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Can parallel lines meet?

There is a person that takes a calculus course with us, and every time we ask him for something he answers us with I'll do it when two parallel lines meet each other. So I decided to give him a proof of this so that he won't say it anymore (it is annoying).

I've heard that two parallel lines can meet each other at infinity, is this true?

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marked as duplicate by Chris Godsil, Amzoti, Micah, Davide Giraudo, Henry T. Horton Aug 16 '13 at 20:38

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Related : math.stackexchange.com/questions/200212/… ? –  lab bhattacharjee Aug 16 '13 at 19:13
In projective geometry: Yes! –  Sami Ben Romdhane Aug 16 '13 at 19:14
extremely related. Discusses what a parallel line. math.stackexchange.com/questions/411249/what-is-a-parallel-line –  gekkostate Aug 16 '13 at 19:25
Whether two parallel lines never meet or whether they meet at infinity, it doesn't help you get the other person to do things in finite time. –  Rahul Aug 16 '13 at 19:27
Roughly related - there are finite projective spaces, including projective planes over finite fields. Finite projective planes can still have points and a line "at infinity" (properly defined) –  Mark Bennet Aug 16 '13 at 19:48

2 Answers 2

As Daniel Rust notes, the definition of parallel is that two lines don't meet. What some people are trying to point out as examples are situations where lines cannot be parallel. These settings help regularize the geometry. For example, spherical geometry takes place on the surface of a sphere. The "lines" in spherical geometry are the "great circles": the circles which have the diameter of the sphere. Note then that two lines always intersect in a "point" (which in spherical geometry is defined as the two points opposite each other on the sphere).

Spherical geometry regularizes plane geometry in several ways. First, it elminates parallel lines: now every two lines intersect in a point, and every two points define a line (exercise!). Second, it unifies the treatment of lines and circles: everything is now a circle, in effect.

So "parallel" does strictly mean two lines that do not meet, but there are ways to eliminate the concept with a suitable geometry. Projective geometry is another very useful but more complex way to do this.

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I should add, there are other definitions of 'parallel' lines in the plane ($\mathbb{R}^2$) which are equivalent, but which necessitate a proof that two parallel lines do not intersect. For instance, '$l_1$ and $l_2$ are parallel if, $l_1\neq l_2$ and for all $x_1,x_2\in l_1$ we have $\inf_{y\in l_2}\{|x_1-y|\}=\inf_{y\in l_2}\{|x_2-y|\}$', or Euclid's celebrated parallel postulate. –  Daniel Rust Aug 16 '13 at 19:33

Parallel lines cannot meet as by definition, parallel lines are lines that remain the same distance apart, no matter what part of the lines are compared.

Him saying "I'll do it when two parallel lines will meet each other" is another of saying he'll never do it.

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