$k$-dimensional manifold in $k^2$-dimensional space that passes through 3 points

Question

I'm very unsure about the mathematical terminology, so any help with that would be appreciated too.

$d$ is a $k$-dimensional space and $D$ is a $k^2$-dimensional space. $x$, $y$, and $z$ are points in $D$. $P: D \to d$ is an orthogonal projection.

I am looking for an $f: d \to D$ function that passes through $x$, $y$, and $z$ and $P(f(p)) = p$. So $f$ is kind of an inverse of $P$ that picks a manifold in $D$ that goes through $x$, $y$, and $z$.

I think many such manifolds are possible. I'm looking for something simple and smooth.

The application is that a user can see and manipulate a simplified view ($d$) in an application and I want to map it back to the full thing ($D$) while making sure some special points in $D$ are accessible.

Answer 1

I spoke to a mathematician! (My wife got home.)

So I think $f(p_1 ... p_k) = (p_1 ... p_k, g_{k+1}(p) ... g_{k^2}(p))$ where $g_{k+1} ... g_{k^2}$ are quadratic (?) polynomials that make sure $x,y,z$ are covered.