6

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?

Gil
  • 707
  • Related : http://math.stackexchange.com/questions/200212/the-intersection-of-two-parallel-lines ? – lab bhattacharjee Aug 16 '13 at 19:13
  • 1
    In projective geometry: Yes! –  Aug 16 '13 at 19:14
  • For some people, the definition for two lines in the plane to be parallel is that they do not intersect so, given this definition, it is trivially true. – Dan Rust Aug 16 '13 at 19:16
  • @SamiBenRomdhane That's one way of looking at projective geometry, but globally there are no distinguished lines or points. [ie you can add points/lines/planes at infinity and get a model of projective geometry, but the axioms of projective geometry do not distinguish an "infinity"] – Mark Bennet Aug 16 '13 at 19:23
  • 1
    extremely related. Discusses what a parallel line. http://math.stackexchange.com/questions/411249/what-is-a-parallel-line – Jeel Shah Aug 16 '13 at 19:25
  • 1
    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. –  Aug 16 '13 at 19:27
  • @RahulNarain, whether or not the universe is infinite, the student's life is not. Getting them to recognize that their "infinity" is, from the outside, finite, and may be nearer or further depending on the way they treat others, could have some effect. – dfeuer Aug 16 '13 at 19:29
  • @MarkBennet Yes you're right the point at infinity is not a purely projective notion, but it allows to go from projective to affine and affine to projective as I know. –  Aug 16 '13 at 19:35
  • @SamiBenRomdhane Granted. – Mark Bennet Aug 16 '13 at 19:42
  • 1
    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 Answers2

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.

  • 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. – Dan Rust Aug 16 '13 at 19:33
1

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.

m.smakg
  • 303