How to draw spherical geometries with TeX?I am working in the field of Astrodynamics and have to do many computations using spherical geometry. To visualize some of the concepts I would like to prepare drawings of things like spherical triangles, planes around the earth, or circles projected onto the sphere. Here are some examples (1):
How is this possible to do in TeX? I had a look whether this could be done with TikZ but could not find any comparable examples. I am convinced that in theory it should be possible to prepare a TikZ extension that does things like this rather simply, but at the moment I am looking for an already existing solution to accomplish such tasks. When I get more experienced with making such plots I might consider writing a basic package.
The question is thus actually threefold:
Please note that this is not the kind of question you sometimes get here that asks "Please do this for me!" I honestly tried solving this problem but did not get far at all. It would be really great to get an answer, a lot of people in my field would benefit from this.
(1) Wertz, James R. (2009). Orbit & Constellation Design & Management. New York: Springer.
This is a long answer since there are good tools for spherical geometry scattered all around, so I created a few sections addressing those tools.
tikz-3dplot:especially tdplotdrawarc
I suggest to use \tdplotdrawarc . This is explained in the TikZ and PGF Manual. You need to define three angles $\alpha$, $\beta$ and $\gamma$ for the arc, theen the radius, origin, initial and final angle. I include here and example with the angles used. With this example you can build new examples explaining other angle combinations.
\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage{tikz-3dplot}
\usepackage{hyperref}
\usepackage{pgfplots}
\usetikzlibrary{calc,3d,intersections, positioning,intersections,shapes}
\newcommand{\InterSec}[3]{%
\path[name intersections={of=#1 and #2, by=#3, sort by=#1,total=\t}]
\pgfextra{\xdef\InterNb{\t}}; }
\begin{document}
\begin{center}
\begin{tikzpicture}[scale=2]
\pgfmathsetmacro\R{sqrt(3)}
\fill[ball color=white!10, opacity=0.1] (0,0,0) circle (\R); % 3D lighting effect
\tdplotsetmaincoords{80}{110}
\begin{scope}[tdplot_main_coords, shift={(0,0)}]
\coordinate (O) at (0,0,0);
% circle around Cp
% rotate circle to make it look better.
\pgfmathsetmacro{\thetavec}{0}
\pgfmathsetmacro{\phivec}{0}
\tdplotsetrotatedcoords{\phivec}{\thetavec}{0}
\tdplotdrawarc[tdplot_rotated_coords,color=blue]{(O)}{\R}{-70}{110}{}{}
\tdplotdrawarc[tdplot_rotated_coords,color=blue, dashed]{(O)}{\R}{110}{290}{}{}
\node[] at (-1,2,1) {\textcolor{blue}{\scriptsize
$\alpha=\thetavec \, , \, $\beta=\phivec}};
\pgfmathsetmacro{\thetavec}{90};
\tdplotsetrotatedcoords{\phivec}{\thetavec}{0};
\tdplotdrawarc[tdplot_rotated_coords,color=brown]{(O)}{\R}{0}{180}{}{};
\tdplotdrawarc[tdplot_rotated_coords,color=brown, dashed]{(O)}{\R}{180}{360}{}{};
\node[yshift=4 mm] at (-1,2,1) {\textcolor{brown}{\scriptsize $\alpha=\thetavec \, , \,
$\beta=\phivec}};
\pgfmathsetmacro{\phivec}{90}
\tdplotsetrotatedcoords{\phivec}{\thetavec}{0};
\tdplotdrawarc[tdplot_rotated_coords,color=red]{(O)}{\R}{0}{180}{}{};
\tdplotdrawarc[tdplot_rotated_coords,color=red, dashed]{(O)}{\R}{180}{360}{}{};
\node[yshift=8 mm] at (-1,2,1) {\textcolor{red}{\scriptsize $\alpha=\thetavec \, , \,
$\beta=\phivec}};
%axis
\coordinate (X) at (5,0,0) ;
\coordinate (Y) at (0,3,0) ;
\coordinate (Z) at (0,0,3) ;
\draw[-latex] (O) -- (X) node[anchor=west] {$X$};
\draw[-latex] (O) -- (Y) node[anchor=west] {$Y$};
\draw[-latex] (O) -- (Z) node[anchor=west] {$Z$};
\end{scope}
\end{tikzpicture}
\end{center}
\end{document}
The corresponding figure is:
Here is a post which addresses the drawing an equator when the north pole is given. A simple macro to speed up coding draw an equator when north pole is known [1] .
Fake 2D, intersections package, and the instruction to [bend right], to [bend left]
Sometimes is better to say away from thinking and trying to do 3D. So I am contradicting here myself with the advise of using tikz-3dplot . Think how to draw a 3D thinking 2D (that is ellipses and arcs).
