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Subject: "Parallel Lines"     Previous Topic | Next Topic
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Conferences The CTK Exchange Middle school Topic #121
Reading Topic #121
abhineetkaul
Member since Sep-1-06
Sep-01-06, 07:32 PM (EST)
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"Parallel Lines"
 
   Hi,

Could you please clarify my doubt that in a 2-D plane, where do parallel lines meet? Do they meet at + infinity or - infinty or both?


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alexbadmin
Charter Member
1891 posts
Sep-04-06, 12:39 PM (EST)
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1. "RE: Parallel Lines"
In response to message #0
 
   I am certain that the notions of ±infinity is not applicable to a plane. When it comes to that, they usually consider a whole line at infinity, not just two points.

Each point at a line at infinity corresponds to a direction in the plane, so that all lines parallel to that direction meet at that one point.


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Jack A. Cohen
guest
Sep-29-06, 07:18 AM (EST)
 
2. "RE: Parallel Lines"
In response to message #1
 
   If two parallel lines intersected in a point that would violate Euclid's 5th postulate and we would no longer be doing "plane geometry."

I believe that it was the replacement of this postulate (which had been viewed for centuries as too complex to be a postulate, and as qualitatively different in structure from the other postulates) with the assumption that hey intersected in a point which led to the first non-Euclidean geometry, which I believe is the geometry of the projective sphere:

Establish x-y axes in the plane.
Set a unit'sphere on the origin.
Map each point, p, in the plane to a unique point, s, on the sphere by drawing a straight line from the "north pole", N, of the sphere to p; s is the point where that line intersects the sphere.

Under this mapping all lines eventually intersect at "the point at infinity", which maps into N, the north pole of the sphere. (Visualize the line from N to the origin, then watch as you move p, the point in the plane, along the x-axis. You can "see" s, the point of intersection with the sphere, moving steadily upward. And then realize that it doesn't matter what direction the point is moving in.)

But this is all WITHOUT the 5th postulate, and consequently it requires a perspective from outside the plane (namely, from N).


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alexbadmin
Charter Member
1891 posts
Sep-30-06, 07:07 AM (EST)
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3. "RE: Parallel Lines"
In response to message #2
 
   >If two parallel lines intersected in a point

By defintion, two lines are parallel if they do not intersect. Thus paralle lines can't intersect.


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