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I've stumbled upon an exercise regarding the projective space and I have no idea how to proceed. At first I thought it was so easy, but not many minutes have passed and I found myself clueless. Perhaps, I'm not understanding the projective plane as well as I thought. Every little hint and idea will be helpful! Thank you!

The exercise:

If $Π=\left\{ (x,y,z)\in\mathbb{R}^3: x+2y-z=1 \right\}$ show that $(U_Π,φ_Π)\in \mathcal{A}^*$

What you might need to know:

We define the projective plane as $\mathbb{RP}^2 := \left\{ l:l \space line \space in \space \mathbb{R}^3 \space such \space that \space 0 \in l\right\}$ and a metric $d(l_1, l_2)=\widehat{(l_1,l_2)} \in \left[ 0,\frac{\pi}{2} \right]$ on it.

Now, for every plane $Π \subset \mathbb{R}^3$, with $0\notinΠ$ we define a chart $(U_Π, φ_Π)$ with $U_Π=\left\{l\in\mathbb{RP}^2:l \nparallelΠ\right\}=B_d(Π^{\perp}, \frac{\pi}{2})$ and $φ_Π = coordinates\space of\space l\capΠ\space on\space the\space cartesian\space system\space of\space Π$.

The basic charts of $\mathbb{RP}^2$ are $\left\{(U_i,φ_i)\right\}_{i=1,2,3}$ which correspond to the planes $x=1,\space y=1,\space z=1$.

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    What is $\mathcal A^*$ ? – 2017-01-16
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    The maximal atlas produced by $(U_i, φ_i), i=1,2,3$ – 2017-01-16
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    Did you try to compute the transitions functions ? What did you obtain ? – 2017-01-16
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    The first thing I've tried! But I can't get nowhere, because I can't write down $φ_Π$ as clear as $φ_1$ is! :/ – 2017-01-16
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    You can write a line passing by $(x,y,z)$ as $[x:y:z]$. Notice that by definition $[\lambda x : \lambda y : \lambda z] = [x:y:z]$ where $\lambda \in \mathbb R^*$. For example, your third chart is defined as $[x:y:z] \to (x/z,y/z)$. Its is defined if $l$ pass by the plane $z=1$. The transition function for the third chart is $\phi_1 \circ \phi_3^{-1} = (a^{-1}b, a)$. Is this clear ? – 2017-01-16
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    Point in your chart will be the point with $(1 - 2y + z, y, z)$, and you can compute everything exactly the same way as before. – 2017-01-16
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    I know how to make $(U_i, φ_i), i=1,2,3$ but I can't see how I can use this construction to make $φ_Π, φ_Π^{-1}$. Also, Ι don't know how to describe $φ_1(U_1\cap U_Π)$ and $φ_Π(U_1\cap U_Π)$ (although I believe it's just $\mathbb{R}^2$ without some line). – 2017-01-17
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    What I've managed so far: $1.\space U_1\cap U_Π = {[x,y,z]:x\neq 0, x\neq z-2y+1}$ and $2.\space φ_1\circ φ_Π^{-1}(u,v) = φ_1([v-2u+1,u,v]) = (\frac{u}{v-2u+1},\frac{v}{v-2u+1}) $ – 2017-01-17
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/51950/discussion-between-sotiris-simos-and-n-h). – 2017-01-17

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