Create a bijection with continuous function between finite subspace of $\mathbb R^3$ and predefined finite subspace of $\mathbb R^2$

Question

Not sure if this is possible, but I want to make a continuous function that serves as a bijection between any finite subspace of $\mathbb R^3$, and a predefined subspace of $\mathbb R^2$. I'm looking for a practical/realizable (in the sense of software implementation) method to fit a continuous function approximator (like a neural network) as opposed to some really crazy mathematical trick that is super ugly and hardly possible to implement in software. I found some good comments and answers in this following question that make it seem like in the more general case ($\mathbb R^3$ and $\mathbb R^2$) is very ugly and probably not going to have a very implementable solution in real-time in software, but was hoping that a finite subspace of $\mathbb R^3$ might be constrained enough that there is a nice solution Does same cardinality imply a bijection?