Parallel lines and space curvature

[wildo69]

ok, i know that this has in someway been answered before, and i am new here, i also am by far not a geometry major, but that is why i am asking here, because you peopl eknow this stuff.
so here is the question
the standard definition of parallel lines are two lines on the same plane that are of an equal distance (hence they will never intersect)
i know there are non-standard or different definitions but i am trying to stay relativly basic here.
i remember reading or hearing somewhere that if space had a positive or negative curvature to it that two parallel lines may actual separate or intersect. so i have to ask if there is anyone that might be able to explain this, and alsoif they intersect how are they parallel, a part of me thinks that as soon as the distance between them increases or decrease or they intersect, they are not parallel lines so its ok if they intersect.

sorry i know i am getting a bit on a tangent,
i would appreciate any insight, or links so i could learn a bit more, or understand this a bit more.

 

[hello3719]

Well by definition parallel lines are lines which never intersect, it doesn't have anything to do with how the space is or isn;t. Unless you don't like the word "parallel" or its definition there is no problem or contradiction.

 

[AKG]

ok, i know that this has in someway been answered before, and i am new here, i also am by far not a geometry major, but that is why i am asking here, because you peopl eknow this stuff.
so here is the question
the standard definition of parallel lines are two lines on the same plane that are of an equal distance (hence they will never intersect)
i know there are non-standard or different definitions but i am trying to stay relativly basic here.
i remember reading or hearing somewhere that if space had a positive or negative curvature to it that two parallel lines may actual separate or intersect. so i have to ask if there is anyone that might be able to explain this, and alsoif they intersect how are they parallel, a part of me thinks that as soon as the distance between them increases or decrease or they intersect, they are not parallel lines so its ok if they intersect.

sorry i know i am getting a bit on a tangent,
i would appreciate any insight, or links so i could learn a bit more, or understand this a bit more.

Generally, wikipedia.com or mathworld.com are really good places to look for anything related to.. anything. Wikipedia's almost always got what you need. Anyway, here's a link to a wikipedia article on non-euclidean geometries (those which would described a curved space, and thus one where "parallel" lines may meet).

http://en.wikipedia.org/wiki/Non-Euclidean_geometry

Anyways, according to the mathworld site, the definition for parallel lines was altered slightly when it's discussed with respect to non-euclidean gemotries. It basically calls parallel lines those which, at some point, are both at right angles to a third line. This becomes pretty clear if you see the diagram at the wikipedia page. Now our normal parallel lines have this property. The lines in non-euclidean geometries with this property are also called parallel, and these types of lines can be shown to meet in some situations. In normal cases, we call parallel lines those that do not intersect, and they happen to be mutually at right angles to another line, in fact infinitely many lines, so they kind of took that aspect of parallel lines and made it part of a non-euclidean gemotrical definition for "parallel." I could be wrong, this is just the impression I get from what I've read.

 

[chroot]

Consider the surface of the earth, which is a positively curved two-dimensional space. Start on the equator. Two lines which cross the equator at right angles are both known as lines of longitude. Since the two lines both cross the equator at right angles, they can be said to be parallel. However, as you follow the two lines north (or south), you'll find that they grow closer and closer together until they eventually intersect at the north and south poles.

If you consider a very small neighborhood of the equator, you can treat it as if it were a plane -- the curvature of the Earth is insignificant over a small enough area. In that small region, the lines could be considered parallel; when you consider the entire surface of the earth, however, those parallel lines actually eventually intersect.

- Warren

 

[HallsofIvy]

"two lines on the same plane that are of an constant distance (hence they will never intersect)" is not the "standard" definition of parallel! The only definition of parallel is "two lines that never intersect". It is only in Euclidean geometry (on surfaces of 0 curvature) that parallel lines are always an constant distance apart.
Back when I was in high school (the dark ages!) I asked why not simply define parallel lines as being equidistant. The answer is that you cannot prove that the "set of points at a fixed distance from line l" is a line. Chroots example is excellent- the "lines" on a sphere (of positive curvature) are the "great circles" (a circle on the sphere whose center is the center of the sphere). The Earth's equator is an example. The set of all points of latitude 1 degree north is a "set of points equidistant" from the equator but are NOT on a great circle.

 

[jtolliver]

"two lines on the same plane that are of an constant distance (hence they will never intersect)" is not the "standard" definition of parallel! The only definition of parallel is "two lines that never intersect".

are these two lines parallel?

x=s
y=-2s
z=0

x=3t
y=2t
z=1

it is easy to see that they do not intersect(look at the z-coordinates).

It is only in Euclidean geometry (on surfaces of 0 curvature) that parallel lines are always an constant distance apart.

my example above was in euclidean geometry, and by your definition they are parallel.
however they are not a constant distance apart. the closest distance any points on the lines can get is 1 (due to the difference of the z-coordinates). one example of such points is (0,0,0) and (0,0,1).
consider the case where s=1. if there exists a point on the second line at a distance 1 from the point corresponding to s=1, (1,-2,0) then its value of t satisfies

(3t-1)^2+(2t+2)^2+1=1

and thus 3t=1 and 2t=-2
since these equations are contradictory the lines are not at a constant distance.