\documentclass[a4paper,10pt]{article}
 
 % Use UTF8 since the laptop is configured so and use ngerman for word breaking.
 % If you encounter an encoding problem, remove the line with utf8.
 \usepackage[utf8]{inputenc}
 \usepackage[ngerman]{babel}
 \usepackage[pdftex]{graphicx}
 
 % TikZ für schöne Grafiken und so.
 \usepackage{tikz}
 \usetikzlibrary{positioning,calc,fadings,decorations.pathreplacing,arrows}
 \usepackage{pgfplots}

% \usepackage{hyperref}
 \usepackage{colortbl}
 \usepackage{tabularx}
 \usepackage{color}
 \usepackage{amsfonts}
 \usepackage{amsmath}
 \usepackage{listings}
 \usepackage{multirow}
\usepackage{fancyhdr}
 \usepackage{multicol}
 \usepackage{pdfpages}
 \usepackage[colorlinks=true,
  linkcolor=black,
  citecolor=black,
  filecolor=black,
  pagecolor=black,
  urlcolor=black,
  bookmarks=true,
  bookmarksopen=true,
  bookmarksopenlevel=3,
  plainpages=false,
  pdfpagelabels=true]{hyperref}

% Hack for german umlaute
\lstset{
  literate={ö}{{\"o}}1
           {ä}{{\"a}}1
           {ü}{{\"u}}1
}

 % I set here a different sidemargin because the original margin looks not so
 % good for normal documents. Additionally I have to enlarge the textwidth.
 \setlength{\oddsidemargin}{0cm}
 \setlength{\evensidemargin}{0cm}
 \addtolength{\textwidth}{4cm}
\newcolumntype{C}[1]{>{\centering\arraybackslash}m{#1}} 

\begin{document} 
 % Here I use the up-to-date font encoding T1 and the font familly Computer Modern
 % Sans Serif (since I don't like the standard font) medium and normal (non-italic or so).
 \usefont{T1}{cmss}{m}{n}
\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}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,thin,black] (\angVis:1) arc (\angVis:\angVis+180:1);
  \draw[current plane,thin,dashed] (\angVis-180:1) arc (\angVis-180:\angVis:1);
}%this is fake: for drawing the grid


\newcommand\DrawLongitudeCirclered[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,red,thick] (150:1) arc (150:180:1);
  %\draw[current plane,dashed] (-50:1) arc (-50:-35:1);
}%for drawing the grid


\newcommand\DLongredd[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,black,dashed, ultra thick] (150:1) arc (150:180:1);
}


\newcommand\DLatred[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane,dashed,black,ultra thick] (-50:1) arc (-50:-35:1);
}


\newcommand\fillred[2][1]{
  \LongitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
   % angle of "visibility"
  \pgfmathsetmacro\angVis{atan(sin(#2)*cos(\angEl)/sin(\angEl))} %
  \draw[current plane,red,thin] (\angVis:1) arc (\angVis:\angVis+180:1);
}

\newcommand\DrawLatitudeCircle[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  \draw[current plane,thin,black] (\angVis:1) arc (\angVis:-\angVis-180:1);
  \draw[current plane,thin,dashed] (180-\angVis:1) arc (180-\angVis:\angVis:1);
}%Defining functions to draw limited latitude circles (for the red mesh)


\newcommand\DrawLatitudeCirclered[2][1]{
  \LatitudePlane{\angEl}{#2}
  \tikzset{current plane/.prefix style={scale=#1}}
  \pgfmathsetmacro\sinVis{sin(#2)/cos(#2)*sin(\angEl)/cos(\angEl)}
  % angle of "visibility"
  \pgfmathsetmacro\angVis{asin(min(1,max(\sinVis,-1)))}
  %\draw[current plane,red,thick] (-\angVis-50:1) arc (-\angVis-50:-\angVis-20:1);
\draw[current plane,red,thick] (-50:1) arc (-50:-35:1);
}


\tikzset{
  >=latex,
  inner sep=0pt,
  outer sep=2pt,
  mark coordinate/.style={inner sep=0pt,outer sep=0pt,minimum size=3pt, fill=black,circle}
}

\begin{tikzpicture}[scale=1,every node/.style={minimum size=1cm}]
	%% some definitions
	
	\def\R{4} % sphere radius
	
	\def\angEl{25} % elevation angle
	\def\angAz{-100} % azimuth angle
	\def\angPhiOne{-110} % longitude of point P
	\def\angPhiTwo{-45} % longitude of point Q
	\def\angBeta{30} % latitude of point P and Q
	
	%% working planes
	
	\pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole
	\LongitudePlane[xzplane]{\angEl}{\angAz}
	\LongitudePlane[pzplane]{\angEl}{\angPhiOne}
	\LongitudePlane[qzplane]{\angEl}{\angPhiTwo}
        \LongitudePlane[nzplane]{\angEl}{-86}
	\LatitudePlane[equator]{\angEl}{0}
	\fill[ball color=white!10] (0,0) circle (\R); % 3D lighting effect
	\coordinate (O) at (0,0);
	\coordinate[mark coordinate] (N) at (0,\H);
	\coordinate[mark coordinate] (S) at (0,-\H);
	
    \DrawLongitudeCircle[\R]{\angPhiOne} % pzplane
    \DrawLongitudeCircle[\R]{\angPhiTwo} % qzplane
    \DrawLatitudeCircle[\R]{\angBeta}
    \DrawLatitudeCircle[\R]{0} % equator
	%labelling north and south
	\node[above=8pt] at (N) {$\mathbf{N}$};
	\node[below=8pt] at (S) {$\mathbf{S}$};
        \draw[-,dashed, thick] (N) -- (S);

    %setup coordinates P and Q
    \path[pzplane] (0:\R) coordinate (P);
    \draw[->] (O) -- node[above=4pt] {$\overrightarrow{P}$} (P);
    \path[qzplane] (\angBeta:\R) coordinate (Q);
    \draw[->] (O) -- node[above=2pt] {$\overrightarrow{Q}$} (Q);
    \path[nzplane] (153:\R) coordinate (N);
    \draw[->,color=red] (O) -- node[right=2pt] {$\overrightarrow{N}$} (N);
    \draw (P) arc (-110:-45:\R) (Q);	
\end{tikzpicture}

 \end{document}