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.


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.

  • $\begingroup$ I really appreciate your input! Even though I was already aware of this charaterization, I upvoted your answer! However, I won't accept it since the question is still not settled. $\endgroup$
    – C. Falcon
    Jun 28 '16 at 18:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.