What is mean , "consider $\mathbb R^{n+1}$ as the cone over $S^n$ with vertex at the origin"

Question

What is mean , "consider $\mathbb R^{n+1}$ as the cone over $S^n$ with vertex at the origin" where $S^n$ is the n-sphere in $\mathbb R^{n+1}$

I know that one cone $K\subset \mathbb R^{n}$ is a set such that $\alpha K\subset K$ for all $\alpha\in\mathbb R_{+}$,.

I know that is neccesary consider polar coordinate for understand this in a best way, so i need to find a function $\psi: S^n \to \mathbb R^{n+1}$?

I have trouble with the meaning of this statement consider $\mathbb R^{n+1}$ as the cone over $S^n$ with vertex at the origin

Please, can you give me a hint or what i need to do?

Thanks a lot

Answer 1

It means that you should consider the function $$\psi : S^{n-1} \times [0,\infty) \mapsto \mathbb R^n $$ defined by $f(p,r) = rp$ (i.e. scalar multiplication).

This function maps $S^{n-1} \times 0$ to the origin of $\mathbb R^n$.

Also, for each $p \in S^{n-1}$ this function maps $p \times [0,\infty)$ to the ray in $\mathbb R^n$ that is based at the origin and that passes through $p$.