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I'm learning projective geometry and I'm having difficulties understanding the notion of homogeneous coordinates. Here's my problem :

We define the real projective line $\mathbb{RP}^1$ as the set of equivalence classes of $\mathbb{R^2} \setminus (0, 0)$, i.e., $2$-space without the origin, where two points $P = (x, y)$ and $P' = (x', y')$ are equivalent iff there is a nonzero real number $\lambda$ such that $P = \lambda P'$, i.e., $x = \lambda x', y = \lambda y'$.

The way I understand this is that $\mathbb{RP}^1$ is in fact the space of all lines through the origin in $\mathbb{R^2}$ (or equivalently the set of all one-dimensional subspaces of $\mathbb{R}^2$).

Now for the part which I don't understand. It is said (in my notes) that, the usual way to write an element of the projective line, i.e., the equivalence class corresponding to an point $(x, y)$ in $\mathbb{R}^2$, is $[x : y]$ called the homogeneous coordinates associated to the point $(x, y)$. However, later on, it is said that to make $2$D homogeneous coordinates we simply add an additional variable $z$ into the existing coordinates. Therefore, a point in $(x,y) \in \mathbb{R^2}$ becomes $[x : y : z]$ in homogeneous coordinates.

This is very confusing. On one hand we define $[x : y]$ to be the homogeneous coordinates associated to the point $(x, y)$ and a few lines later the author is saying that any point $(x,y) \in \mathbb{R^2}$ becomes $[x : y : z]$ in homogeneous coordinates. At the end of this I don't understand anymore what homogeneous coordinates are. Maybe someone can clarify my misunderstanding.

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  • $\begingroup$ $[x:y:z]$ is typically a homogeneous coordinates for $\mathbb{RP}^2$, i.e. a space of all lines through the origin of $\mathbb{R}^3$ $\endgroup$ – user113988 Sep 4 '17 at 16:41
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I’d guess that the “later on” part of the notes is talking about the projective plane $\mathbb{RP}^2$, not $\mathbb{RP}^1$. Similarly to the construction that was used for homogeneous coordinates in $\mathbb{RP}^1$, we identify points in $\mathbb{RP}^2$ with lines through the origin in $\mathbb R^3$. The coordinates of the points that lie on any such line are scalar multiples of some triple $(x,y,z)$, so we can take $[x:y:z]$ as the homogeneous coordinates of the corresponding point in $\mathbb{RP}^2$.

In case this hasn’t been mentioned in your course, an important thing to note is that the significant things in homogeneous coordinates are the ratios between the components (which is what motivates the use of a colon as a separator); the specific values of the coordinates aren’t as important. That’s one reason why $[x:y:z]$ and $[kx:ky:kz]$ (for $k\ne0$) represent the same point. So, even though there are three numbers involved in $[x:y:z]$, there are only two degrees of freedom, which is what one would expect for a two-dimensional space.

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