[Tex/LaTex] Drawing ellipse that passes through specified coordinates

tikz-pgf

I'm trying to draw an ellipse that must pass through (0, -1), (0, 1), (-1, 0), (1, 0), (1, -1) and (-1, 1). My MWE is below. I know that, I'm missing some math here.

enter image description here

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\begin{document}

\begin{tikzpicture}

\draw [help lines] (-2, -2) grid (2, 2);
\draw[rotate=45] (0, 0) ellipse (0.5cm and 1.5cm);

\filldraw[black] (0, 0) circle(1.5pt);
\filldraw[black] (0, 1) circle(1.5pt);
\filldraw[black] (1, 0) circle(1.5pt);
\filldraw[black] (-1, 0) circle(1.5pt);
\filldraw[black] (0, -1) circle(1.5pt);
\filldraw[black] (1, -1) circle(1.5pt);
\filldraw[black] (-1, 1) circle(1.5pt);

\draw[<->] (-2.0, 0) -- (2.0, 0) node[right]{\footnotesize $X_{1}$};
\draw[<->] (0, -1.5) -- (0, 1.5) node[above]{\footnotesize $X_{2}$};

\node[above right] at (0, 1)   {\tiny $\left(0, 1\right)$}; 
\node[above right] at (1, 0)   {\tiny $\left(1, 0\right)$}; 
\node[above left]   at (-1, 0)  {\tiny $\left(-1, 0\right)$}; 
\node[below right] at (0, -1)  {\tiny $\left(0, -1\right)$}; 
\node[below right] at (1, -1)  {\tiny $\left(1, -1\right)$}; 
\node[above left]   at (-1,  1)  {\tiny $\left(-1, 1\right)$}; 

\end{tikzpicture}

\end{document}

Edit

The equation of the ellipse is $X_1^2 + X_1 X_2 + X_2^2 = 1$.

Best Answer

Output

enter image description here

You can manually perform the computations as suggested by JoeDub. You can then change

\draw[rotate=45] (0, 0) ellipse (0.5cm and 1.5cm);

into

\draw[rotate=45] (0, 0) ellipse (0.82cm and 1.4cm);

Code

%http://tex.stackexchange.com/questions/99714/drawing-ellipse-that-passes-through-specified-coordinates#99714
\documentclass[border=5,convert={density=150}]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\begin{document}

\begin{tikzpicture}

\draw [help lines] (-2, -2) grid (2, 2);
\draw[rotate=45,color=blue,very thick] (0, 0) ellipse (0.82cm and 1.4cm);

\foreach \x/\y in {0/0, 0/1, 1/0, -1/0, 0/-1, 1/-1, -1/1}
\filldraw[black] (\x, \y) circle(1.5pt);

\draw[<->] (-2.0, 0) -- (2.0, 0) node[right]{\footnotesize $X_{1}$};
\draw[<->] (0, -1.5) -- (0, 1.5) node[above]{\footnotesize $X_{2}$};

\foreach \x/\y/\direction in {0/1/above right, 1/0/above right, -1/0/above left, 0/-1/below left, 1/-1/below right, -1/1/above left}
\node [\direction] at (\x,\y) {\tiny $\left(\x, \y\right)$}; 
\end{tikzpicture}

\end{document}

You can also do this without rotation by specifying your major axis and minor axis (first axis and second axis in the manual) through the use of the \pgfpathellipse command. Here it is with your MWE.

\documentclass[border=5,convert={density=150}]{standalone}
\usepackage{tikz}
\usetikzlibrary{shapes.geometric}

\begin{document}

\begin{tikzpicture}

\draw [help lines] (-2, -2) grid (2, 2);
\pgfpathellipse{\pgfpointorigin}{\pgfpointxy{1}{-1}}{\pgfpointxy{0.58}{0.58}} % Create an ellipse with one end of major axis passing through (1,-1) and with one end of minor axis at (0.58,0.58)
\pgfusepath{stroke}

\foreach \x/\y in {0/0, 0/1, 1/0, -1/0, 0/-1, 1/-1, -1/1}
\filldraw[black] (\x, \y) circle(1.5pt);

\draw[<->] (-2.0, 0) -- (2.0, 0) node[right]{\footnotesize $X_{1}$};
\draw[<->] (0, -1.5) -- (0, 1.5) node[above]{\footnotesize $X_{2}$};

\foreach \x/\y/\direction in {0/1/above right, 1/0/above right, -1/0/above left, 0/-1/below left, 1/-1/below right, -1/1/above left}
\node [\direction] at (\x,\y) {\tiny $\left(\x, \y\right)$}; 
\end{tikzpicture}

\end{document}

Note here that the length of the semi-minor axis is approximately the square root of 0.58^2 + 0.58^2.

