How do I build a homeomorphism between $\mathbb{R} ^ n \setminus \mathbb{R} ^ k$ and $S ^ {n-k-1} \times \mathbb{R} ^ {k+1} $

Let n, k be positive integers such that k<n. How do i build a homeomorphism between $\mathbb{R} ^ n \setminus \mathbb{R} ^ k$ and $S ^ {n-k-1} \times \mathbb{R} ^ {k+1} $?

I can show that $\mathbb{R} ^ n \setminus \mathbb{R} ^ 0 \cong S ^ {n-1} \times \mathbb{R} $. A map that shows the isomorphism is $ x \mapsto (x/||x||, ||x||)$ and I can prove the homeomorphism by induction. But I'm stuck on building such bijection.

Exactly the same procedure: let $\mathbb R^k = \mathbb R^k \times \{0\} \subset \mathbb R^n$. Write $z\in \mathbb R^n$ as $z=(x, y) \in \mathbb R^k \times \mathbb R^{n-k}$, we have

$$\mathbb R^n \setminus \mathbb R^k = \{(x, y) : y\neq 0\}$$

Thus we define

$$ F : \mathbb R^n \setminus \mathbb R^k \to \mathbb R^k \times S^{n-k-1} \times \mathbb R$$

with $F(x, y) = (x, y/\|y\|, \log \|y\|)$. It is clear that $F$ has inverse $F^{-1}(x, v, t) = (x, e^t v)$.

(Note that your isomorphism shows $\mathbb{R} ^ n \setminus \mathbb{R} ^ 0 \cong S ^ {n-1} \times (0,\infty)$, but not $\mathbb{R} ^ n \setminus \mathbb{R} ^ 0 \cong S ^ {n-1} \times \mathbb{R}$).