In a book that I am reading, Polygon Mesh Processing (page 1, last paragraph), the authors say this:
[…] implicit definition is only available for planar curves, i.e., $\mathcal{C} = \{x \in \mathbb{R}^2 | F(x)=0\}$ with $F:\mathbb{R}^2 \to \mathbb{R}$.
in the context of defining parametric and implicit surface representations for curves. On the other hand, the authors specify no such restriction of planar curves when talking about parametric surfaces.
What I fail to reason, or get an intuition for is why we are unable to define a 3D non-planar curve in implicit representation, while being able to come up with a parametric definition of the same?
Best Answer
Every closed subset of ${\mathbb R}^n$ is the zero level set of an infinitely differentiable function, see here. In particular, according to the definition you gave in a comment, every curve (with the mild restriction that it is a closed subset, that is also necessary) admits an implicit description.
However, likely, the authors of your test are simply sloppy with their language and in the definition of an implicit description of a subset $C$ as the zero-level set of a function $f: {\mathbb R}^n \to {\mathbb R}$, they also (implicitly) require that $0$ is a regular value of the function $f$, meaning that $$ \nabla f(x)\ne {\mathbf 0}, \forall x\in C. $$ The implicit function theorem will imply that the set $C$ will be a manifold of dimension $n-1$. Hence, $C$ cannot be a (nonempty) curve unless $n=2$.