I am trying to solve the following exercise:
Prove that the Real Projective Line $\mathbb R\mathrm P^1$ is diffemorphic to certain submanifold $M$ of $\mathbb R^3$ via the map $f:\mathbb R\mathrm P^1\to \mathbb R^3$ given by: $$f(x:y)=\Big(\frac{x^2}{x^2+y^2},\frac{xy}{x^2+y^2},\frac{y^2}{x^2+y^2}\Big)$$
It is a simple exercise show that $f$ is injective and thus $f:\mathbb R\mathrm P^1\to M=f(\mathbb R\mathrm P^1)$ must be a bijection.
Also, I can show that the Jacobian of $f$ has rank 1 respecto to the local charts on $\mathbb R\mathrm P^1$: $$ \begin{array}{cll} \mathcal U_1=\{(x:y)\in \mathbb R\mathrm P^1|x\ne 0\}&\longrightarrow&\mathbb R\\ (x:y)&\longmapsto& y/x \end{array}\qquad \begin{array}{cll} \mathcal U_2=\{(x:y)\in \mathbb R\mathrm P^1|y\ne 0\}&\longrightarrow&\mathbb R\\ (x:y)&\longmapsto& x/y \end{array}$$
Unfortunately I am not able to find which submanifold is $f(\mathbb R\mathrm R^1)$.
Any help?
$\textbf{Claim:}$ The image $f(\mathbb{R}P^1)$ is a circle of radius $\frac{1}{2}$ lying in the plane $u+w = 1$ with center $\left(\frac{1}{2},0\right)$.
We will denote the coordinates on $\mathbb{R}^3$ as $(u,v,w)$ as suggested by Thomas in the comments, and derive the suggested relations. Now $u(x:y) = \frac{x^2}{x^2+y^2}$ and $w(x:y) = \frac{y^2}{x^2+y^2}$. Hence $u(x:y) + w(x:y) = 1$ for all $(x:y)\in \mathbb{R}P^1$. So, the image of $f$ has to be contained in the plane $u+w = 1$. A similar calculation verifies that the image also satisfies $v^2 = uw$.
Now rearranging the equation for the plane we get $w = 1-u$. Replacing $w$ in the equation $v^2 = uw$ and rearranging, we get, $$v^2 +u^2 - u = 0$$ Finally, by completing the square we arrive at $$v^2 +\left(u-\frac{1}{2}\right)^2 = \frac{1}{4} $$