Smaller index sets that define all circles (centered at (0,0))

Question

Circles can be defined with the index set {Ar|r∈{R}} if Ar={(x,y)|x^2+y^2=r^2}.

However, most circles would be repeated in this indexed family. How could I define the same collection of circles without repetition?

Answer 1

$\{Ar:r\in\mathbb{R}^+\}$ seems to work as the only repeats in your construction is that $r$ and $-r$ give rise to the same circle.

It's worth noting that this isn't every Circle in $\mathbb{R}^2$, merely the ones centered at $(0,0)$. To get all the circles you would want $\{A(r,a,b):a,b,r\in\mathbb{R},r\geq 0\}$ where $$A(r,a,b)=\{(x,y)\in\mathbb{R}^2:(x-a)^2+(y-b)^2=r^2\}$$