Sorry for such question, But I don't know what exactly is the Euclidean metric on $\Bbb R^n\setminus\{0\}$? Induced by $\iota:\Bbb R^n\setminus\{0\}\to \Bbb R^n$ or $\varphi :\Bbb R^n\setminus\{0\}\to\Bbb R\times \Bbb S^{n-1}$ or something else?
Update: It seems that my problem is misunderstanding the "induced" maybe. This is my thought: because $x,y\in \Bbb R^n\setminus\{0\}$ are also in $\Bbb R^n$ so $d(x,y)=d_{can}(x,y)$. but this do not work for $x=(1,0,0...,0)$ and $y=(−1,0,0...,0)$!!
By definition, a Euclidean space is an affine space, often confused with a vector space. Since $\mathbb{R}^n \setminus \{0\}$ is no longer an affine space, there is no standard definition of the Euclidean distance on it.