# Exactly fitting an ellipsoid in $\mathbb{R}^n$

I'm trying to understand the constraints on exactly fitting an ellipse/ellipsoid to a small set of points as the number of dimensions increases. The ellipse/ellipsoid is always centred at the origin.

I conjecture that in $\mathbb{R}^n$, any $n$ points can have an infinite number of ellipsoids centred at the origin fitted through them so long as the $n$ points satisfy some constraint to avoid colinearity/coplanarity, which would lead to degenerate solutions (circles/spheres/spheroids) or no solution at all. I'm not clear on the precise nature of this constraint as $n$ increases.

Suppose we identify and satisfy these constraints so that special cases are excluded. Can this infinite number of ellipsoids be reduced to a unique ellipsoid by adding 1 more point, subject to additional constraints? What are those additional constraints? And is 1 more point always enough?

For example, if you have three points (plus origin) in $\mathbb{R}^2$, you can select the two points that make the widest angle with respect to the origin, and draw a line between them. The remaining point can only be fitted to an ellipse through the other two points if it lies "outside" the line with respect to the origin. I imagine that some similar constraint should exist in $\mathbb{R}^n$ - what is this constraint?

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An general ellipsoid (as well as hyperboloids and other quadrics) can be described by a quadratic form. You can think of this as a symmetric $r\times r$ matrix, with $r=n+1$. It has $\frac{r(r+1)}2$ different entries. But you actually have one less degree of freedom due to homogenity. The centering at the origin eats another $n$ degrees of freedom, since you can achieve all other quadrics using translation in $n$ dimensions. So you have
$$\frac{r(r+1)}2-r = \frac{r(r-1)}2 = \frac{n(n+1)}2$$
real degrees of freedom. Each defining point contributes one real degree of freedom: it can be freely moved in $n$ dimensions, but even for a given quadric, it can still move insie the cuadric, i.e. a $(n-1)$-dimensional manifold, without changing its definition.
So in 2D, you have 3 defining points, in 3D you have 6, and so on. So no, $n+1$ points won't be enough to uniquely describe a quadric.