Edit

I have used \foreach in the above codes to make them shorter.

Related Solutions

[Tex/LaTex] Rotate a node but not its content: the case of the ellipse decoration

The source of the difficulty is that ellipses are constructed in a particular way in TikZ. They are paths that start from the x-axis and proceed counter-clockwise around their centre. The vast majority of the time, the exact parametrisation doesn't matter. You appear to have found the one situation where it does!

In the actual question, you only want to be able to mirror the ellipse, and so draw it starting from the negative x-axis (the title of the question suggests a more flexible approach). That's actually not too hard since we can exploit the symmetry of the ellipse. The key is to provide it with a negative x-radius, since then it will start from the negative x-axis (and proceed clockwise, but we could correct for that by negating the y-radius as well). To do this, we interrupt the call from the node shape to the drawing command and flip the sign of the x-radius. The simplest way to do this is to redefine the \pgfpathellipse macro to do the negation and then call the original macro. The following code does this.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations,shapes,decorations.markings}
\makeatletter
\let\origpgfpathellipse=\pgfpathellipse
\def\revpgfpathellipse#1#2#3{%
  #2%
  \pgf@xa=-\pgf@x
  \origpgfpathellipse{#1}{\pgfqpoint{\pgf@xa}{0pt}}{#3}}
\makeatother

\tikzset{
  reversed ellipse/.style={
    ellipse,
    reverse the ellipse%
  },
  reverse the ellipse/.code={
    \let\pgfpathellipse=\revpgfpathellipse
  }
}
\begin{document}
\begin{tikzpicture}
\node[ellipse,
  draw,
  postaction={
    decorate,
    decoration={
      markings,
      mark=at position 1 with {
        \arrow[line width=5pt,blue]{>}
      }
    }
  }
] at (0,0) {hello world};

\node[reversed ellipse,
  draw,
  postaction={
    decorate,
    decoration={
      markings,
      mark=at position 1 with {
        \arrow[line width=5pt,blue]{>}
      }
    }
  }
] at (0,-2) {hello world};
\end{tikzpicture}

\end{document}

Here's the result:

rotated ellipse

(the arrow got clipped, but you can see where it lies)

[Tex/LaTex] How to define the default vertical distance between nodes

I don't really get the question so I hope this is what you wanted. If you include a full document (such that we copy paste and see the problem on our systems) things are much more easier.

Here, you can change the default setting within a scope but your block style had a node distance which was resetting every time it is issued. I've made it 2mm such that we can see the difference easier.

\documentclass[tikz]{standalone}
\usetikzlibrary{arrows,shapes.geometric,positioning}

\begin{document}
\begin{tikzpicture}[decision/.style={diamond, draw, text width=4.5em, text badly centered, node distance=3.5cm, inner sep=0pt},
                    block/.style   ={rectangle, draw, text width=6em, text centered, rounded corners, minimum height=4em, minimum height=2em},
                    cloud/.style   ={draw, ellipse, minimum height=2em},
                    line/.style    ={draw,-latex'},
                    node distance = 1cm, 
                    auto]
    \node [block] (1st) {1st};
    \node [block, right= of 1st]  (2nd1) {2nd1};
\begin{scope}[node distance=2mm and 10mm]%Here we change it for everything inside this scope
    \node [block, above= of 2nd1] (2nd2) {2nd2};
    \node [block, below= of 2nd1] (2nd3) {2nd3};
    \node [block, right= of 2nd1] (3rd1) {3rd1};
    \node [block, above= of 3rd1] (3rd2) {3rd2};
    \node [block, above= of 3rd2] (3rd3) {3rd3};
\end{scope}
    \node [block, below= of 3rd1] (3rd4) {3rd4};
    \node [block, below= of 3rd4] (3rd5) {3rd5};

    \path [line] (1st) -- (2nd1);
    \path [line] (2nd1) -- (2nd2);
    \path [line] (2nd1) -- (2nd3);
    \path [line] (2nd2) -- (3rd3);
    \path [line] (2nd1) -- (3rd1);
    \path [line] (1st) -- (2nd1);
\end{tikzpicture}
\end{document}

enter image description here

Best Answer

Output

Code

Edit

Related Solutions

[Tex/LaTex] Rotate a node but not its content: the case of the ellipse decoration

[Tex/LaTex] How to define the default vertical distance between nodes

Related Question