What Are Parallel Lines in Geometry?

[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]

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

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]

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.

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/java/elements/bookXI/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]

That's nice

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]

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.

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]

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.

Canute, That's is really Great !

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

like of:

www.gurdjieff-internet.com/books_template.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!

 

[moshek]

Well as far as i can see There are no parallel line not in the real world and not outside any real world ( Plato) it just two words " Parallel " and "line".
and the question is way these words relate to mathematics language ?

 

[Edwin]

Draw a set of 4 horizontal parallel "x" lines, and 4 vertical "y" lines that intersect to form a square grid. This should make 16 squares. Within each sqaure, draw a circle that is just large enough to fit snuggly in each sqaure. Then pick a point on the paper somewhere within the squares and mark it with a pencil or pen. Next, using a ruler, measure a straight line from the center of every circle through the point you marked on the paper and where your ruler intersects the circumference of the circle, mark a pencil dot. Do this for all 16 circles. After you have completed this step, connect all the dots on each circle within each collumn and row you made on each circle's circumference. You will notice that there exists an underlying series of varying curves by the pattern of dots you made on the circles' circumferences that get more extreme as you approach the point on the piece of paper. This curvature is pretty obvious with just sixteen circles, but if you were to fill in 1,000 overlapping circles within that square grid and repeat the steps above, you would notice a pronounced curvature, that almost seems like Einstein's curved space. If this grid were without bounds, then the degree of curvature of space would approach zero as you go further and further from the point in space you marked on the paper. It is presumable that any where within a finite distance of the point, the space that we formed by the method mentioned above, has negative curvature. It is presumable that as you approach an infinite distance from the point on the paper, the curvature of the space we made, approaches zero. The question I have is as follows: is there a solution such that the arbitrary negative curvature of the mathematical space mentioned above exactly equals the degree of negative curvature of space-time predicted for an object composed of x number of oscillators with a mass m kilograms?

Inquistively,

Edwin G. Schasteen

 

[Canute]

Canute, That's is really Great !

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

like of:

www.gurdjieff-internet.com/books_template.php?authID=121


Moshek

Checked the link but don't know which bit you meant me to read. Interesting that Gurdjieff's name should come up here. What's the connection?

I'm no mathematician but at least know what hyperbolic and (er, what's the other one called) are, roughly.

I asked the question out of a growing interest in Euclid's fifth postulate.

I'm going to take it that parallel lines are lines on the same plane that never meet when extended infinitely in both directions. Is that ok?

Edwin - solution to follow shortly.

 

[moshek]

You gave a good definition. In a projective geometry the direction of the line is also a point of the line so there are no parallel line and a very nice duality between points and lines.

Are you familiar with the Holy dance of Gurdjieff's ?

Moshek

 

[Gokul43201]

I'm pretty certain that parallel lines need to be coplanar. "Is parallel to" is considered an equivalence relation, so it has to be transitive. This would not be true for skew lines, i.e : A does not intersect B and B does not intersect C, doesn't require that A not intersect C.

 

[fourier jr]

Ok straight from the horse's mouth:

Euclid's Postulate 5, the only one that's a real mouthful compared with the others
That, 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.
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#posts
Playfair proved that this is equivalent to
Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

Here's a short history of non-Euclidean geometry, including a discussion of the parallel postulate:
http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html

 

[Canute]

You gave a good definition. In a projective geometry the direction of the line is also a point of the line so there are no parallel line and a very nice duality between points and lines.

Yep.

Are you familiar with the Holy dance of Gurdjieff's ?

Moshek

No I don't know that. What's it all about? I only know Gurdjieff via Ouspensky, (but I mostly agree with what he says).

 

[moshek]

I am glad you agree "Gurdjieff via Ouspensky" as you wrote.

Please look at:

www.gurdjieffmovements.org/


Moshek