift=-5mm, xshift=6mm] at (\dx,\dy,\dz) 
        { $( x=\x, y=\y, z=\z)$};


    \draw[fill=red, opacity=0.4] (A) to [bend right] (B) 
        to [bend right] (C) to [bend right]  (A);






    \draw[-latex, color=red, line width=1] (O)--(A) node[anchor=east] {\tiny $A$};
    \draw[-latex, color=red, line width=1] (O)--(B) node[anchor=west] {\tiny $B$};
    \draw[-latex, color=red, line width=1] (O)--(C) node[anchor=south] {\tiny $C$};

    %legend


    % axis
    \coordinate (XX) at (3,0,0) ;
    \coordinate (YY) at (0,3,0) ;
    \coordinate (ZZ) at (0,0,3) ;


    \draw[-latex] (O) -- (XX) node[anchor=east] {$X$};
    \draw[-latex] (O) -- (YY) node[anchor=north] {$Y$};
    \draw[-latex] (O) -- (ZZ) node[anchor=south] {$Z$};

  \end{scope}

\end{tikzpicture}

\end{document}

The figure is next:

Please observe that I included legends showing the coordinates in both systems: Spherical and Cartesians.