Define $M$ as the set of all pairs $(p, r)$ where $r$ is an affine line $r \subset \mathbb{R}^2$ and $p \in r$.

I am given a map $P \colon \mathbb{R}^3 \to M$ where $(u_1, u_2, u) \mapsto (p,r)$ where $p = (u_1, u_2)$ and $r = \{(x,y) \in \mathbb{R}^2 \mid y-u_2 = u(x-u_1)\}$ and I am asked the following

Show that $P$ is injective, find its image $\mathcal{U}\subset M$, define a chart $(\mathcal{U}, \varphi = P^{-1})$ of $M$ (and similarly define another chart $(\mathcal{V}, \psi)$ so that $\{(\mathcal{U}, \varphi), (\mathcal{V}, \psi)\}$ is an atlas. Prove that there is a unique riemannian metric $g$ on $M$ such that its local expression in the chart $(\mathcal{U}, \varphi)$ si $$g^{\varphi} = du_1^2 + du_2^2 + \frac{1}{(1+u^2)^2}du^2$$

Showing that $P$ is injective has been no problem, but I am unable to find it's image. Also I can't find the inverse map $P^{-1}$. I believe that if I had at least $P^{-1}$ I could prove the last question about the local representation of the metric $g$, but I am stuck at the beginning. Thank you.

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    $\begingroup$ Given an arpitrary $(p,r)\in M$, can you find $(u_1,u_2,u)\in \Bbb R^3$ that masp to $(p,r)$? What is a possible obstacle? $\endgroup$ – Hagen von Eitzen Jun 26 '18 at 18:29
  • $\begingroup$ This raises the interesting question of endowing the set of affine lines with a « natural topology ». Some points on that question here. $\endgroup$ – mathcounterexamples.net Jun 26 '18 at 20:20
  • $\begingroup$ @HagenvonEitzen Well, the first two coordinates $(u_1, u_2)$ have to be precisely the coordinates of the point $p$. Since $r$ is a line that goes through $p$, the las coordinate should be the slope of $r$, maybe? $\endgroup$ – user313212 Jun 27 '18 at 7:22

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