tikz-3dplot, geodesics and adressing endpoints in polar coordinates

Question

I am trying to draw geodesics on a sphere. While many approaches, like for example this one, i am more interested in another way. If i specify two points, say qand p on the Sphere, i would like to join them by a geodesic, i.e. arc. Whether thats the longer or shorter arc, might be a problem for further stuff, i managed to do the following

\documentclass[a4paper]{standalone}
\usepackage{tikz,tikz-3dplot}
\tikzstyle{point}=[inner sep=0pt, outer sep=0pt,%
    minimum size=2pt,fill=black,shape=circle]
\begin{document}
    \tdplotsetmaincoords{70}{110}
\begin{tikzpicture}
    \begin{scope}[tdplot_main_coords]
%draw sphere
        \tdplotsphericalsurfaceplot{72}{36}{1}{black!75!white}{blue!20!white}%
        {\draw[color=black,thick,->] (1,0,0) -- (1.5,0,0) node[anchor=north east]{$x$};}%
        {\draw[color=black,thick,->] (0,1,0) -- (0,1.5,0) node[anchor=north west]{$y$};}%
        {\draw[color=black,thick,->] (0,0,1) -- (0,0,1.5) node[anchor=south]{$z$};}%
% draw geodesics
        \tdplotdefinepoints(0,0,0)(0.7071,-0.7071,0)(0,0.7071,0.7071)
        \tdplotdrawpolytopearc[thick,red!50!black]{1}{}{}
        \tdplotdefinepoints(0,0,0)(0.7071,-0.7071,0)(0.7071,0.7071,0)
        \tdplotdrawpolytopearc[thick,blue!50!black]{1}{}{}
        \tdplotdefinepoints(0,0,0)(0.7071,0.7071,0)(0,0.7071,0.7071)
        \tdplotdrawpolytopearc[thick,green!50!black]{1}{}{}
%draw point
        \tdplotsetcoord{P}{1}{30}{60}
        \node[point,label={0:\(p\)}] at (P) {};

    \end{scope}
\end{tikzpicture}
\end{document}

which yields

an MWE not a beauty

and leads me to my two questions:

1) I would like to be able to specify both “spanning” points (the second and third of \tdplotdefinepoints in polar coordinates. It would be enough, to have a function performing theta,phi into px,py,pz, similar to the function used for the point P. Or maybe one could also kind of extract these coordinates from a label; is either of that possible?

2) when drawing an arc, is there a possibility – similar to the usual \draw to access the mid point? It would be enough to be able to place a node (with style) and label there, otherwhise a \coordinate would of course do the job, too. Any ideas on how to get that?

Would you be interested in a solution that is entirely based on spherical coordinates? (IMHO it does not no make too much sense to use cartesian coordinates to parametrize points on the surface of a sphere.) — , Dec 30 '18 at 00:22
In most of my calculations its easier to have cartesian coordinates, since for geodesics (great arcs) you would have to calculate several modulo cases in spherical coordinates; tangent vectors are cartesian anyways. However, I would be interested in any solution :) — Ronny, Jan 04 '19 at 07:52

score 2 · Answer 1 · answered Jan 05 '19 at 05:31

This is at best 50% of an answer because I simply do not understand the first request. \tdplotsetcoord{P}{1}{30}{60} does define a point in 3d. Could you please rephrase the first request?

The second point is straightforward. decorations.markings allows you to mark a point at any position of the path, of course including the middle. The style add coordinate={<name> at <pos>} does that in the following MWE.

\documentclass[a4paper]{standalone}
\usepackage{tikz,tikz-3dplot}
\usetikzlibrary{decorations.markings}
\tikzset{point/.style={inner sep=0pt, outer sep=0pt,%
    minimum size=2pt,fill=black,shape=circle},
    add coordinate/.style args={#1 at #2}{postaction={decorate,
    decoration={markings,mark=at position #2 with {\coordinate (#1);}}}}}
\begin{document}
    \tdplotsetmaincoords{70}{110}
\begin{tikzpicture}
    \begin{scope}[tdplot_main_coords]
%draw sphere
        \tdplotsphericalsurfaceplot{72}{36}{1}{black!75!white}{blue!20!white}%
        {\draw[color=black,thick,->] (1,0,0) -- (1.5,0,0) node[anchor=north east]{$x$};}%
        {\draw[color=black,thick,->] (0,1,0) -- (0,1.5,0) node[anchor=north west]{$y$};}%
        {\draw[color=black,thick,->] (0,0,1) -- (0,0,1.5) node[anchor=south]{$z$};}%
% draw geodesics
        \tdplotdefinepoints(0,0,0)(0.7071,-0.7071,0)(0,0.7071,0.7071)
        \tdplotdrawpolytopearc[thick,red!50!black,add coordinate={M1 at 0.5}]{1}{}{}
        \tdplotdefinepoints(0,0,0)(0.7071,-0.7071,0)(0.7071,0.7071,0)
        \tdplotdrawpolytopearc[thick,blue!50!black,add coordinate={M2 at 0.5}]{1}{}{}
        \tdplotdefinepoints(0,0,0)(0.7071,0.7071,0)(0,0.7071,0.7071)
        \tdplotdrawpolytopearc[thick,green!50!black,add coordinate={M3 at 0.5}]{1}{}{}
%draw point
        \tdplotsetcoord{P}{1}{30}{60}
        \node[point,label={0:\(p\)}] at (P) {};
        \node[point,label={90:{$M_1$}}] at (M1){};
        \node[point,label={-90:{$M_2$}}] at (M2){};
        \node[point,label={0:{$M_3$}}] at (M3){};
    \end{scope}
\end{tikzpicture}
\end{document}

tikz-3dplot, geodesics and adressing endpoints in polar coordinates

1 Answers1