Regarding homogeneous coordinates for real projective plane (specially point $(0,0,1)$) We extend the Euclidean plane $\mathbb{R}^2$ by introducing points a infinity and a line at infinity.
Then we represent the extended Euclidean plane by homogeneous coordinates in $\mathbb{R}^3$.
(a) The points $(x,y)\in \mathbb{R}^2$ are represented by points $(x,y,1)$.
(b) The points at infinity are represented by points of the form $(x,y,0)$.
(c) The line at infinity is represented by the point $(0,0,1)$.
The point $(0,0,1)$ also belongs to point (a) above.
So does $(0,0,1)$ represent both the point $(0,0)$ of $\mathbb{R}^2$ and the line at infinity?
I wanted to relate it to the homogeneous coordinates in $\mathbb{R}^2$ for the extended real line $\mathbb{R} \cup \{\infty\}$. Where $x\in \mathbb{R}$ Corresponds to the point $(1,x)$ in $\mathbb{R}^2$ and the point at infinity corresponds to the point $(0,1)$.
 A: You can think of the projective plane as the set of all lines through the origin in $\mathbb R^3$. So every line through the origin in $\mathbb R^3$ is a point in $\mathbb RP^2$, and every plane through the origin in $\mathbb R^3$ is a line in $\mathbb RP^2$.
Every line in $\mathbb R^3$ is defined by two points. In this case, the origin and one additional point $(x_1,x_2,x_3)\neq(0,0,0)$. The coordinates of this additional point are the homogenous coordinates of that line as a point in $\mathbb RP^2$. We can embed the affine plane $\mathbb R^2$ into this projective plane by identifying $(x,y)$ with the line that goes through the point $(x,y,1)$. In other words, the homogenous coordinates of the point $(x,y)$ from the affine plane are $(x,y,1)$. The newly added points at infinity are the lines in $\mathbb R^3$ which do not go through a point of the form $(x,y,1)$, so the ones which do not intersect the plane $z=1$. In other words, the lines which are parallel to the $x-y$-plane, and thus go through a point of the form $(x,y,0)$. This is why the points at infinity have homogenous coordinates $(x,y,0)$.
Now to the line at infinity: As I said at the beginning, lines in $\mathbb RP^2$ are planes through the origin in $\mathbb R^3$. But planes through the origin in $\mathbb R^3$ are uniquely defined by a line through the origin which is perpendicular to said plane. But those lines are already the points in $\mathbb RP^2$. So there is a certain duality between points and lines in $\mathbb RP^2$: If we take a line in $\mathbb RP^2$, we can view it as a plane in $\mathbb R^3$, which is definable through a line in $\mathbb R^3$ perpendicular to said plane, and the line can again be interpreted as a point in $\mathbb RP^2$.
The line at infinity consists of all the points at infinity. In $\mathbb R^3$, this corresponds to all lines through the origin parallel to the $x-y$-plane. Together, they form the $x-y$-plane. The line perpendicular to this plane is the one pointing only in the $z$-direction, in other words, the one through the point $(0,0,1)$. So the line at infinity in $\mathbb RP^2$ can be identified with the line through the origin and $(0,0,1)$ in $\mathbb R^3$, which in turn can be identified with the point $(0,0,1)$ (homogenous coordinates) in $\mathbb RP^2$.
And that's why both the line at infinity and the origin in the projective plane have the same homogenous coordinates: $(0,0,1)$.
