Drawing functions on spheres

Question

In a paper, I have a class of functions S¹→S² depending on a real parameter t>0, and I would like to represent some of them on the sphere as curves to illustrate their behavior as t goes to 0, but I never used TikZ or anything else, so I have no clue how to do so.

Answer 1

I learned a bunch from figuring this out. So thanks for asking! Here’s the nearly-MWE I came up with for pgfplotset:

\PassOptionsToPackage{svgnames}{xcolor}
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8x]{inputenc}
\usepackage{mathtools}
\usepackage[Symbolsmallscale]{upgreek}
\usepackage{textcomp}
\usepackage{pgfplots}

\pgfplotsset{width=\textwidth,compat=1.12}
\pgfplotsset{colormap={ends}{gray(0cm)=(0.875) gray(1cm)=(0.125) gray(2cm)=(0.875)}}

\DeclareRobustCommand\degree{\ensuremath{^\circ}}

\begin{document}

Here's a MWE.  Let's use as our example the parameterized path that we would use to fly from the North Pole, facing Greenwich, to the South Pole, circumnavigating the globe \(t\) times clockwise.  For convenience, we'll use spherical coordinates \( (r,\theta,\phi) \) and use as our units radians and the Earth's radius, so the North Pole is at \( (1,\theta,0) \) and the South pole at \( (1,\theta,\uppi) \).  Each clockwise trip around the Earth is \(2 \uppi\) radians, and we make \(t\) of them.  Parameterizing the path in \( u \in [0,1] \):

\[ r = 1 \qquad \theta = 2\uppi t u \qquad \phi = \uppi u \]

Converting to Cartesian coordinates:

\begin{align*}
x &=& r \cos \theta \sin \phi &=& \cos 2\uppi t u \cdot \sin \uppi u \\
y &=& r \sin \theta \sin \phi &=& \sin 2\uppi t u \cdot \sin \uppi u \\
z &=& r \cos \phi &=& \cos \uppi u
\end{align*}

Here are some graphs of these parametric functions with \texttt{pgfplotset}.  The case where \(t=0.5\) is in green, \(t=1\) in blue and \(t=2\) in red.  These two plots are rotated \(90\degree\) from each other:

\begin{tikzpicture}
\begin{axis}[view={30}{30},
             xmin=-1, xmax=1, ymin=-1, ymax=1,
             xlabel=$x$, ylabel=$y$, zlabel=$z$,
             unit vector ratio = 1 1 1
            ]
  \addplot3[blue,line width=1pt,variable=\u,domain=0:1,samples=45]( {cos(360*1*u)*sin(180*u))}, {sin(360*1*u)*sin(180*u)}, {cos(180*u)} );
  \addplot3[red,line width=1pt,variable=\u,domain=0:1,samples=45]( {cos(360*2*u)*sin(180*u))}, {sin(360*2*u)*sin(180*u)}, {cos(180*u)} );
  \addplot3[green,line width=1pt,variable=\u,domain=0:1,samples=45]( {cos(360*0.5*u)*sin(180*u))}, {sin(360*0.5*u)*sin(180*u)}, {cos(180*u)} );
\addplot3[mesh,z buffer=sort,samples=20,variable=\u,domain=-1:1,variable y=\v,y domain=0:2*pi,colormap name=ends,line width=0.1pt]({sqrt(1-u^2) * cos(deg(v))},{sqrt( 1-u^2 ) * sin(deg(v))},u);
\end{axis}
\end{tikzpicture}

\begin{tikzpicture}
\begin{axis}[view={120}{30},
             xmin=-1, xmax=1, ymin=-1, ymax=1,
             xlabel=$x$, ylabel=$y$, zlabel=$z$,
             unit vector ratio = 1 1 1
            ]
  \addplot3[blue,line width=1pt,variable=\u,domain=0:1,samples=45]( {cos(360*1*u)*sin(180*u))}, {sin(360*1*u)*sin(180*u)}, {cos(180*u)} );
  \addplot3[red,line width=1pt,variable=\u,domain=0:1,samples=45]( {cos(360*2*u)*sin(180*u))}, {sin(360*2*u)*sin(180*u)}, {cos(180*u)} );
  \addplot3[green,line width=1pt,variable=\u,domain=0:1,samples=45]( {cos(360*0.5*u)*sin(180*u))}, {sin(360*0.5*u)*sin(180*u)}, {cos(180*u)} );
\addplot3[mesh,z buffer=sort,samples=20,variable=\u,domain=-1:1,variable y=\v,y domain=0:2*pi,colormap name=ends,line width=0.1pt]({sqrt(1-u^2) * cos(deg(v))},{sqrt( 1-u^2 ) * sin(deg(v))},u);
\end{axis}
\end{tikzpicture}

This one is from overhead:

\begin{tikzpicture}
\begin{axis}[view={0}{90},
%             xmin=-1, xmax=1, ymin=-1, ymax=1,
             xlabel=$x$, ylabel=$y$, zlabel=$z$,
             unit vector ratio = 1 1 1]
  \addplot3[blue,variable=\u,domain=0:1,samples=45]( {cos(360*1*u)*sin(180*u))}, {sin(360*1*u)*sin(180*u)}, {cos(180*u)} );
  \addplot3[red,variable=\u,domain=0:1,samples=45]( {cos(360*2*u)*sin(180*u))}, {sin(360*2*u)*sin(180*u)}, {cos(180*u)} );
  \addplot3[green,variable=\u,domain=0:1,samples=45]( {cos(360*0.5*u)*sin(180*u))}, {sin(360*0.5*u)*sin(180*u)}, {cos(180*u)} );
\end{axis}
\end{tikzpicture}

\end{document}

(Image composited for space.)

A larger version of the first plot:

An alternative which might perform better and give you prettier output is to import a graph from Maple or GNU Octave, which you can even use to generate a png, tikz code or a SVG (which you can compress with gzip --best -c curves.svg > curves.svgz):

clf
colormap(gray);
[x,y,z] = sphere(20);
mesh(x,y,z);
hold on
t = [0:0.01:1];
plot3 (cos (2*pi*1*t) .* sin (pi*t), sin(2*pi*1*t) .* sin (pi*t), cos(pi*t), cos (2*pi*0.5*t) .* sin (pi*t), sin(2*pi*0.5*t) .* sin (pi*t), cos(pi*t), cos (2*pi*2*t) .* sin (pi*t), sin(2*pi*2*t) .* sin (pi*t), cos(pi*t) );