Stereographic and cylindrical map projections

Examples inspired by the thread at comp.text.tex about how to convert some hand drawn pictures into programmatic 3D sketches.

The sketches present stereographic and cylindrical map projections and they pose some interesting challenges for doing them with a 2D drawing package PGF/TikZ.

The main idea is to draw in selected 3D planes and then project onto the canvas coordinate system with an appriopriate transformation. Some highlights:

• usage of pgf math engine for calculation of projection transformations and transitions points from visible (solid lines) to invisible (dashed lines) on meridians and latitude circles
• definition of 3D plane transformation with expanded styles so that they are robust against redefinition of macros used in their construction
• usage of named coordinates (nodes) for definition of characteristic points in local coordinate systems so that they are accessible outside of their plane of definition
• calculation of intersections points with TikZ intersection coordinate system
• usage of 'to' path operation instead of 'arc' for marking angles to allow for easy positioning of text labels on the curve
• 3D lighting effects with shading

Edit and compile if you like:

% Stereographic and cylindrical map projections
% Author: Tomasz M. Trzeciak
% Source: LaTeX-Community.org 
%         
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}


%% helper macros

\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{2.5} % sphere radius
\def\angEl{35} % elevation angle
\filldraw[ball color=white] (0,0) circle (\R);
\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} }
\foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} }

\end{tikzpicture}

\begin{tikzpicture} % CENT

%% some definitions

\def\R{2.5} % sphere radius
\def\angEl{35} % elevation angle
\def\angAz{-105} % azimuth angle
\def\angPhi{-40} % longitude of point P
\def\angBeta{19} % latitude of point P

%% working planes

\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\tikzset{xyplane/.style={cm={cos(\angAz),sin(\angAz)*sin(\angEl),-sin(\angAz),
                              cos(\angAz)*sin(\angEl),(0,-\H)}}}
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhi}
\LatitudePlane[equator]{\angEl}{0}

%% draw xyplane and sphere

\draw[xyplane] (-2*\R,-2*\R) rectangle (2.2*\R,2.8*\R);
\fill[ball color=white] (0,0) circle (\R); % 3D lighting effect
\draw (0,0) circle (\R);

%% characteristic points

\coordinate (O) at (0,0);
\coordinate[mark coordinate] (N) at (0,\H);
\coordinate[mark coordinate] (S) at (0,-\H);
\path[pzplane] (\angBeta:\R) coordinate[mark coordinate] (P);
\path[pzplane] (\R,0) coordinate (PE);
\path[xzplane] (\R,0) coordinate (XE);
\path (PE) ++(0,-\H) coordinate (Paux); % to aid Phat calculation
\coordinate[mark coordinate] (Phat) at (intersection cs: first line={(N)--(P)},
                                        second line={(S)--(Paux)});

%% draw meridians and latitude circles

\DrawLatitudeCircle[\R]{0} % equator
%\DrawLatitudeCircle[\R]{\angBeta}
\DrawLongitudeCircle[\R]{\angAz} % xzplane
\DrawLongitudeCircle[\R]{\angAz+90} % yzplane
\DrawLongitudeCircle[\R]{\angPhi} % pzplane

%% draw xyz coordinate system

\draw[xyplane,<->] (1.8*\R,0) node[below] {$x,\xi$} -- (0,0) -- (0,2.4*\R)
    node[right] {$y,\eta$};
\draw[->] (0,-\H) -- (0,1.6*\R) node[above] {$z,\zeta$};

%% draw lines and put labels

\draw[dashed] (P) -- (N) +(0.3ex,0.6ex) node[above left] {$\mathbf{N}$};
\draw (P) -- (Phat) node[above right] {$\mathbf{\hat{P}}$};
\path (S) +(0.4ex,-0.4ex) node[below] {$\mathbf{S}$};
\draw[->] (O) -- (P) node[above right] {$\mathbf{P}$};
\draw[dashed] (XE) -- (O) -- (PE);
\draw[pzplane,->,thin] (0:0.5*\R) to[bend right=15]
    node[pos=0.4,right] {$\beta$} (\angBeta:0.5*\R);
\draw[equator,->,thin] (\angAz:0.4*\R) to[bend right=30]
    node[pos=0.4,below] {$\phi$} (\angPhi:0.4*\R);
\draw[thin,decorate,decoration={brace,raise=0.5pt,amplitude=1ex}] (N) -- (O)
    node[midway,right=1ex] {$a$};

