\documentclass{report}

\usepackage{tikz}
\usetikzlibrary{intersections}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{1mm}%
\begin{document}
\begin{tikzpicture}

\end{tikzpicture}
	
\begin{tikzpicture}[]
\draw[green,rotate=-55,name path=ellips1] (3,2.5) ellipse (9cm and 2cm);
\draw[green,rotate=55,name path=ellips2] (-3,2.5)ellipse (9cm and 2cm);


  \draw[gray,very thick,name path=line1] (-5,0) -- (5,0) node[at end,above]{$v_{1}$};‎
  \draw[gray,very thick,name path=line2] (-5,-1) -- (5,-1)node[at end,above]{$v_{2}$};‎
  \draw[gray,very thick,name path=line3] (-5,-2) -- (5,-2)node[at end,above]{$v_{3}$};‎
  \draw[gray,very thick,name path=line4] (-5,-3) -- (5,-3)node[at end,above]{$v_{4}$};‎
  \draw[gray,very thick,name path=line5] (-5,-4) -- (5,-4)node[at end,above]{$v_{5}$};‎

\node[name intersections={of=ellips1 and ellips2}](Center) at(intersection-4){};
\node[name intersections={of=ellips1 and line1}](A1) at(intersection-1){};
\node[name intersections={of=ellips2 and line1}](B1) at(intersection-1){};
  
  \node[name intersections={of=ellips1 and line2}](A2) at(intersection-1){};
  \node[name intersections={of=ellips2 and line2}](B2) at(intersection-1){};
  
  \node[name intersections={of=ellips1 and line3}](A3) at(intersection-1){};
  \node[name intersections={of=ellips2 and line3}](B3) at(intersection-1){};
  
  \node[name intersections={of=ellips1 and line4}](A4) at(intersection-1){};
  \node[name intersections={of=ellips2 and line4}](B4) at(intersection-1){};
  
  \node[name intersections={of=ellips1 and line5}](A5) at(intersection-1){};
  \node[name intersections={of=ellips2 and line5}](B5) at(intersection-1){};
  
  
\draw[fill=red](Center)circle(.6mm);
\draw[-] (B1) to[bend right=20] (A1);
\draw[-] (B2) to[bend right=20] (A2);
\draw[-] (B3) to[bend right=20] (A3);
\draw[-] (B4) to[bend right=20] (A4);
\draw[-] (B5) to[bend right=20] (A5);
  


‎\end{tikzpicture}

\end{document}