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.

```
\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

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

into

## Code

You can also do this without rotation by specifying your major axis and minor axis (

first axisandsecond axisin the manual) through the use of the`\pgfpathellipse`

command. Here it is with your MWE.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.