\end{tikzpicture}

\begin{tikzpicture} % MERC

%% some definitions

\def\R{3} % sphere radius
\def\angEl{25} % elevation angle
\def\angAz{-100} % azimuth angle
\def\angPhiOne{-50} % longitude of point P
\def\angPhiTwo{-35} % longitude of point Q
\def\angBeta{33} % latitude of point P and Q

%% working planes

\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhiOne}
\LongitudePlane[qzplane]{\angEl}{\angPhiTwo}
\LatitudePlane[equator]{\angEl}{0}

%% draw background sphere

\fill[ball color=white] (0,0) circle (\R); % 3D lighting effect
%\fill[white] (0,0) circle (\R); % just a white circle
\draw (0,0) circle (\R);

%% characteristic points

\coordinate (O) at (0,0);
\coordinate[mark coordinate] (N) at (0,\H);
\coordinate[mark coordinate] (S) at (0,-\H);
\path[xzplane] (\R,0) coordinate (XE);
\path[pzplane] (\angBeta:\R) coordinate (P);
\path[pzplane] (\R,0) coordinate (PE);
\path[qzplane] (\angBeta:\R) coordinate (Q);
\path[qzplane] (\R,0) coordinate (QE);

%% meridians and latitude circles

% \DrawLongitudeCircle[\R]{\angAz} % xzplane
% \DrawLongitudeCircle[\R]{\angAz+90} % yzplane
\DrawLongitudeCircle[\R]{\angPhiOne} % pzplane
\DrawLongitudeCircle[\R]{\angPhiTwo} % qzplane
\DrawLatitudeCircle[\R]{\angBeta}
\DrawLatitudeCircle[\R]{0} % equator

% shifted equator in node with nested call to tikz 
% (I didn't know it's possible)
\node at (0,1.6*\R) { \tikz{\DrawLatitudeCircle[\R]{0}} };

%% draw lines and put labels

\draw (-\R,-\H) -- (-\R,2*\R) (\R,-\H) -- (\R,2*\R);
\draw[->] (XE) -- +(0,2*\R) node[above] {$y$};
\node[above=8pt] at (N) {$\mathbf{N}$};
\node[below=8pt] at (S) {$\mathbf{S}$};
\draw[->] (O) -- (P);
\draw[dashed] (XE) -- (O) -- (PE);
\draw[dashed] (O) -- (QE);
\draw[pzplane,->,thin] (0:0.5*\R) to[bend right=15]
    node[midway,right] {$\beta$} (\angBeta:0.5*\R);
\path[pzplane] (0.5*\angBeta:\R) node[right] {$\hat{1}$};
\path[qzplane] (0.5*\angBeta:\R) node[right] {$\hat{2}$};
\draw[equator,->,thin] (\angAz:0.5*\R) to[bend right=30]
    node[pos=0.4,above] {$\phi_1$} (\angPhiOne:0.5*\R);
\draw[equator,->,thin] (\angAz:0.6*\R) to[bend right=35]
    node[midway,below] {$\phi_2$} (\angPhiTwo:0.6*\R);
\draw[equator,->] (-90:\R) arc (-90:-70:\R) node[below=0.3ex] {$x = a\phi$};
\path[xzplane] (0:\R) node[below] {$\beta=0$};
\path[xzplane] (\angBeta:\R) node[below left] {$\beta=\beta_0$};

\end{tikzpicture}


\begin{tikzpicture} % KART

\def\R{2.5}

\node[draw,minimum size=2cm*\R,inner sep=0,outer sep=0,circle] (C) at (0,0) {};
\coordinate (O) at (0,0);
\coordinate[mark coordinate] (Phat) at (20:2.5*\R);
\coordinate (T1) at (tangent cs: node=C, point={(Phat)}, solution=1);
\coordinate (T2) at (tangent cs: node=C, point={(Phat)}, solution=2);
\coordinate[mark coordinate] (P) at ($(T1)!0.5!(T2)$);

\draw[dashed] (T1) -- (O) -- (T2) -- (Phat) -- (T1) -- (T2);
\draw[<->] (0,1.5*\R) node[above] {$y$} |- (2.5*\R,0) node[right] {$x$};
\draw (O) node[below left] {$\mathbf{O}$} -- (P)
    +(1ex,0) node[above=1ex] {$\mathbf{P}$};
\draw (P) -- (Phat) node[above=1ex] {$\mathbf{\hat{P}}$};

\end{tikzpicture}

\end{document}