Why are parallel lines a problem in Euclidean Geometry

I am reading the book Multiple View Geometry in Computer Vision, 2e, 2003 by Hartley and Zisserman. In the introductory 1st chapter, it states "Euclidean geometry is troublesome" and goes on to explain that "two lines almost always meet in a point, but there are pairs of lines that do not do so - those that we call parallel."

Why is this a sticking point? The text doesn't really explain it (except to say that there is a "difficulty with infinity"), and I don't have the necessary mathematical background to understand it.

• It is a good thing not all pairs of lines meet in a point. Traffic is bad enough as it is. – John Habert Feb 7 '14 at 18:50

It would have been nice if there was an intersection point of any two lines. In that case, it wouldn't have been necessary to say things like 'let X be the point of intersection of lines $k$ and $l$ (or $X=k\cap l$), which exists because $k$ and $l$ are not parallel', because the condition is then always true. There are other geometries for which every pair of lines (or the equivalent objects in those geometries) always have exactly one point of intersection, like the projective geometry.

• Projective geometry is the focus of the book in question, and the excerpts I quoted are provided as introductory text leading to an explanation of projective geometry. But why would it be nice to not have to say "... because k and l are not parallel"? Is it simply for being able to ignore an exceptional case? – Sabuncu Feb 7 '14 at 18:44
• @Sabuncu, well, if the book is about projective geometry, I can see why the author would qualify Euclidean geometry as 'imperfect' or 'not so nice'. Projective geometry does have some nice properties that Euclidean geometry doesn't have, but which one is better depends on your personal views. – Ragnar Feb 7 '14 at 18:47

It was suspected for perhaps more than two thousand years that one could prove, using more basic facts, the following:

• If $\ell$ is a line and $P$ is a point not on $\ell$, then there is exactly one line passing through $P$ that is parallel to $\ell$.

If I recall correctly, it was not until about two centuries ago that it was shown that that cannot be done.

The reason can be seen in part by thinking about the Poincaré disk model. If one could prove the proposition above using those putatively more basic axioms, then the same proof would show that the curves in the Poincaré disk model that serve in the role in which lines serve in Euclidean geometry also satisfy the proposition above, since those more basic axioms are still true in the Poincaré disk model with those curves in place of lines. But the proposition above is not true in the Poincaré disk model. Nor is it true in the real projective plane, in which no two lines are parallel, even those the real projective plane also satisfies those more basic axioms.

• I edited your post to separate the parallel postulate and your commentary on it. I hope you don't mind. – Cameron Williams Feb 7 '14 at 18:39
• One point why the parallel postulazte was so much "hated" and considered a candidate of a tehroem provable from the other axioms is the relatively cumbersome formulation it requires in the first place (compare with: "Through any two distinct points, there exists a straight line") – Hagen von Eitzen Feb 7 '14 at 18:44
• @Michael Hardy: Understanding the Poincaré disk model is beyond my skills/intelligence level. Thanks just the same. – Sabuncu Feb 7 '14 at 18:48
• @Sabuncu : It's probably not beyond you; it may be just that that particular article isn't the best thing for you to read. I'll see what else I can find. – Michael Hardy Feb 8 '14 at 0:01