The next example is an improvement over an example shown here Spherical triangles and great circles [2] . The code is based on @Tarass great insight. The example is shown here more to show the capabilities of Tikz and the use of it for other purposes. As I said, it is better to use, in general \tdplotdrawarc .
Here is the piece of code (copied and modified from @Tarass code)
\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage{xcolor}
\usepackage{tikz-3dplot}
\usepackage{hyperref}
\usepackage{pgfplots}
\usetikzlibrary{calc,3d,intersections, positioning,intersections,shapes}
\pgfplotsset{compat=1.11}
\newcommand{\InterSec}[3]{%
\path[name intersections={of=#1 and #2, by=#3, sort by=#1,total=\t}]
\pgfextra{\xdef\InterNb{\t}}; }
\begin{document}
\begin{center}
\begin{tikzpicture}
\pgfmathsetmacro\R{2}
\fill[ball color=white!10, opacity=0.2] (0,0,0) circle (\R); % 3D lighting effect
\foreach \angle[count=\n from 1] in {-5,225,290} {
\begin{scope}[rotate=\angle]
\path[draw,dashed,name path global=d\n] (2,0) arc [start angle=0,
end angle=180,
x radius=2cm,
y radius=1cm] ;
\path[draw,name path global=s\n] (-2,0) arc [start angle=180,
end angle=360,
x radius=2cm,
y radius=1cm] ;
\end{scope}
}
\InterSec{s1}{s2}{I3} ;
\InterSec{s1}{s3}{I2} ;
\InterSec{s3}{s2}{I1} ;
%
\fill[fill=red,opacity=0.5] (I1) to [bend right=8.5] (I2) to [bend left=7]
(I3) to [bend left=6] (I1);
\InterSec{d1}{d2}{J3} ;
\InterSec{d1}{d3}{J2} ;
\InterSec{d3}{d2}{J1} ;
%\fill[blue] (J1)--(J2)--(J3)--cycle ;
\fill[fill=blue,opacity=0.5] (J1) to [bend right=8.5] (J2) to [bend left=7]
(J3) to [bend left=6] (J1);
\end{tikzpicture}
\end{center}
\end{document}
and here the picture.
Drawing lunes could be hard sometimes. I first refer to an StackExchange link with a problem with drawing lunes and solutions both in metapost and TiKz. The link is: How to draw a lune and shade it in TiKz [3]
I offer here another figure showing the duality between a segment and its lune. In this particular example I combine 3D and 2D, so back into suggesting the use of tikz-3dplot: the code is next:
\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc,3d,decorations.markings, backgrounds, positioning,intersections,shapes}
\usepackage{pgfplots}
\newcommand{\InterSec}[3]{%
\path[name intersections={of=#1 and #2, by=#3, sort by=#1,total=\t}]
\pgfextra{\xdef\InterNb{\t}};
}
\newcommand\getEquator[2]
{
\def\yt{#1}
\def\zt{#2}
\pgfmathsetmacro{\betav}{acos(\zt)};
\def\gammav{0}
\ifthenelse{\equal{\betav}{0.0}}
{
\def\alphav{0}
}
{
\pgfmathsetmacro{\alphav}{asin(\yt/(sin(\betav))}
};
}
% to color a line
\tikzset{test/.style={
postaction={
decorate,
decoration={
markings,
mark=at position \pgfdecoratedpathlength-0.5pt with
{\arrow[blue,line width=#1] {>}; },
mark=between positions 0 and \pgfdecoratedpathlength step 0.5pt with {
\pgfmathsetmacro\myval{multiply(divide(
\pgfkeysvalueof{/pgf/decoration/mark info/distance from start},
\pgfdecoratedpathlength),100)};
\pgfsetfillcolor{blue!\myval!green};
\pgfpathcircle{\pgfpointorigin}{#1};
\pgfusepath{fill};}
}
}
}
}
\begin{document}
\begin{tikzpicture}[scale=1.3]
\coordinate (O) at (0,0,0);
\tdplotsetmaincoords{60}{110}
\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};
\def\angA{90}
\def\angB{60}
\pgfmathsetmacro{\ax}{cos(\angA)}
\pgfmathsetmacro{\ay}{sin(\angA)}
\pgfmathsetmacro{\z}{0}
\pgfmathsetmacro{\bx}{cos(\angB)}
\pgfmathsetmacro{\by}{sin(\angB)}
\pgfmathsetmacro{\aax}{\R*cos(\angA)}
\pgfmathsetmacro{\aay}{\R*sin(\angA)}
\pgfmathsetmacro{\bbx}{\R*cos(\angB)}
\pgfmathsetmacro{\bby}{\R*sin(\angB)}
\coordinate (A) at (\aax,\aay,\z);
\coordinate (B) at (\bbx,\bby,\z);
\getEquator{\ay}{\z};
\tdplotsetrotatedcoords{\alphav}{\betav}{\gammav};
\tdplotdrawarc[tdplot_rotated_coords,color=green, name path global=GF, opacity=0]
{(0,0)}{\R}{180}{360}{}{};
\tdplotdrawarc[tdplot_rotated_coords,color=green, name path global=GB, opacity=0]
