# Covering space of implicit vs parametric functions

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
Mentor
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.

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.

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mfb
Mentor
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.

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
Mentor
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