In the usual stereographic projection, the projection point is taken to be $(0, \ldots, 0, \pm 1) \in S^n$ and we project to the plane $x_{n+1} = 0$. Is there a formula for the projection from an arbitrary point $p$ on the sphere to the plane $x_{n+1} = 0$?
I have found this thread: Stereographic Projection from an Arbitrary Point, but it seems to be concerned with the case where we are projecting from an arbitrary point $p$ to a plane which is tangent to the sphere at $p$. In contrast, I'm interested in both $p$ and the plane being in general position to one another.
In particular, I'm interested in computing the area element on the hyperplane $H$ to which we are projecting. I know for the 'standard' projection this looks like $$ \left(\frac{2}{1 + r^2}\right)^{2n}dA $$
I'm also interested in what happens when we do this kind of projection from the sphere to a more general surface $\mathcal{M}$ such as the graph $x_{n+1} = F(x_1, \ldots, x_n)$.