When do $n+2$ points in $\mathbb{R}^n$ lie on a same $(n-1)$-sphere?

Question

When $n=2$, the following results are well-known:

Proposition 1. Let $A,B,C,D$ be $4$ distinct points in $\mathbb{R}^2$. They are aligned or cocyclic if and only if: $$\left(\overrightarrow{CA},\overrightarrow{CB}\right)\equiv\left(\overrightarrow{DA},\overrightarrow{DB}\right)\mod \pi.$$

Proposition 2. (Ptolemy's theorem) Let $A,B,C,D$ be $4$ distinct points in $\mathbb{R}^2$. They are cocyclic if and only if one of the following equalities holds true: $$AB.CD\pm AC.DB\pm AD.BC=0.$$

In this recent question it is proven that whenever $n+1$ points in $\mathbb{R}^n$ do not lie in any affine hyperplane, they are on a unique $(n-1)$-sphere, which leads me to ask the following:

Question. Is there a necessary and sufficient condition to determine when $n+2$ points in $\mathbb{R}^n$ are on a same affine hyperplane or on a same hypersphere?

I easily derived from the equations of an affine hyperplane and of an hypersphere that if $x_i:=(x_{i,j})$ are $n+2$ points of $\mathbb{R}^n$, the $x_i$s are on a same hyperplane or lie on a same hypersphere if and only if: $$\left|\begin{matrix}{x_{1,1}}^2+\cdots+{x_{1,n}}^2&x_{1,1}&\cdots&x_{1,n}&1\\{x_{2,n}}^2+\cdots+{x_{2,n}}^2&x_{2,1}&\cdots&x_{2,n}&1\\\vdots&\vdots&\ddots&\vdots&\vdots\\{x_{n+1,1}}^2+\cdots+{x_{n+1,n}}^2&x_{n+1,1}&\cdots&x_{n+1,n}&1\\{x_{n+2,1}}^2+\cdots+{x_{n+2,n}}^2&x_{n+2,1}&\cdots&x_{n+2,n}&1\end{matrix}\right|=0.$$ However, I am more interested in a characterization involving angles in the same way as in proposition 1. or distances like in proposition 2. In particular, in the case $n=3$ is there a necessary and sufficient condition expressing a relation between solid angles?

Regarding the case $n=3$, my guess would be to determine the set of points from where one can observe a given circle with a constant solid angle.

Answer 1

Assume $x_0 = 0$ for simplicity and let $x_i' = \frac{x_i}{|x_i|^2}$ be the images of $x_i$'s under an inversion centered at $x_0$. By a well-known property of inversions, $x_0,\ldots,x_{n+1}$ lie on an affine $n-1$-plane or an $n-1$-sphere if and only if $x_1',\ldots,x_{n+1}'$ lie on an affine $n-1$-plane.

When the latter is expressed using the determinant, this probably yields a condition analogous to the one you stated. However, I feel that this point of view is more geometric in nature.

Answer 2

This is an old questions, but I will answer it anyway, it may be useful to someone.

There is a well known necessary and sufficient conditions (in terms of mutual distances) for $n+2$ points ($P_1, \ldots, P_{n+2}$) in $\mathbb{R}^n$ to lie on a $(n-1)$-sphere or an hyperplane, namely,
$$\begin{vmatrix} 0 &d_{12}^2 & \cdots & d_{1n+2}^2\\ d_{12}^2 &0 &\cdots & d_{2(n+2)}^2\\ \vdots&\vdots&\vdots&\vdots\\ d_{1(n+2)}^2& d_{2(n+2)}^2&\cdots&0 \end{vmatrix}=0 $$ where $d_{ij}$ is the distance between the point $ P_i$ and the point $P_j$. This statement with proof is Proposition 9.7.3.7 in the book Geometry I by Marcel Berger.