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Let $E: (u,v) \mapsto (3,1,1)+u(1,0,2)+v(1,0,1)$ an affine plane. How do I get its completion in $\mathbb{P}_{3}(\mathbb{R})$.

I started with projective geometry and have my difficulties with understanding.

I know that $\mathbb{P}_{3}(\mathbb{R}) = \mathbb{R}^3 \cup H_{\infty} = \mathbb{R}^3 \cup \mathbb{P}_{2}(\mathbb{R}) $. And for an affine subspace $E \subset \mathbb{R^3}$ its completion $\overline{E}= E \cup \mathbb{P}(E_{Y})$.

Well $E_{Y}= u(1,0,2)+v(1,0,1)$ and $\mathbb{P}(E_{Y})$ should be all lines through origin of $E_{Y}$. This would be $[t_1:0:t_2]$.

I think I know what $E$ and $\mathbb{P}(E_{Y})$ are but I don't know how to unify them. The projective completion should be in $\mathbb{R}^4$ so I propably need to add $x_4$.

Can someone help me out?

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There is a bijection that takes points from $\mathbb{R}^3 \cup H_{\infty}$ to points in $\mathbb{P}^{3}$. Let $\{O;B\}$ be an affine reference with $B$ the base of the vector space associated to the affine space and let

$$\psi_o: \mathbb{R}^3 \cup H_{\infty} \rightarrow \mathbb{P}^{3}$$

be a bijection such that $$A\rightarrow [1,\vec{OA}]$$ if A is a point, and $$v\rightarrow [0,v]$$ if v is a vector. This is all you need to complete the affine space.

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