Angle problem between points on a unit sphere

Question

I am dealing with an applied question with space angles correction but have no clue. The question: Given n points are denoted by $\{V_1,V_2,\ldots,V_{n-1},V_n\}$ on the unit sphere whose center is at the original point $O$. The projections of $\{\vec{OV_1},\ldots,\vec{OV_n}\}$ onto the $XOY$ plane sorted counterclockwise. Now there is a point $V_{n+1}$. It is known that the angle between the vector $\vec{OV_1}$ and the vector $\vec{OV_{n+1}}$ is $\alpha_1$. And the $n-1$ angles between vector $\vec{OV_{n+1}}$ and vectors $\{\vec{OV_2},\ldots\vec{OV_n}\}$ are denoted by $\alpha_2,\ldots,\alpha_n$. Then I want to ask what are the relationships between the $\alpha_1$ and $\alpha_2,\ldots,\alpha_n$ and what are the range of the $\alpha_2,\ldots,\alpha_n$. I am confused about using angles to describe the spatial position of points. Thank you so much for your attention and answers.

Answer 1

It is known that the angle between the vector $\vec{OV_1}$ and the vector $\vec{OV_{n+1}}$ is $\alpha_1$.

Ok, so $$\textbf{V}_{n+1}\cdot \textbf{V}_1 = \cos(\alpha_1).$$

And the $n-1$ angles between vector $\vec{OV_{n+1}}$ and vectors $\{\vec{OV_2},\ldots\vec{OV_n}\}$ are denoted by $\alpha_2,\ldots,\alpha_n$.

Meaning $$ \textbf{V}_{n+1}\cdot \textbf{V}_{k} = \cos(\alpha_k),$$ $k=1,\dots,n$.

Then I want to ask what are the relationships between the $\alpha_1$ and $\alpha_2,\ldots,\alpha_n$...

Well, for example, the difference between $\alpha_k$ and $\alpha_l$ will be $$ \delta \alpha_{k l} = \cos^{-1}(\textbf{V}_{n+1}\cdot \textbf{V}_{k})- \cos^{-1}(\textbf{V}_{n+1}\cdot \textbf{V}_{l}). $$

... and what are the range of the $\alpha_2,\ldots,\alpha_n$.

Your points are sorted only in the XOY plane, so it is not obvious to me that there's any simple way to get the max or minimum $\alpha$ by just selecting the correct two $\textbf{V_k}$ and comparing them with $V_{n+1}$. Depending on your application, it may be suitable to just brute force calculate the maximum and minimum of $\alpha_k = \cos^{-1}(\textbf{V}_{n+1}\cdot \textbf{V}_k). $ across all $k$.