What? 2 parallel lines meet at infinity?

[brum]

anyone know of a proof for this?


here's my best guess...
if you think of "infinity" as an actual "place" (for example, you could say that x is at infinity), then i could kinda see how 2 parallel lines could meet.

if you think of "infinity" as only "approachable" (ie x is approaching infinity, meaning it's just getting bigger and bigger and bigger and bigger) then it doesn't seem reasonable that 2 parallel lines would meet.

in other words, if you think of "infinity" as a PLACE where the rules of geometry n' such are different, then yes, i can kind of see 2 parallel lines meeting.

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anyone got a proof / a way of thinking about it?

 

[StephenPrivitera]

Here's my thoughts. Suppose we have y=mx+b for one line and y=mx+c for another. Saying that these two lines meet at infinity is equivalent to saying lim(x-->oo) [(mx+b)-(mx+c)]=0, but we know that the limit is b-c. So I say these two lines will never meet unless b=c.
Also, I've heard you can think of a straight line as a circle of infinite radius. If you have two circles of infinite radius, then they can be considered concentric. If you have two circles which are concentric, they either overlap at all points or at no points. So two parallel line can never meet unless they are originally collinear. I'm not too sure about the second argument, but it sounds decent to me.
(edit: supposing they weren't concentric, they could feasibly intersect at two points, perhaps infinity and negative infinity - in this case, there should be some points in which y1>y2 and some points where y2>y1 - OW! infinity hurts!)

 

[KLscilevothma]

2 parallel lines meet at infinity?

I don't think it is true for Euclidean Geometry. According to the fifth postulate of Euclid's Element, "if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

Since the interior angles on the same side of 2 parallel lines equal 90 degrees, the lines won't meet even if prodeced to infinity.

 

[HallsofIvy]

You can't give any proof of it- it's not true unless you specifically define infinity so that it is true. And in that case you don't have Euclidean geometry anymore.

 

[jonnylane]

Originally posted by HallsofIvy
You can't give any proof of it- it's not true unless you specifically define infinity so that it is true. And in that case you don't have Euclidean geometry anymore.

Absolutely right. I have had this same debate before.

 

[selfAdjoint]

The form of geometry where this is assumed to be true is projective geometry. It has many beautiful results and is just as "true" as Euclidean geometry ("Of course parallel line meet at infinity - just look down railroad tracks").

 

[theriddler876]

lines do not meet at infinity, because infinity is a value, not a location, anyways, infinity does not cannot exsist by any technical purposes, it's just when people decided to stop counting

 

[quantum]

Infinity isn't a number, nor is it some sort of physical place, it is simply... large... Infinity is difficult to define, it simply means that it is large beyond comprehension.

 

[Hurkyl]

Infinity means precisely what one defines it to mean; nothing more, nothing less.

 

[rdecker]

ok!

I've been waiting for this. Infinity is a bunny rabbit with blue fur.

 

[Hurkyl]

Aww, how cute! Can I pet it?

 

[Organic]

Please think of an infinitely long carpet (hurkyl the pet in the car is yours).

Will its both sides ever meet ?

 

[quartodeciman]

Just for fun, you can seduce a geometry off a flat plane onto the surface of a sphere that makes it act like a space with a singular point (my preferred term).

Let S² be the surface of a sphere with radius 1. That provides the points and curves of the system. Let E² be a flat two-dimensional plane, like an endless rug that S² is sitting on. Set up 3D coordinates for the sphere S² and its rug E², letting the third coordinate be fixed at value 0 in defining the plane. Let point <0, 0, 0> be the one and only point shared by sphere and plane. The center of the sphere is at <0, 0, 1>, but this is not a point of either space. The top of the sphere (antipode to <0, 0, 0>) is <0, 0, 2>. This is a point of S².

Here is how we make the special geometry for S². For any point <x, y, z> of S², draw a 3D line through <0, 0, 2> and that point <x, y, z>. That line will meet the plane E² at one point <x', y', 0>. Likewise, for any point <x', y', 0> of E², draw a line through <0, 0, 2> and <x', y', 0>. That will cross the sphere S² at one different point <x, y, z> of S². Therefore, there is a 1-1 mapping that carries points of S² to points of E², except for <0, 0, 2>. <0, 0, 2> is our singular point for the coming geometry on S² and there is no line through it that meets E² at some point without crossing through the sphere S². This mapping is called a stereographic mapping of the sphere (minus one point) to the plane. It is invertible.

Here are some coordinate transformation equations for the stereographic mapping function:

mapping points of S² to points of E² --

x' = 2x/(2-z)
y' = 2y/(2-z)
z' = 0

(notice that this fails when z = 2, the singular point on the sphere)

mapping points of E² back to points of S² --

x = 4x'/(x'² + y'² + 4)
y = 4y'/(x'² + y'² + 4)
z = 2 - 8/(x'² + y'² + 4)

(notice that z is constrained to have a value between 0 and 2, except 2 itself is unallowed. This is because the denominator x'² + y'² + 4 is constrained to be value 4 or larger, so 8/(x'² + y'² + 4) is constrained to be value 0 to 2. But the only way this last term could have value 0 is if the denominator is not a finite value. Our points <x', y', 0> all have finite coordinates. So z must be 0-or-larger, up to, but NOT including, 2. The top point of the sphere is never reached.)

Alway remember that the points <x, y, z> of S² are constrained to lie on the unit sphere, so the coordinates must always obey the sphere equation:

(x-0)² + (y-0)² + (z-1)² = 1²
. That just boils down to:

x² + y² + z² = 2z
. The <x', y', 0> points are only constrained to have finite-valued coordinates, with z' always equal to 0.

The stereographic mapping allows us to map euclidean curves and figures from S² to the flat space E². We look at what the images of these subsets are in the 2D geometry of E² and attribute these geometric attributes to the originals on S². So, a straight line on S² will be something that maps to a euclidean line on E² and a circle of S² will be something that maps to a euclidean circle on E². The same rule holds for line segments, figures, etc. This is the stereographic geometry on S².

It turns out that the circles of this geometry are those euclidean circles (not only so-called "great circles") on the sphere that do not cross point <0, 0, 2>. Also, the lines of this geometry are those euclidean circles on the sphere that DO cross point <0, 0, 2>. Remember that <0, 0, 2> itself is NOT mapped to anything by the stereographic mapping. This geometry is a non-euclidean geometry, since there are no truly parallel lines (lines always contain point <0, 0, 2>). <0, 0, 2> is the one and only singular point of this space.

 

[hawaiidude]

Originally posted by quantum
Infinity isn't a number, nor is it some sort of physical place, it is simply... large... Infinity is difficult to define, it simply means that it is large beyond comprehension.

Like he said, infinity is not a place or a number...it's just an idea and a mathematical concept of things getting bigger and bigger but i don't think lines would meet at a certain point especially infinity...even the lim a number~~>infinit ( mx+b) this just means a number is getting bigger