Let's say we have a $d$-dimensional unit hypersphere. On the surface of the hypersphere, we have points that have a value of either $\{-1, 1\}$. I wish to create a function $\Phi$ that interpolates between these points, such that if I pick any point on the hyper-sphere's surface I get a value $[-1, 1]$.
Hopefully the above image demonstrates the problem (in 2 dimensions). As far as I can tell, we can define a unique point on the surface of the $d$ hypersphere using $d-1$ variables provided we have a fixed reference point $u$. If we consider a sample point on the surface to be defined as a vector $v$ that starts at the centre of the sphere and has length $1$, then we can simply pick an arbitary reference vector $u$ and choose $d-1$ planes such that each plane is orthogonal to all other planes and measure the angle of $v$ from $u$ in each of these planes.
Therefore, the surface of the hyper-sphere from the reference point $u$ is simply $[0, 2\pi]\times...\times[0, 2\pi]$ $d-1$ times. The interpolant then should be $\Phi:[0, 2\pi]\times...\times[0, 2\pi] \mapsto [-1, 1]$. I need to find a tidy way to create this $d-1$ interpolant that takes into account the continuous/looping/infinite nature of hyperspheres. In other words, how do I go about constructing this interpolant (which sits on the boundary of a high dimensional sphere) when each point on the boundary needs to agree with the derivative and value of the opposite boundary? Or is there a better way to frame this problem?
Thanks in advance!