Can a two variable parametric equation ever determine a 3-D surface? Is it possible to concoct a set of two-variable parametric equations (let us say just three equations for simplicity) that determines a 3-dimensional surface / a solid geometric object in $\mathbb{R}$$^3$? I suppose the $2$-dimensional analog to this question would be, could we find two equations $x$ $=$ $f(t)$ and $y$ $=$ $g(t)$ where $t$ $\in$ $I$ for some subset $I$ $\subseteq$ $\mathbb{R}$, such that the set { ($f(t)$, $g(t)$) $\in$ $\mathbb{R}$$^2$ $|$ $t$ $\in$ $I$ } is a 2-dimensional surface?

Even a function with 1D domain is enough: there are space-filling curves, continuous functions $$f:[a,b]\longrightarrow\Bbb R^n$$ with n-dimensional image set.

**But** the space-filling curves are very bad behaved: not differentiable and not injective.