I'm interested in the topological properties of certain real algebraic curves in high-dimensional spaces. I want to visualize these curves (say, [like this](https://mathematica.stackexchange.com/questions/5968/plotting-implicitly-defined-space-curves)), and so I'm pursuing dimensionality reduction into $\mathbb{R}^3$. (Considering the proof of Theorem 3.1 in [this paper](https://arxiv.org/abs/1103.5479), I expect a generic projection of this sort to successfully embed the curve.)


**Is the generic projection of a real algebraic curve into $\mathbb{R}^3$ again a real algebraic curve? If so, is there an efficient procedure to derive the corresponding polynomials?**


[This question is related](https://mathoverflow.net/questions/86099/projections-of-real-algebraic-curves), but my setting should be fundamentally different, since [the failure of Tarski–Seidenberg with algebraic sets](https://en.wikipedia.org/wiki/Tarski%E2%80%93Seidenberg_theorem#Failure_with_algebraic_sets) appears to stem from a failure to embed.



Does generic projection into $\mathbb{R}^3$ preserve real-algebraic-curve-ness?
The following is pretty much the standard argument due, I think, to Whitney (from the proof of his embedding theorem in the "stable range"). Suppose that $V\subset {\mathbb C}^N$ is an affine complex-algebraic subset defined over the real numbers (I.e. by polynomials with real coefficients), where $dim(V)=m$ and $2m+1< N$. Consider the map 
$V\times V\times {\mathbb C}\to {\mathbb C}^N$, $(x,y,t)\mapsto t(x-y)$. Due to our dimension assumptions, this map is not surjective. For the same dimension reasons, its image $W$ (a complex algebraic subvariety) does not contain ${\mathbb R}^N$ and, moreover, is "small" in any meaningful sense (its dimension is $<N$). Take any (necessarily nonzero) $v\in {\mathbb R}^N \setminus W$ and let $p: {\mathbb C}^N\to {\mathbb C}^N/Span(v)$ denote the quotient map (you can think of it as the orthogonal projection to the subspace normal to $v$ with respect to the standard inner product). Then this map is injective. It is also defined over the real numbers, hence, $p(V({\mathbb R}))$ (image of the set of real points in $V$) is contained in the set of real points of ${\mathbb C}^N/Span(v)$. I claim that the set of real points of $p(V)$ equals $p(V({\mathbb R}))$. Otherwise, there is a pair of distinct complex-conjugate points $u, v\in V$ such that $p(u)=p(v)$, contradicting injectivity of $p$. Thus, $p(V({\mathbb R}))\subset {\mathbb R}^N/Span\_{\mathbb R}(v)\cong {\mathbb R}^{N-1}$ is an affine real-algebraic set. 


In your case, $m=1$ and hence, we can apply the projection argument inductively until the dimension of the ambient space drops to $3$. 


