Projective completion of an affine plane

Question

I know that a infinite point in a projective space can be defined as a equivalence class of $||$ relation in affine space.
So if i have an affine space $(\mathcal{A},\mathcal{D}, \phi)$, let $\mathcal{D}\subset\mathcal{P(\mathcal{A})}$ a set of "lines". Because in affine space parallelism is an equivalence relation, then we can define $ [d]=\{d'|d'\subset$$\mathcal{D}$$, d||d'\}$. And why infinity points are defined as $\{[d]| d\in\mathcal{D}\}$ ? $[d]$ shouldn't be a line? Why it is a point?

Answer 1

$d$ is a line, but $[d]$ is a set of lines.

Two lines define a point, namely their point of intersection. In affine geometry you need the lines to not be parallel for this to be the case, but in projective geometry any two distinct lines define a point.

You can also add more lines to the definition, as long as they meet in a single point. You might describe each finite point by the set of all lines through that point. And you might define each point at infinity by all the parallels in that direction.

Whether such a set of concurrent lines describes the point in question (in a bijective way), or whether it actually is that point, is a matter of perspective. The former is certainly more aligned with intuition, while the latter is closer to the formulas in your case. An axiomatic system would typically use point as a term that is only defined by the properties of those axioms. In that sense, as long as you can describe geometric operations on them you might as well consider the sets of lines as being points, in your model of the plane.

I would recommend treating finite and infinite points the same. If you say a point at infinity is a set of parallel lines, then a finite point should also be considered a set of concurrent lines. If you say the point is an entity in its own right, with incidence to lines as a property, then a point at infinity should likewise be an object in its own right, outside the affine plane, and the parallels incident with it are merely a property of that new kind of object.