Tangent of ellipse

In this post, we would like to give a formula to plot the tangent plane of an ellipse. The full course with relevant topics is available on my Github.

An n-dimensional ellipse can be defined by the following equation
$$\langle x-c, P^{-1}(x-c)\rangle=1.$$

where $P\in \mathbb R^{m\times m}$ is a positive definite matrix and $c\in \mathbb R^m$ is the center of ellipse.

Then for any point $x_0\in \mathbb R^m$ belonging to this ellipse, the tangent plane at $x_0$ is given by
$$\langle x-c, P^{-1}(x_0-c)\rangle=1.$$

Note. For a general point $g$, we may construct a point $x_0$ lying on the ellipse by using the following formula
$$x_0=\frac{g}{\sqrt{\langle g, P^{-1}g\rangle}}+c.$$

The following figure is our experiment using Python.