Latitude and longitude coordinates on a sphere using tikz

Question

I am trying from Stereographic and cylindrical map projections by Tomas M. Trzeciak to reproduce the two spheres below. However, I am having hard time to understand correctly the code and so far, I was at least able to obtain the paralleles and meridians.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\usepackage{verbatim}

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}
\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}

%% document-wide tikz options and styles

\tikzset{%
  >=latex, % option for nice arrows
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}

\begin{document}

\begin{tikzpicture} % "THE GLOBE" showcase
\def\R{4 } % sphere radius
\def\angEl{20} % elevation angle
\def\angAz{-20} % azimuth angle
\filldraw[ball color=white] (0,0) circle (\R);
\filldraw[fill=white] (0,0) circle (\R);

\foreach \t in {0,30} { \DrawLatitudeCircle[\R]{\t} }
\foreach \t in {-120} { \DrawLongitudeCircle[\R]{\t} }

\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate (O) at (0,0);
\node[circle,draw,black,scale=0.3] at (0,0) {};
\draw[left] node at (0,0){O};
\coordinate[mark coordinate] (N) at (0,\H);
\draw[left] node at (0,\H){N};
\coordinate[mark coordinate] (S) at (0,-\H);
\draw[left] node at (0,-\H){S};
\draw[thick, dashed, black](N)--(S);
\end{tikzpicture}

\end{document}

Would you help me to get correctly the points and draw the corresponding pictures ?

Indeed, I have looked at it before but the trouble was about the code. — tcsd1, Dec 19 '17 at 13:08

marmot · Accepted Answer · 2017-12-19 15:44:05Z

UPDATE I encountered some minor problems with the above, otherwise fantastic macros as well as those under Henri Menkes link. The main issue is that I find that the yshift is only correct for a sphere radius of 1. In what follows, I present 3 codes, the last of which might be the most convenient. Needless to say that they are at best minor amendments to the fantastic routines by Alain Matthes.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\usepackage{verbatim}

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}

\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\NewLatitudePlane[4][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#3} % elevation
  \pgfmathsinandcos\sint\cost{#4} % latitude
  \pgfmathsetmacro\yshift{#2*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}

%% document-wide tikz options and styles

\tikzset{%
  >=latex, % option for nice arrows
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}

\begin{document}

\begin{tikzpicture} % "THE GLOBE" showcase
\def\R{4 } % sphere radius
\def\angEl{20} % elevation angle
\def\angAz{-20} % azimuth angle
\filldraw[ball color=white] (0,0) circle (\R);
\filldraw[fill=white] (0,0) circle (\R);

\foreach \t in {0,30} { \DrawLatitudeCircle[\R]{\t} }
\foreach \t in {-120} { \DrawLongitudeCircle[\R]{\t} }


\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate (O) at (0,0);
\node[circle,draw,black,scale=0.3] at (0,0) {};
\draw[right] node at (0,0){O};
\coordinate[mark coordinate] (N) at (0,\H);
\draw[left] node at (0,\H){N};
\coordinate[mark coordinate] (S) at (0,-\H);
\draw[left] node at (0,-\H){S};
\draw[thick, dashed, black](N)--(S);

\NewLatitudePlane[planeP]{\R}{\angEl}{30};
\path[planeP] (-120:\R) coordinate (P);
\draw[left] node at (P){$P$};

\NewLatitudePlane[equator]{\R}{\angEl}{00};
\path[equator] (-120:\R) coordinate (Pprime);
\draw[left] node at (Pprime){$P'$};

\draw[-,dashed] (O)--(P);
\draw[-,dashed] (O)--(Pprime);

\LongitudePlane[angle]{\angEl}{-120};
\draw[angle,-] (0:1) arc (0:30:1);

\end{tikzpicture}

\end{document}

The only thing I did is to introduce a command \NewLatitudePlane, which differs from \LatitudePlane in that it also takes a radius. This is necessary in order to compute the correct yshift. Then you can define longitude planes and latitude planes and draw all the features within them. The above code yields

I introduced two latitude planes, one at 30 degrees called planeP, in which I put the point P at -120 degrees longitude, and an equator plane, in which P' is placed at -120 degrees. Then I defined a longitudinal plane for drawing the angle. I did not spend any energy in placing P and P' more nicely.

For the other picture, you may start from

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\usepackage{verbatim}

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}

\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\NewLatitudePlane[4][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#3} % elevation
  \pgfmathsinandcos\sint\cost{#4} % latitude
  \pgfmathsetmacro\yshift{#2*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}

%% document-wide tikz options and styles

\tikzset{%
  >=latex, % option for nice arrows
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}

\begin{document}

\begin{tikzpicture} % "THE GLOBE" showcase
\def\R{4 } % sphere radius
\def\angEl{20} % elevation angle
\def\angAz{-20} % azimuth angle
\filldraw[ball color=white] (0,0) circle (\R);
\filldraw[fill=white] (0,0) circle (\R);



\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate (O) at (0,0);
\node[circle,draw,black,scale=0.3] at (0,0) {};
\draw[right] node at (0,0){O};
\coordinate[mark coordinate] (N) at (0,\H);
\draw[left] node at (0,\H){N};
\coordinate[mark coordinate] (S) at (0,-\H);
\draw[left] node at (0,-\H){S};
\draw[thick, dashed, black](N)--(S);

\tikzset{
    every path/.style={
        color=green
    }
}
\DrawLatitudeCircle[\R]{0}
\tikzset{
    every path/.style={
        color=black
    }
}


