6

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.

Latitude and longitude

\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 ?

8

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

enter image description here

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}

enter image description here

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}

enter image description here

  • 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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