# Plotting ellipse intersection points

I'm trying to plot the intersection points of two ellipses, but they appear to be off.

My code:

\documentclass [10pt] {article}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}
\begin{scope}
\clip (0,0) ellipse (1cm and 3cm);
\fill [blue,pattern=north east lines] (0,0) ellipse (3cm and 1cm);
\end{scope}
\draw[red] (0,0) ellipse (3cm and 1cm);
\draw[blue] (0,0) ellipse (1cm and 3cm);
\draw[step=1cm,gray,very thin,dashed,] (-2.9,-2.9) grid (2.9,2.9);
\draw[<->,thick] (-3.2,0) -- (3.2,0);
\draw[<->,thick] (0,-3.2) -- (0,3.2);
\draw (0.866,0.866) circle (0.5ex)[fill=black]node[anchor=south west, black]{$(\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2})$};
\end{tikzpicture}
\end{document}


And a picture of the issue:

And just to make sure I wasn't going crazy, I checked the intersection points on WolframAlpha.

How can I accurately plot the points I need?

You should check your math again. I don't know what you really want, but here is one correct solution, which matches the formulas you used on WolframAlpha.

\documentclass[10pt]{article}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{patterns}
\begin{document}
\begin{tikzpicture}
\begin{scope}
\clip (0,0) ellipse (1 and 1.732050808);
\fill [blue,pattern=north east lines] (0,0) ellipse (1.732050808 and 1);
\end{scope}
\draw[red] (0,0) ellipse (1.732050808 and 1);
\draw[blue] (0,0) ellipse (1 and 1.732050808);
\draw[step=1,gray,very thin,dashed,] (-2.9,-2.9) grid (2.9,2.9);
\draw[<->,thick] (-3.2,0) -- (3.2,0);
\draw[<->,thick] (0,-3.2) -- (0,3.2);
\draw (0.866,0.866) circle (0.5ex)[fill=black]node[anchor=south west, black]{$\left(\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}\right)$};
\end{tikzpicture}
\end{document}


Output:

• Ah right, the ellipses aren't 3 long, they're √3 long. My mistake, thank you. Feb 28 '16 at 0:20

You can use the intersections library of tikz to get the real intersection point. I recommend you double-check the math to ensure the calculated coordinates are accurate.

\documentclass [10pt] {article}
\usepackage{tikz}
\usetikzlibrary{patterns,intersections}
\begin{document}
\begin{tikzpicture}[x=1cm,y=1cm]
\pgfmathsetmacro{\intp}{sqrt(3)/2.0};
\begin{scope}
\clip (0,0) ellipse (1cm and 3cm);
\fill [blue,pattern=north east lines] (0,0) ellipse (3cm and 1cm);
\end{scope}
\path[draw,red,name path=p1] (0,0) ellipse (3cm and 1cm);
\path[draw,blue,name path=p2] (0,0) ellipse (1cm and 3cm);
\draw[step=1cm,gray,very thin,dashed,] (-2.9,-2.9) grid (2.9,2.9);
\draw[<->,thick] (-3.2,0) -- (3.2,0);
\draw[<->,thick] (0,-3.2) -- (0,3.2);
\path [name intersections={of=p1 and p2}];
\draw (intersection-1) circle (0.5ex)[fill=black]node[anchor=south west, black]{$(\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2})$};
\end{tikzpicture}
\end{document}