Btukfyl

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

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?

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}