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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.

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  • $\begingroup$ 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 $\mathbb{P}^1(\mathbb{C}) = \{ [z:w], (z,w) \in \mathbb{C}^2 \setminus (0,0),\forall \lambda \in \mathbb{C}^*, [\lambda z:\lambda w] = [z:w]\}$ $\endgroup$ – reuns Aug 20 '17 at 17:21
  • $\begingroup$ 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. $\endgroup$ – Carrascado Aug 20 '17 at 17:28
  • $\begingroup$ 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. $$\mathbb{P}^1(\mathbb{R}) = \{ [x:1], x \in \mathbb{R} \} \cup \{ [1:0]\}$$ $\endgroup$ – reuns Aug 20 '17 at 18:02
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Maybe think of it like this. Start with euclidean space, now for each family of parallel lines add ONE point which lies on all those lines, (and on no other lines). So now you have a bunch of new points at infinity. Now add one line which passes thorough all those points, (or say that those new points FORM a line, if that language is better for you.) Now you have all "original" lines intersecting the line at infinity, and intersecting each other either in an "original" point or in a new point at infinity. Now pass a bunch of laws declaring all lines are equal. (political commentary). This gives projective space.

To go backward, look at your homogeneous projective space pick any line, remove it and all points on it, and what is left is Euclidean space. Hope it helps.

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Consider looking at the ground, at a straight road that continues forever. The sides of the road are parallel, but as the road goes off into the distance, the sides of the road appear to meet at a single point on the horizon. This horizon line is like your line at infinity. (Yes I'm pretending the Earth is flat for this example.)

To answer your question 1: No there are no parallel lines in this new geometry, if you draw parallel lines on your sheet of paper then when they extend out to infinity they will meet at a point (on the line at infinity).

Question 2 is a little more technical, but essentially in this new geometry the line at infinity is not structurally different than any other line. Your notion of which lines are parallel will change according to which line you are calling the line at infinity, but this is really a nonissue because when you are focusing on the projective space and not the affine space, there are no parallel lines.

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