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# Talk:Line (Euclidean geometry)/Draft

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## Rather cryptic

"The following demonstrates a line:

Given a line AC
Point B is on AC
ABC is the same line AC"


— rather cryptic; what could it mean? Boris Tsirelson 19:39, 27 March 2010 (UTC)

Boris, could you provide a short lead? If I do it then I will probably not be allowed to approve it. Somewhere the long form "straight line" should also be mentioned. --Peter Schmitt 23:07, 11 May 2010 (UTC)

I shall try. Boris Tsirelson 05:53, 12 May 2010 (UTC)
The following needs some clarification, I think:
"a straight curve having no thickness" -- this is really difficult. But "straight" has to be defined (and all curves have no thickness)
do you think that "an infinite (uniform) curve that looks the same everywhere", or "that can be shifted along itself" could work?
"can be defined in terms of distances" -- assuming some (intuitive?) notion of space
the definitions use sets of points with a given property (similar plane)
--Peter Schmitt 16:48, 14 May 2010 (UTC)
Now it is really a problem. It reminds me of my exchange with Paul about "Plane":
About the "flat surface in which a straight line joining any of its two points lies entirely on that surface" I wonder: does "flat" already mean "in which a straight line joining any of its two points lies entirely on that surface", or does it mean something different? Boris Tsirelson 08:50, 2 April 2010
The first sentence is from the Oxford Dictionary and hence "flat" is as the non-mathematician perceives it. (Undefined, intuitively clear, and perhaps somewhat redundant). --Paul Wormer 09:37, 2 April 2010
Ah, yes, this is why I am reluctant to write introductions; I know that I am too much mathematical for this business. Boris Tsirelson 12:24, 2 April 2010
As for me, the lead is hopelessly non-mathematical, but probably this is what it should be. Being reluctant but almost forced, I've collected a number of such texts scattered on the Internet (see the collection here) and did something similar. If you really want some local corrections to the lead, why not, I'll do. But if you want it to be mathematical then it is better to delete it and start from scratch. Boris Tsirelson 18:03, 15 May 2010 (UTC)

(unindent)
Probably one could write (more than) an article about the notion of straight line from a philosophical or epistemological point of view, but here we only need an introduction to what follows, so some modification should suffice, at least for now. (I also checked some encyclopedia entries -- they usually shy away and start with something approximately mathematical.

Explaining the concept is difficult, and in this article you have described how to define lines given a previous notion of space and plane. Why not say this? The following shows what I mean by this:

In Euclidean geometry, a line (sometimes called, more explicitly, a straight line) is an abstract concept that models the common notion of a (uniform) curve that does not bend, has no thickness and extends infinitely in both directions.
It is closely related to other basic concepts of geometry, especially, distance: it provides the shortest path between any two of its points. Moreover, in space it can also be described as the intersection of two planes.
It is, however, diffult to give a self-cntained definition of straight lines. Assuming an (intuitive or physical) idea of the geometry of a plane, "line" can be defined in terms of distances, orthogonality, coordinates etc. (as we shall do below).
In a more abstract approach (vector spaces) lines are defined as one-dimensional affine subspaces.
In an axiomatic approach, "line", together with "point", is a basic concept of elementary geometry. It is an undefined primitive.

What do you think? --Peter Schmitt 15:49, 26 May 2010 (UTC)

I like it. It is implemented; please look. I did some minor changes:
(uniform) — deleted, since for now the article does not touch this idea (of a transitive group action...);
Moreover, in space... — why "moreover"?
It is, however, difficult to give... — difficult to whom, and in which sense? Deleted.
an (intuitive or physical) idea... — "or" is not apt, since our geometric intuition is (very probably) induced by some properties of the physical world.
(as we shall do below) — ideally, this should hold (by default) for everything said in the lead;
is a basic concept of elementary geometry. It is an undefined primitive. — a bit unclear, is it undefined primitive because it is a basic concept, or what?

(unindent)
The changes are fine for me. It is even better that you made changes. So nobody can accuse me of ordering them ... :-) I just nominated this for approval. --Peter Schmitt 23:28, 26 May 2010 (UTC)

## Betweenness

The two conditions assume "three different points", thus the "at least" could be omitted. --Peter Schmitt 23:18, 11 May 2010 (UTC)

Sorry, I do not understand which "at least" do you mean.
"If three different points belong to the given set then at least one of them lies between the others" — if you mean this, well, I can delete "at least"; it will be a bit less formal but still clear.
"If one of three different points lies between the others, and at least two of the three points belong to the given set, then the third point also belongs to the given set" — well, really it is not mine "at least", I took it from WP (if I remember correctly). I can remove "at least", but probably the fear is that someone may say: it is contradictory, it cannot be that the number of these points on the given set is both two and three. No, after thinking more I see another fear related to Remark 3. If "at least" will be removed maybe we should say "some two of the three points" or maybe "any two of the three points"? The fear is that the reader can interpret the phrase as the weaker condition of Remark 3. Boris Tsirelson 06:27, 12 May 2010 (UTC)
First condition: For (pairwise) distinct points it will always be one point (or none). Thus the "less formal" version is equivalent and easier to read for non-mathematicians.
In the second case, too, I cannot see how "at least" may help. There is the danger that it is confused with convexity, but what has the "at least" to do with it?
I would vote for the least formal but still correct version, but I do not insist on omitting the "at least" (or any other change).
Perhaps "Among three distinct points of the given set there is always one that lies between the two others."?
Afterthought: the "distinct/different" is not needed either (even though the "at least" is then less formal).
And "If one of three distinct given points lies between the two others, and if any two of these three points belong to the given set, then the third point also belongs to the given set."
Or "If one of three given points lies between the two others, and if at least two of these three points belong to the given set, then the third point also belongs to the given set."
(Replacing the assumption "distinct" by the condition "at least"). --Peter Schmitt 10:55, 12 May 2010 (UTC)