\LongitudePlane[angle]{\angEl}{-80};
\draw[angle,-,red] (-70:\R) arc (-70:90:\R); % note : -70 could also be computed! 
\draw[angle,-,red,dashed] (-90:\R) arc (-90:-70:\R); % note : -70 could also be computed! 
\path[angle] (00:\R) coordinate (Pprime);
\draw[right] node at (Pprime){$P'$};


\LongitudePlane[angel]{\angEl}{-120};
\draw[angel,-,blue] (-70:\R) arc (-70:90:\R); % note : -70 could also be computed! 
\draw[angel,-,blue,dashed] (-90:\R) arc (-90:-70:\R); % note : -70 could also be computed! 
\path[angel] (00:\R) coordinate (Oprime);
\draw[left] node at (Oprime){$O'$};

\def\arcrad{2}
\NewLatitudePlane[equator]{\R}{\angEl}{00};
\draw[equator,-,red,dashed] (-120:\arcrad) arc (-120:-80:\arcrad);
\path[equator] (-120:\arcrad) coordinate (m);
\draw[left] node at (m){$m$};
\path[equator] (-80:\arcrad) coordinate (mprime);
\draw[right] node at (mprime){$m'$};

\draw[-,dashed] (Oprime) -- (O) -- (Pprime);

\end{tikzpicture}

\end{document}

Here is a version that supports latitude and longitude arcs.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing,shadings}
\usepackage{verbatim}

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}
\newcommand\LongitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % azimuth
  \tikzset{#1/.style={cm={\cost,\sint*\sinEl,0,\cosEl,(0,0)}}}
}

\newcommand\LatitudePlane[3][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#2} % elevation
  \pgfmathsinandcos\sint\cost{#3} % latitude
  \pgfmathsetmacro\yshift{\RadiusSphere*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\NewLatitudePlane[4][current plane]{%
  \pgfmathsinandcos\sinEl\cosEl{#3} % elevation
  \pgfmathsinandcos\sint\cost{#4} % latitude
  \pgfmathsetmacro\yshift{#2*\cosEl*\sint}
  \tikzset{#1/.style={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,opacity=0.4] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}
\newcommand\DrawLongitudeArc[4][black]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \pgfmathsetmacro\angA{mod(max(\angVis,#3),360)} %
  \pgfmathsetmacro\angB{mod(min(\angVis+180,#4),360} %
  \draw[current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \draw[current plane,#1]  (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}%
\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,opacity=0.4] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}

\newcommand\DrawLatitudeArc[4][black]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \pgfmathsetmacro\angA{max(min(\angVis,#3),-\angVis-180)} %
  \pgfmathsetmacro\angB{min(\angVis,#4)} %
  \draw[current plane,#1,opacity=0.4] (#3:\RadiusSphere) arc (#3:#4:\RadiusSphere);
  \draw[current plane,#1] (\angA:\RadiusSphere) arc (\angA:\angB:\RadiusSphere);
}

%% document-wide tikz options and styles

\tikzset{%
  >=latex, % option for nice arrows
  inner sep=0pt,%
  outer sep=2pt,%
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt,
    fill=black,circle}%
}

\begin{document}

\begin{tikzpicture} % "THE GLOBE" showcase
\def\RadiusSphere{4} % sphere radius
\def\angEl{20} % elevation angle
\def\angAz{-20} % azimuth angle

\shade[ball color = gray!40, opacity = 0.5] (0,0) circle (\RadiusSphere);

\pgfmathsetmacro\H{\RadiusSphere*cos(\angEl)} % distance to north pole
\coordinate (O) at (0,0);
\node[circle,draw,black,scale=0.3] at (0,0) {};
\draw[right] node at (0,0){O};
\coordinate[mark coordinate] (N) at (0,\H);
\draw[left] node at (0,\H){N};
\coordinate[mark coordinate] (S) at (0,-\H);
\draw[left] node at (0,-\H){S};
\draw[thick, dashed, black](N)--(S);

\tikzset{
    every path/.style={
        color=green!50!black
    }
}
\DrawLatitudeCircle[\RadiusSphere]{0}
\tikzset{
    every path/.style={
        color=black
    }
}


\DrawLatitudeArc[blue]{30}{-90}{90}

\DrawLatitudeArc[blue]{20}{-200}{20}



\LongitudePlane[angle]{\angEl}{-80};
\DrawLongitudeArc[red]{-80}{-90}{90}
\path[angle] (00:\RadiusSphere) coordinate (Pprime);
\draw[right] node at (Pprime){$P'$};


\LongitudePlane[angel]{\angEl}{-120};
\DrawLongitudeArc[blue]{-120}{-90}{90}
\path[angel] (00:\RadiusSphere) coordinate (Oprime);
\draw[left] node at (Oprime){$O'$};

\def\arcrad{2}
\NewLatitudePlane[equator]{\RadiusSphere}{\angEl}{00};
\draw[equator,-,red,dashed] (-120:\arcrad) arc (-120:-80:\arcrad);
\path[equator] (-120:\arcrad) coordinate (m);
\draw[left] node at (m){$m$};
\path[equator] (-80:\arcrad) coordinate (mprime);
\draw[right] node at (mprime){$m'$};

\draw[-,dashed] (Oprime) -- (O) -- (Pprime);

\end{tikzpicture}

\end{document}

Thanks a lot @marmot for your example and explanations ! I understand better the way it works. Exactly what I was looking for. — tcsd1, Dec 19 '17 at 13:12

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