Take the cubic bezier graphed below. This is a type of basic parametric function. It uses the functions input (time) and outputs coordinates (x and y)
(My graph: https://www.desmos.com/calculator/ipw4mewbvw)
My question is... Are 2D functions like SDFs a type of parametric? Theyre different, almost the reverse of the example above. They take in coordinates (x and y) and output a value.
For example, A spherical SDF gradient (magnitude(x, y) - 1) limited by a less-than to get a solid object:
If its not considered parametric... What should it be called?
A planar parametric curve is a mapping from some interval in $\mathbb R$ to $\mathbb R^2$.
An SDF (signed distance function) is a mapping from $\mathbb R^2$ to $\mathbb R$.
Signed distance functions are a subset of the larger class of implicit functions, which are again mappings from $\mathbb R^2$ to $\mathbb R$. So, as you said, this is the “reverse” of the parametric case, in some sense.
Parametric curves and implicit curves are two different things. If your parametric curve can be parameterized using rational functions (like the Bézier curve you showed), then it can be converted to implicit form, though the implicit form is typically not an SDF. It’s easy to find examples of implicit curves that can not be expressed in parametric form.
$F(x,y) = x^4 + y^4 - 1$ is an implicit curve that is not an SDF.
$F(x,y) = xy$ is an implicit curve that can not be expressed in parametric form (though portions of it can).