Covering space of implicit vs parametric functions

[ellipsis]

Hello PF, I've got a curiosity question someone may be able to indulge me on:

The set of implicit functions covers a certain function-space - the set of all functions that can be represented by an implicit relation. Parametric functions also covers a function-space, that at least overlaps implicit functions in some points.

Is it the case that every implicit function can be converted to a parametric form? Or are there implicit functions that cannot be represented as a parametric function, and vice versa? Is the set of implicit functions a subset of parametric functions?

 

[mfb]

How exactly do you define "set of implicit functions" and "set of parametric functions"?

As an example, you can always define U(x,y) to be zero for an arbitrary subset of R^2, then the implicit function U(x,y)=0 can be everything.

 

[ellipsis]

I don't have the vocabulary to define these things the proper way. There are functions which just can't be written explicitly in an analytic, closed form. For a quick example, the solution of y in x=y^(y+1). Likewise, there are parametric functions that cannot be written in closed form, such as y=sin(t), x = cos(t). But that parametric function can be converted to implicit form - x^2+y^2 = 1.

Are you saying that the implicit function U(x,y) = 0 is a way of describing any function? I think there might be a function you can write parametrically but not implicitly e.g. x(t) = sin(t)+t, y(t) = cos(t)+t^2

Usually to go from parametric -> implicit, I would solve for T and substitute, but T can be unsolvable for both.

EDIT: I have discovered some information on the topic here. Whether or not an implicit function has a parameterization seems to be related to its 'genus'. Whatever that is.

 

[mfb]

Are you saying that the implicit function U(x,y) = 0 is a way of describing any function?

If you do not put additional constraints on U (continuous, or even differentiable, ...), yes.

 

[ellipsis]

So the answer to my question: Any parametric function can be described as an implicit function U(x,y), but not every implicit function can be described as a parametric one.

 

[mfb]

You can do the same with parametric functions. x=U(t), y=V(t). As long as you do not add more requirements for U and V, this can give every possible function.
This is not limited to R^2, of course, it works for all sets.

 

[WWGD]

Try looking at the implicit function theorem, which gives conditions for the existence of a(n) (at least) local representation f(x,y)=0 for a given function.