This is the 3D version of this question. The responses to that question contained a lot of complaints about fuzzy definition of the problem, so I made this new question very narrow and explicit.

For a given fixed degree, $n$, I'm interested in constructing parametric polynomial approximations of degree $n$ of certain curves, for example those that arise as intersections of two surfaces. This is important in engineering and manufacturing computing, where parametric polynomials are usually the only curves available (in the form of Bézier curves).

A typical example is as follows: Let $f(x,y,z) = x^2 + y^2 - 16$, and let $S(f)$ be the cylindrical surface where $f(x,y,z)=0$. Similarly, let $g(x,y,z) = y^2 + z^2 - 25$, and let $S(g)$ be the cylindrical surface where $g(x,y,z)=0$. Let's focus on the curve $C$ that is the portion of the intersection of these two surfaces lying in the positive octant where $x,y,z \ge 0$. A few simple calculations show that $C$ has end-points $\mathbf{p}_0 = (4,0,5)$ and $\mathbf{p}_1 = (0,4,3)$.

I want to construct a parametric polynomial curve $t \mapsto \mathbf{x}(t) = \big(x(t), y(t), z(t)\big)$ of degree $n$ that approximates $C$ and such that $\mathbf{x}(0) = \mathbf{p}_0$ and $\mathbf{x}(1) = \mathbf{p}_1$. The error in the approximation will be measured by
$$
E(\mathbf{x}) = \sup\big\{ f(\mathbf{x}(t))^2 + g(\mathbf{x}(t))^2 : 0 \le t \le 1\big\}
$$
**First question:** for a given $n$ how can I find the polynomial curve $\mathbf{x}$ of degree $n$ that minimizes $E(\mathbf{x})$?

If we were approximating a real-valued function, there would be a great deal of approximation machinery that would help us: the Weierstrass approximation theorem, the equi-oscillation criterion, the Remez algorithm, etc. If I parameterize $C$, then each of its three components will be a real-valued function, and I can apply these known techniques. But the result will depend on how I parameterize $C$, and will not be optimal.

**Second question:** Has any of the above-mentioned approximation machinery been generalized beyond the real-valued case?