38

I'm trying to adapt this example from texample to produce a sphere with arrows indicating the effect of a elliptic Moebius map.

In fact, I'm trying to redraw the spheres below, but for a while I'm working only with two of them.

enter image description here

Using the file below I got this:

enter image description here

But I don't know how to decrease the length of the arrow and how to move it to other circles. I tried to change the angles but no success. Helps are welcome.

Here is the code

% Stereographic and cylindrical map projections
% Author: Tomasz M. Trzeciak
% Source: LaTeX-Community.org 
%         <http://www.latex-community.org/viewtopic.php?f=4&t=2111>
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,fadings,decorations.pathreplacing}
\usepackage{verbatim}

%% 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/.estyle={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/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,dashed}}
   % 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,line width=.7pt}}
  \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[red,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}[scale=1.4] % "THE GLOBE" showcase

\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{0} % latitude of point P
%
\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} }
%
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\draw (N)  node[inner sep=1pt,below=0pt] {$\infty$}; %+(0.3ex,0.6ex)
%
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhi}
\LatitudePlane[equator]{\angEl}{0}
\draw[red,equator,->,thick] (\angAz:\R) to[bend right=30] (\angPhi:\R);
\end{tikzpicture}

\end{document}
Sigur
  • 37,330

1 Answers1

24

Changing the size of the arrows means changing the angular separation between the tip and tail of the arrow, since they appear in polar form. After reducing the angular separation, you also have to reduce the "bending" of the arrow for it to smear in the equator:

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

%% 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/.estyle={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/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,dashed}}
   % 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][2]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,line width=.7pt}}
  \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[red,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}[scale=1.4] % "THE GLOBE" showcase

\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{0} % latitude of point P
%
\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} }
%
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\draw (N)  node[inner sep=1pt,below=0pt] {$\infty$}; %+(0.3ex,0.6ex)
%
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhi}
\LatitudePlane[equator]{\angEl}{0}
\draw[red,equator,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\end{tikzpicture}

\end{document}

enter image description here

To draw the vector field over other latitudes one can define another planes at other latitudes, just as the original code defines "equator". 

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

%% 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/.estyle={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/.estyle={cm={\cost,0,0,\cost*\sinEl,(0,\yshift)}}} %
}
\newcommand\DrawLongitudeCircle[2][4]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,dashed}}
   % 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][5]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1,line width=.7pt}}
  \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[gray,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}[scale=1.4] % "THE GLOBE" showcase

\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{0} % latitude of point P
%
\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} }
%
\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
\coordinate[mark coordinate] (N) at (0,\H);
\draw (N)  node[inner sep=1pt,below=0pt] {$\infty$}; %+(0.3ex,0.6ex)
%
\LongitudePlane[xzplane]{\angEl}{\angAz}
\LongitudePlane[pzplane]{\angEl}{\angPhi}
\LatitudePlane[equator]{\angEl}{0}
\LatitudePlane[capr]{\angEl}{37}
\LatitudePlane[sag]{\angEl}{68}
\draw[red,equator,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[green,sag,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[blue,capr,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\end{tikzpicture}

\end{document}

The result is the following (you can change the colors, I just display them in order for the arrows to be more trackable) :enter image description here

To add more arrows on the same circle you can use this code

\draw[red,equator,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[red,equator,->,thick] (-135:\R) to[bend right=11] (-110:\R);
\draw[red,equator,->,thick] (-75:\R) to[bend right=11] (-50:\R);
\draw[red,equator,->,thick] (-40:\R) to[bend right=8] (-25:\R);
\draw[blue,capr,->,thick] (\angAz:\R) to[bend right=11] (-80:\R);
\draw[blue,capr,->,thick] (-135:2.6) to[bend right=11] (-110:\R);
\draw[blue,capr,->,thick] (-75:\R) to[bend right=11] (-50:2.6);
\draw[blue,capr,->,thick] (-40:2.65) to[bend right=8] (-25:2.7);

enter image description here

c.p.
  • 4,636
  • Thanks. Great! The main problem is to move the arrow to other circles. – Sigur Jan 20 '13 at 15:12
  • @Sigur the length of the arrows needs some tweaking, but I think you can identify which is which in the code, and do a trial and error process to make them have the length you wish. – c.p. Jan 20 '13 at 15:25
  • Nice! The only new code is the definition of the other two planes parallels to equator? I didn't see where you changed the length of the new arrows. Or it is not necessary, it changes automatically? – Sigur Jan 20 '13 at 15:32
  • 1
    May be it's possible to do that automatically, but I don't think its simple. I did it manually: e.g. the original code's large red arrow was draw[red,equator,->,thick] (\angAz:\R) to[bend right=30] (\angPhi:\R) which I changed into \draw[red,equator,->,thick] (\angAz:\R) to[bend right=11] (-80:\R); (here \angAz is and \angPhi are specific values for angles, which you can change manually) and for another arrow on the same circle you keep equator in the code and just change both angles: \draw[red,equator,->,thick] (-40:\R) to[bend right=8] (-25:\R); – c.p. Jan 20 '13 at 15:40
  • To draw arrows on another circles you use the macro and put some names on them (I baptized them sag and capr) and if you replace equator with one of the new planes, you can draw the arrows on them. – c.p. Jan 20 '13 at 15:45
  • let us continue this discussion in chat – c.p. Jan 20 '13 at 16:48