Does generic projection into $\mathbb{R}^3$ preserve real-algebraic-curve-ness? 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), and so I'm pursuing dimensionality reduction into $\mathbb{R}^3$. (Considering the proof of Theorem 3.1 in this paper, 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, but my setting should be fundamentally different, since the failure of Tarski–Seidenberg with algebraic sets appears to stem from a failure to embed.
 A: 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 $3$.  
