differential geometry – Showing \$S^1\$ is submanifold

I’m trying to show that $$S^1={ (x,y) in mathbb{R}^2: x^2+y^2=1}$$ is a submanifold of $$mathbb{R}^2$$ using the following definition:

A subset $$M in mathbb{R}^n$$ is called a smooth k-dimensional submanifold of $$mathbb{R}^n$$, $$k leq n$$, if any point $$x in M$$ has a neighborhood $$O_x$$ in $$mathbb{R}^n$$ and there exists a smooth vector-function $$Phi:O_x to O_0 subset mathbb{R}^{n},$$ onto a neighborhood of the origin $$0 in O_0 subset mathbb{R}^n$$ with $$text{ } rank frac{d Phi}{dx}|_x=n$$
such that $$Phi(O_x cap M)= mathbb{R}^k cap O_0$$.

I know the idea is to make the circle straight locally, but I can’t explain $$Phi$$, my initial proposal would be $$O_x = B_{frac{1}{4}}(x)$$ and $$O_0=B_0(1)$$, $$Phi(x,y)=(frac{x}{sqrt{x^2+y^2}+1}, frac{y}{sqrt{x^2+y^2}+1})$$, but $$Phi(O_x cap M) neq mathbb{R}^k cap O_0$$.How can I write to $$Phi?$$