# How do you transform from affine space to projective space?

I have already seen this question but I still have enormous doubts: Difference between Projective Geometry and Affine Geometry

I'm reading Multiple View Geometry by Hartley and Zisserman, chapter 1. First they say this:

We can get around this by enhancing the Euclidean plane by the addition of these points at infinity where parallel lines meet, and resolving the difficulty with infinity by calling them “ideal points.”

By adding these points at infinity, the familiar Euclidean space is transformed into a new type of geometric object, projective space.

So, they say that when you get rid of "parallelism" in the euclidean space, putting a new line of points at infinity where these parallel lines meet, you get a projective space. Later they say this:

We will take the point of view that the projective space is initially homogeneous, with no particular coordinate frame being preferred. In such a space there is no concept of parallelism of lines (...) Thus, start with a blank sheet of paper, and imagine that it extends to infinity and forms a projective space IP 2 . What we see is just a small part of the space, that looks a lot like a piece of the ordinary Euclidean plane. Now, let us draw a straight line on the paper, and declare that this is the line at infinity. Next, we draw two other lines that intersect at this distinguished line. Since they meet at the “line at infinity” we define them as being parallel.

Okay, but:

1. The new space we have created (with the artificial, distinguished line at infinity) can't have really parallel lines now? I mean, what happens if I draw really parallel lines at this sheet of paper that don't intersect in the distinguished line?

2. Can I choose any line for this "distinguised line"? What is the point of doing this exactly? Because doing this I could decide that some lines that aren't really parallel would be defined as parallel if they intersect in this distinguished line in the paper.

From our point of view, the world plane and its image are just alternative ways of viewing the geometry of a projective plane, plus a distinguished line.

Wait, wasn't the projective space an euclidean space where you add a new line, called "line at infinity"? Now they say that the euclidean space is a projective space with an additional line called "distinguished line"? So, which space has more lines?

Later they say that when you put projective geometry plus a distinguised line you get affine geometry... wasn't this euclidean geometry? What is the exact difference?

I really don't understand anything, thanks in advance.

• I think it is easier to discuss of the difference projective/affine by taking examples (curves, varieties, subgroups of $GL,PGL,SL$). The Riemann sphere or complex projective line is $\mathbb{P}^1(\mathbb{C}) = \{ [z:w], (z,w) \in \mathbb{C}^2 \setminus (0,0),\forall \lambda \in \mathbb{C}^*, [\lambda z:\lambda w] = [z:w]\}$ – reuns Aug 20 '17 at 17:21
• Thank you for your answer but my level in math is not so high so I wouldn't understand the difference that way. I need sort of an "intuitive explanation" more than an algebraic one. – Carrascado Aug 20 '17 at 17:28
• Then the real projective line $\mathbb{P}^1(\mathbb{R})$ is the set of couples $[x:y]$ with $x,y \in \mathbb{R}$ not both zero, except we decree that $[\lambda x:\lambda y] = [x:y]$ for every $\lambda \in \mathbb{R}^*$. Thus if $y \ne 0$ then $[x:y] = [\frac{x}{y}:1]$, and every point of $\mathbb{P}^1(\mathbb{R})$ is of this form except the one with $y=0$, ie. $$\mathbb{P}^1(\mathbb{R}) = \{ [x:1], x \in \mathbb{R} \} \cup \{ [1:0]\}$$ – reuns Aug 20 '17 at 18:02