Line... circle?

[mendhak]

Is a line basically a circle of infinite radius?

[The_Duck]

No I don't think so. You can mathmatically prove that the two ends of a linear line will never meet !

[mendhak]

Yes, and and the same goes for a circle of infinite radius, doesn't it?

[sciguyryan]

Yes, and and the same goes for a circle of infinite radius, doesn't it?

You are right there.

It may be or it may not be - there is just no way to prove this.

Cheers,

RyanJ

[sql_lall]

I didn't think a circle of infinite radius had ends...
Anyway, it's probably not important, but you can prove that an ellipse, hyperbola and a parabola are the same thing, by using homogeneous coordinates and a 'line at infinity'. Maybe this'd work, but i'm not sure how you could fit in a definition of an infinite radius.
Something like: Assume the centre is in R2. Therefore, it is all the points an inifinite distance from it's centre. Therefore, the circle with infinite radius is the line at infinity.
Somehow I don't think that's right though.

[zaza]

Geometrically, a line isn't a circle. But, when you start dealing with infinities, geometry starts getting into trouble. If by "line", you mean one that you can draw on a piece of paper with a ruler, then you could define that as having a curvature of 0. And since curvature is 1/r, then a radius of infinity would give a curvature of 0, so in that sense, yes.

However, what happens if you roll the paper into a cylinder so that the two ends of the line meet? Do you have a line or a circle? In that instance, the dimension in which the curvature acts is not the same as that in which the line acts. This is different from drawing a circle on the paper using a compass. General relativity.... .

So you have to be a bit careful. I'm not really sure why you'd want to use a circle of infinite radius anyway.

HTH

zaza

PS: but here's one to think about: if a line was a circle of infinite radius, then rather than being on the edge of the circle, imagine you are now at the centre. Where is the edge and what does it look like?

[wossname]

Is a line basically a circle of infinite radius?

That is indeed a recognised and respected analogous model of a straight line.

My esteemed and greatly respected work colleague Clive has been a civil engineer for upwards of 40 years and was a student of some of the classical mathematical models. He has on several occasions described a line as an infinitely large circle's edge. If Clive says it then it's gospel as far as I'm concerned.

[wossname]

However, what happens if you roll the paper into a cylinder so that the two ends of the line meet? Do you have a line or a circle? In that instance, the dimension in which the curvature acts is not the same as that in which the line acts. This is different from drawing a circle on the paper using a compass. General relativity....

Interesting but misleading I think. By rolling up the paper you are effectively changing the axis along which the centre-point of the circle is found.

Lets say you draw the line near the top of the A4 page, the centrepoint would be down the page (and keep going forever in that direction). BUT if you roll up the paper while still on the desk then the centrepoint will be vertically (a couple of inches) above the desk. So you have altered the circle's plane by 90 degrees (around the X axis).

The new radius would not be comparable because the original line now occupies 3 dimensions (drum-roll please...). The original line (now curved back on itself) would actually lie along the surface of an infinitely large SPHERE!! The centre of this sphere would also be a couple of inches above the desk BUT INFINITELY far down the page.

OMG I surprise myself sometimes.

I hope this makes sense to someone.

[zaza]

Hey Woss-ster,

I agree that I have changed the plane of the circle so that it is no longer in the plane of the page; it was just to try to see what happens if mendhak goes cross-eyed.

[mendhak]

Quick thought: A line with ends is basically a line segment, isn't it? So we'd still be talking about theoretical infinite lines here.

[SLH]

Yeah. A line segment is a line, with some bounds on it.

[wossname]

Hey Woss-ster,

I agree that I have changed the plane of the circle so that it is no longer in the plane of the page; it was just to try to see what happens if mendhak goes cross-eyed.

Mathematically speaking it is impossible to NOT be cross eyed when you have 3 eyes. You would have to be focusing on a point infinitely far away. Since frog's eyes have a finite limit to the amount of detail that is resolvable, a mendhak's natural state is crosseyedness. Ipso Factotum, Quod Erat Demonstradum.

[zaza]

if the eyes were to be focusing on a point infinitely far away, and that point was the centre of a circle, mendhak would in fact be walking a very fine line.


Incidentally, surely all three eyes can be looking at the same point in space at any given time, as long as it is sufficiently far away from each eye that the eye can focus on it. Or does he have eyes in the back of his head?

[mendhak]

Very enlightening, that was.

But take note that the eyes can move independent of each other, so the point can be at two locations at the same instant of time. I hope that gives a tasty quantum twist to the conclusion.

[zaza]

zaza

[moeur]

Back to the original question.

Is a line basically a circle of infinite radius?

The answer is yes.

Consider a circle of radius r
As r -> infinity the curvature of the circle -> 0 which is a line.

[zaza]

Actually, wrt moeur's post, I would have written "See post #6"

I guess we've come "full circle". Hohoho.

zaza