{(0,0)}{\R}{0}{180}{}{};
\tdplotdrawarc[tdplot_rotated_coords,color=yellow, name path=YB, opacity=0]
{(0,0)}{\R}{90}{180}{}{};
\getEquator{\by}{\z};
\tdplotsetrotatedcoords{\alphav}{\betav}{\gammav};
\tdplotdrawarc[tdplot_rotated_coords,color=blue, name path=BF, opacity=0]
{(0,0)}{\R}{180}{360}{}{};
\tdplotdrawarc[tdplot_rotated_coords,color=blue, name path=BB, opacity=0]
{(0,0)}{\R}{0}{180}{}{};
\tdplotdrawarc[tdplot_rotated_coords,color=red, name path=RB, opacity=0]
{(0,0)}{\R}{90}{180}{}{};
%\draw[color=red] (A) arc (\angA:\angB:\R);
\draw[test=0.2mm] (A) arc (\angA:\angB:\R);
\InterSec{GF}{BF}{F};
\InterSec{GB}{BB}{B};
\InterSec{C}{GF}{CG};
\InterSec{C}{BF}{CB};
\InterSec{C}{RB}{RC};
\InterSec{GB}{RB}{RBF};
\InterSec{YB}{C}{T};
%\draw[] (F) circle (1pt) node[] {\; \; \tiny F};
%\draw[] (CG) circle (1pt) node[] {\tiny CG};
%\draw[] (CB) circle (1pt) node[] {\tiny CB};
%\draw[] (B) circle (1pt) node[] {\tiny B};
%\draw[] (RBF) circle (1pt) node[] {\; \; \tiny RBF};
%\draw[] (T) circle (1pt) node[] {\tiny T};
%\draw[] (RC) circle (1pt) node[] {\tiny RC};
%axis
\coordinate (X) at (4,0,0) ;
\coordinate (Y) at (0,3,0) ;
\coordinate (Z) at (0,0,3) ;
\draw[-latex] (O) -- (X) node[anchor=east] {\; \; $X$};
\draw[-latex] (O) -- (Y) node[anchor=north] {$Y$};
\draw[-latex] (O) -- (Z) node[anchor=south west] {$Z$};
\shade[left color=blue, right color=green, opacity=0.8] (F) to [bend right=50] (CB) to
[bend right=10] (CG) to [bend left] (F);
\shade[left color=blue, right color=green, opacity=0.3] (CB) to [bend right=10] (CG) to
[bend right] (B) to [bend left] (CB);
\shade[left color=green, right color=blue, opacity=0.3] (B) to [bend right=60] (RC) to
[bend right=10] (RBF) to [bend left ] (B);
\shade[left color=green, right color=blue, opacity=0.8] (F) to [bend left=10] (RC) to
[bend right=10] (T) to [bend right] (F);
\end{scope}
\end{tikzpicture}
\end{document}
and the figure is here:
Conversion of Coordinates and alternatives for drawing arcs
In spherical geometry understanding where coordinates (a point) are and how to draw arcs is a fundamental issue.
There could be confusion because spherical coordinates for mathematicians and phycisist use different symbols, the following link provides macros for conversion between spherical (azimuth, polar) and cartesian coordinates and addresses conversions in terms of geografic (latidue, altitude) coordinates as well: spherical coordinates in 3d [4] .
Finally since TiKz do not seem to have tools to draw arcs given a center and a radius I wrote a macro and posted here [5] .
[1] https://tex.stackexchange.com/questions/277810/draw-an-ecuator-circle-given-a-unit-3d-vector-the-north-pole\InterSec, \tdplotsetrotatedcoords etc., not to mention the tikzpicture environment.... - cfr
The R package GeoMap will create Spherical project of the earth with the continent maps. I have not used it except to verify it loads and builds a map. If you combine with the package tikzDevice you will get tikz code which could be modified. Be aware that it will be a large file due to the extensive use of points for plotting.
Once this is working you should be able to implement with Sweave so that all the code is contained within the LaTeX file.
I would consider this just a workaround until a tikz package was built with pure tikz.
pgfplotssuch as\documentclass{article} \usepackage{pgfplots} \begin{document} \begin{tikzpicture} \begin{axis} \addplot3[mesh, samples=20, domain=0:2*pi,y domain=0:pi, ] ({cos(deg(x)) * sin(deg(y))}, {sin(deg(x)) * sin(deg(y))}, {cos(deg(y))}); \end{axis} \end{tikzpicture} \end{document}not sure about the other bits though :) - cmhughes25vote mark! Does anyone have any news? Sadly I could not find much time to work on this issue and resorted tomayavifor such purposes, which admittedly is a much less elegant solution. - Ingo