Is there a standard notation for a set which contains all the coordinates that are a part of a circle with a given radius?

Question

Let's say that we have a circle with a radius of $R$ units with it's centre coinciding with the origin of the Cartesian Plane.

It's equation will be : $x^2 + y^2 = R^2$
Now, does there exist a standard notation of a set that contains the coordinates of all the points that lie on the circle ($C_R$, for example)? If I ever use such a set in any statement, do I consider it to be predefined or define it whilst writing the statement? I have stated an example of a situation where I might use the statement below. $$\text{If } x \in \Bbb R \text{, } (\cos x, \sin x) \in C_1$$ If I need to define $C_R$, I will define it as follows : $$C_R = \{ (x,y) : x^2 + y^2 = R^2 \}$$ and hence, I would define $C_1$ as follows : $$C_1 = \{ (x,y) : x^2 + y^2 = 1^2 = 1 \}$$ I know that this seems pretty obvious but I'm asking this question as I've not seen this notation being used anywhere yet.
Also, $n(C_R) = \infty$, right?

Thanks!

Answer 1

I'm going to answer the question you asked:

Now, does there exist a standard notation of a set that contains the coordinates of all the points that lie on the circle ($_$, for example)?

The answer is "yes," and the standard notation is $[-R, R]$, because that set of real numbers contains all coordinates of all points that lie on that circle.

The question I think you meant to ask was "Is there a standard notation of the set of all points whose coordinates satisfy the equation $x^2 + y^2 = R^2$?"

The answer there is "not that I know of." But as other have said, you can say: "Let $C_R$ denote the set of points of the circle of radius $R$" or, more simply, "Let $C_R$ be the circle of radius $R$", because a circle is a set of points.

For $R = 1$, there is a very widely accepted notation, namely $S^1$ (the "one dimensional sphere"). You might argue that because it lives in the plane, it's two-dimensional, but that ship has sailed: $S^1$ means exactly the set of points whose distance from the origin in the plane is $1$. I've occasionally seen people write $3S^1$ to mean "the circle of radius 3", or even $rS^1$ to indicate the circle of radius $r$, but it's uncommon and looks ugly to me, and I cannot recommend it.

Short summary: use a short sentence to define the set $C_R$, and move on.