On another matter: What about mentioning that the "betweenness" (as defined here) is related to the notion of "shortest path"? --Peter Schmitt 10:55, 12 May 2010 (UTC)

Yes, I agree, and did. I only disagree with the last version: "If one of three given points lies between the two others, and if at least two of these three points belong to the given set, then the third point also belongs to the given set." What if A=B, C is different, and A, B are on the line? Well, it seems, the matter is settled anyway. Boris Tsirelson 14:42, 12 May 2010 (UTC)
This whole article seems very poorly thought out. A line isn't necessarily a straight line. Even for straight lines, the article is far too narrow. It seems to be confined to Euclidean geometry. Straight lines can be generalized as far as projective geometry, where even the concept of betweenness doesn't apply. Peter Jackson 14:45, 12 May 2010 (UTC)
For every mathematical notion there exist generalizations (usually, already known, but at least, to appear in a future). You are welcome to extend the article (and I promise you that I always will be able to add something else). But why "very poorly thought out"? For now it is geared toward Euclidean geometry only. Is it stupid? Boris Tsirelson 14:55, 12 May 2010 (UTC)

(unindent)
Boris: As for the second formulation: Of course, I was sloppy (I realized my mistake just now, while listening to a talk ...)
Peter: You are right, this is about the "elementary" (basic) concept of a straight line, as in Euclidean or physical (non-relativistic) space. The lead (to be added) certainly will make this clear.
There are many valid ways in which this topic can be approached. This is one of them. It explains -- in a consistent and well-organized way -- "line" based on the intuitive concept of length. I would have written a very different article (and, may be, some time in the future I will). There is only one mathematics, but it can be told in many different stories.
Of course, there are generalizations of the line concept (projective, geodesic, combinatorial, etc.), but it is a bad idea to treat them all in a single article (except, perhaps, in a survey on the idea of a line). Pages should always have a digestable length.
--Peter Schmitt 15:38, 12 May 2010 (UTC)

It intrigues me that you would have written a very different article; I'll be glad to look if this will happen. Boris Tsirelson 15:54, 12 May 2010 (UTC)

## Definition via right angles

A link to Pythagorean theorem seems to be appropriate. --Peter Schmitt 23:41, 11 May 2010 (UTC)

Done. Boris Tsirelson 06:11, 12 May 2010 (UTC)

Some lead is written; please look. (I do not think I am a good leadwriter.) Boris Tsirelson 15:52, 12 May 2010 (UTC)

## Figure

In the caption: the figure shows only a part of a line. --Peter Schmitt 23:03, 13 May 2010 (UTC)

Fixed. Boris Tsirelson 10:57, 14 May 2010 (UTC)

## convex disk

Not every disk is circular, I think. --Peter Schmitt 16:50, 14 May 2010 (UTC)

Really? In the context of elementary geometry it is, I believe. WP agrees.
To my surprise, you are right. (As for the leads - line and plane: I have to think about them, too. They should provide adequate information for those who are not willing to read the whole page -- and perhaps make them curious. It is indeed not easy ...) --Peter Schmitt 00:14, 16 May 2010 (UTC)

Now #5 with quotes (with or without brackets). Boris Tsirelson 20:09, 11 October 2010 (UTC)

## Toward Approval

This edit changes the meaning of the sentence. Which sentence is correct? D. Matt Innis 01:19, 31 May 2010 (UTC)

There is always some tension between "a" and "the" in mathematical texts in such cases. However, I do not think it is a problem here. The coefficients a, b, c did not appear before, thus, "the linear equation" meant "the linear equation written just below", not "the equation introduced before", while "a linear equation" means "some linear equation (to be introduced just now)". This is my understanding, but of course, my English is poor; if doubts remain then it is better to ask someone else. Boris Tsirelson 05:57, 31 May 2010 (UTC)
Okay, thanks Boris, I'll accept that in this context it is a copy-edit meaning about the same thing. D. Matt Innis 12:26, 31 May 2010 (UTC)

## Projective geometry

Not mentioned, which it should be, I think. Peter Jackson (talk) 09:46, 5 September 2020 (UTC)

On second thoughts, as the article defines line as a concept in Euclidean geometry, I suggest renaming as Line (Euclidean geometry). Peter Jackson (talk) 09:21, 7 September 2020 (UTC)