Definition of Parallel Lines

Canute

Can someone take a moment to give me the technical definition of a parallel line?

moshek

As far as i can remember in the Euclidian geometry
line A is parallel to line B if they don’t have any common point.
Maybe there is another or even better definition to that.

Moshek

www.physicsforums.com/showthread.php?t=17243

honestrosewater

I would just add that in Euclidean (aka parabolic) geometry, given line B, there is exactly one line parallel to it. But there are other non-Euclidean geometries where there is no line parallel to line B, or at least two lines parallel to line B.

You may be interested in reading about Euclid's fifth postulate and all the doomed attempts to prove it- just google Euclid's fifth postulate ;)

Happy thoughts
Rachel

happybunnyfood

If I'm not completely daft, the definition of a parallel line is a line that lies in the same plane as another but they will never meet.

honestrosewater

Wow, it's so easy to make a mistake if you're not careful. I need to correct myself. In Euclidean geometry, given line B and a point P not on line B, exactly one line can be drawn through point P that is both parallel to line B and lies in the same plane as line B.
I think that is correct.
Rachel

Imparcticle

xcellent job! I think that is a really good, simple definition. Also, parallel lines have the same slope.

honestrosewater

Is that correct? I didn't think parallel lines had to lie in the same plane. I thought it was just that two lines are parallel if they never intersect, as Moshek said.
Rachel

honestrosewater

http://aleph0.clarku.edu/~djoyce/jav...I/propXI9.html

Hurkyl

(at least in the text I've been looking at recently) two lines are said to be parallel if they don't intersect or are the same line.


This can be refined by saying two lines are skew parallel if they don't lie in the same plane, and antiparallel if you have directed lines and they point in opposite directions.

honestrosewater

It's different in this way?- Ex.
Draw a circle and its origin. Imagine the origin is a point on a line extending toward and away from you, perpendicular to the screen/page. Every line tangent to the circle will never intersect the line through the origin, and by my definition is parallel. Now, extend the tangent lines to planes in the same way the origin was extended to a line, perpendicular to the page. Every line in these "tangent planes" is also parallel to the origin line.
However, by adding the "in the same plane" requirement, in each tangent plane, only the lines perpendicular to the page would be parallel to the line through the origin.?
Please check, I make mistakes effortlessly
Rachel
(Not to mention I already wrote this post and lost it! so am annoyed)

moshek

Hurkyl Hi:

I like the idea that two line that are the same
are parallel, it's nice and i did not know that.

Thank you.

Moshek

HallsofIvy

Interesting. I would have sworn that every book I've looked in used "parallel" only to mean lines in the same plane that do not intersect and "skew" for lines that do not lie in one plane.

Canute

Dammit, I only wanted the orthodox textbook definition and already we're into non-Euclidean geometry and differences of opinion. That's what I love about this place. Thanks for all this.

master_coda

I agree. All the definitions I've ever seen include the requirement that the direction vectors of the lines are linearly dependent (and thus the vectors lie on a single plane). I've never heard "skew parallel" either. The definitions of skew that I've seen explictly require the lines to be not parallel; thus a pair of lines that do not intersect are either parallel or skew but not both.

moshek

Canute, That's is really Great !

And what do you know or think about
non-Euclidian mathematics ?

like of:

http://www.gurdjieff-internet.com/bo...php?authID=121


Moshek

Hurkyl

I have seen the definitions go the other way as well. I think the convention adopted depends on if you're taking a synthetic approach or an analytical approach.

HallsofIvy

It's probably another of those blasted "physicists versus mathematicians" things!