Given $k$ points in $n$-dimensions, such that $n\geq3$, is there a polytime algorithm for finding a curve that splits them into 2 sets of points?

Question

So in this math exchange question I asked, it was proven that for $n>2$ dimensions, you can always find a curve that separates $k$ points in $n$-dimensional space into $2$ arbitrary sets that you pre-defined. What I wanted to know is if there is a polytime algorithm (polynomial in $k$) for determining this curve.

If so, is there a known way to transform this curve into a function using some form of linear or non-linear transformation?