Given a set of coordinates in $N$-dimensional euclidean space with infinity norm less than or equal to $l$ (i.e. we have a cube-like bounded region in $N$ dimensions), is there a known one-to-one transformation that can map each of these coordinates to a unique point on the surface of the unit hyper-sphere in $M$-dimensional space (assuming no constraints on $N$ and $M$)?
Formally speaking, if the set of euclidean coordinates is $\mathcal{S} \in \mathbb{R}^N$ s.t. $||x||_\infty \leq l \hspace{3pt} \forall \hspace{3pt} x \in \mathcal{S}$, is there a known mapping function $\mathcal{T}: \mathbb{R^N} \rightarrow \mathcal{X}$ where $\mathcal{X} \in \mathbb{R}^M$ s.t. $||x|| = 1 \hspace{3pt} \forall \hspace{3pt} x \in \mathcal{X} $ ?
For the sake of clarity: by "one-to-one transformation", I mean that it should be possible to convert any coordinate from the bounded region $N$-dimensional euclidean space defined above to the $M$-dimensional hyper-sphere surface point and back again without loss of information.
P.S. In particular, I am interested in the case where $N \in \{2,3\}$.
Spherical coordinates should work and will map an $N$-dimensional cartesian coordinate to an $N+1$ dimensional sphere.
Suppose we have an input point $P$ in the cartesian space, with $P=(x_1,\ldots,x_n)$. For each $x_i$, we know that $-l\le x_i\le l$. We can scale and shift this cube:
$$u_i=\begin{cases}\frac{x_i+l}{2l}\pi &\mbox{if } i<n \\ \frac{x_i+l}{l}\pi &\mbox{if } i=n\end{cases}$$
We now have a rectangular prism with $0\le u_i\le\pi$ for $i<n$ and $0\le u_n\le2\pi$. This is exactly the range over which spherical coordinates operate, so $u_i$ specifies a unique point on the surface of the $N+1$-dimensional unit hypersphere.