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|Line (geometry) has been listed as a level-3 vital article in Mathematics. If you can improve it, please do. This article has been rated as B-Class.|
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|Wikipedia Version 1.0 Editorial Team / Vital|
- 1 Collinear vs Coplanar?
- 2 Too advanced?
- 3 Side of a line?
- 4 Multiple concepts in one article?
- 5 Two links to Bulgarian (Български)
- 6 Question
- 7 Notation
- 8 Discussion of more abstract definitions
- 9 POLYLINE
- 10 3D Lines
- 11 Title
- 12 y=mx+c to y=mx+b
- 13 Requested move
- 14 Colinear or Collinear?
- 15 Line with two origins
- 16 lines history
- 17 Shortest Distance Between Points
- 18 Nature of lines
- 19 Second sentence
- 20 half line?
- 21 Disambiguation of the term "Lineaments"
- 22 Identities
- 23 Collinearity
- 24 Straight curve
- 25 Four definitions: 1 informal, 2 different, obsolete, and obscure, 3 incomplete, and 4 circular
- 26 Is a curve a line?
- 27 Definitions and Descriptions
- 28 Ray nonsense in the current article
- 29 Definitions and descriptions II
Collinear vs Coplanar?
- Yes. Any three points are coplanar. The fact that three collinear points doesn't define a unique plane containing them doesn't mean they aren't coplanar, it just means there is more than one plane containing them. Coplanar doesn't mean "exactly one" plane contains them, it just means at least one plane contains them. --Cheeser1 00:06, 23 September 2007 (UTC)
This page sucks balls! What of all the middle school geometry students who stop by to learn about the line? We should introduce things a little more gently, in the context of ordinary Euclidean geometry in the plane, and then in 3D, before getting into the fancier abstract concepts. I'll make these changes if there's no objection and no one else does first. Deco 04:09, 2 Jun 2005 (UTC)
- Please do. I've linked to the somewhat simpler Linear equation (which should be consulted by editors to prevent inordinate duplication), but I'm afraid it's not very prominent. - dcljr (talk) 7 July 2005 06:48 (UTC)
Side of a line?
Given a point and a line in a plane, how do you determine what side of the line the point is on?
- The easiest way is with the cross product. If two points a,b are on the line, and p is your point, all with z coordinate zero, then the z component of (p-a) × (b-a) will be positive or negative, depending on which side of the line z falls. Deco 04:07, 5 November 2005 (UTC)
- How is a 3D concept such as cross product (which will stop almost all high school students dead in their tracks) the "easiest way?" How many years was it between when you learned the 2D concept of area of a triangle and the 3D concept of cross product?
- A much easier way conceptually (though admittedly not operationally, the actual arithmetic isn't any easier) is to pick any coordinate frame and define the area under a line segment AB to be its width (the x coordinate of B minus that of A, which will be negative if and only if B is properly to the left of A) times its average height above the X-axis (half the sum of the y-coordinates of A and B). (This is equivalent to the concept of area under a curve being integrated, including getting the sign of integration right.) The area of ABC is then simply the sum of the areas under the sides AB, BC, CA (don't reverse any of these segments!).
- The area of ABC is equal to the area of BCA and of CAB, and the area of CBA is equal to the area of BAC and ACB, and these two areas are the negations of each other. (There being only 3! = 6 possible ways of feeding the three points to this method, this accounts for all possibilities.) The sign of the area tells you that the three vertices you gave to the algorithm, when followed around the triangle in the order you gave them, run clockwise around the triangle if positive and counterclockwise if negative.
- To remember this, assume the order ABC and picture C to the right of A. If B is above AC (the clockwise case) the areas under AB and BC will obviously be greater than that under AC and hence the total area will be positive (because the algorithm uses CA rather than AC and therefore subtracts the area under AC by virtue of adding the area under CA). The reverse is true when B is below AC. If B is on AC the area will be zero. Although a number of other configurations are possible all you need to remember is ABC with C to the right of A and everything else follows automagically! --Vaughan Pratt (talk) 07:21, 6 December 2008 (UTC)
How about first figuring out where is "front" and where is "back"? And if you live in a 3-dimensional world you should start wondering where is "up" and where is "down". Not not mention the "second up" and "second down" and so forth if your mind is not bound by our common day experience of space and time. Good question, though, since it already lured an answer or two. Lapasotka (talk) 23:36, 8 June 2011 (UTC)
Multiple concepts in one article?
Here Line and Line segment are treated as part of the same article. In Polish Wikipedia - they are separated. How can it refer to both articles in Polish now?
- I think all we can do is crosslink between Polish "Line" and English "Line", and leave Polish "Line segment" without an en: link. There's a similar situation on English with Addition and Summation; all but a few languages treat these in a single article (including Polish), so there are lots of interlanguage links for Addition but few for Summation. Melchoir 21:07, 31 January 2006
This is because the content of this article is in two bg articles.
No where in the article does it say that a line contains infinite number of points. Is this correct, or does Quantum Physics state that this is incorrect? —The preceding unsigned comment was added by 22.214.171.124 (talk • contribs) 14:43, 10 March 2006.
- In Euclidean geometry, which seems to be the context of this article, it's correct: a line contains infinitely many points. I wouldn't worry about quantum physics in the real world, which doesn't affect mathematical models. Why don't you try adding this information in? Melchoir 22:48, 10 March 2006 (UTC)
- Yes, but in theory, does a line actually have infinite number of points? This seems impossible according to Quantum. —The preceding unsigned comment was added by 126.96.36.199 (talk • contribs) 16:05, 10 March 2006.
- I don't see the connection. A line is a mathematical abstraction, and quantum physics has little to say about it. In reality, there is no conceivable experiment that could determine whether an "actual line" has infinitely many "points", so the question is only a philosophical matter. If you're interested in the impact of quantum physics on indivisibility you can read up on Zeno's paradoxes here or an even longer reference at the Stanford Encyclopedia of Philosophy. Melchoir 01:09, 11 March 2006 (UTC)
- Yes, but in theory, does a line actually have infinite number of points? This seems impossible according to Quantum. —The preceding unsigned comment was added by 188.8.131.52 (talk • contribs) 16:05, 10 March 2006.
- Don't get math confused with physics. because physics is almost completely dependent on mathematics does not mean the reverse is true. as a matter of fact, the inverse of the reverse is true--math is completely independent of physics (the only part of math that is affected by physics is what mathematicians choose to explore, but thats not really math being affected, thats mathematicians being affected.) get it? This is the generally accepted view among mathematicians and physicists alike (who are often the same people of course.) Brentt 11:47, 16 August 2006 (UTC)
The article on Interval_(mathematics) includes various notations on number sets, including the French notation I grew up with. This article doesn't include the equivalent notation I learned, which was:
- (A, B) -- line passing through A and B
- [A, B) -- half-line starting at A, continuing through B
- ]A, B) -- half-line starting at, but not including, A, continuing through B
- [A, B] -- segment between A and B
- ]A, B[ -- segment between A and B, excluding both ...
I don't know if that's also used for American notation, but an equivalent reference here would be handy. (I see something vaguely similar, but less complete, in the french version of Segment 184.108.40.206 20:07, 16 March 2006 (UTC)
Discussion of more abstract definitions
Perhaps this article could benefit from some discussion of more abstract definitions of a line? e.g. ("a straight line is a curve, any part of which is similar to the whole" from topology) Brentt 11:52, 16 August 2006 (UTC)
- Do you have references for that? if yes, it could go at the bottom, in a section called "Generalizations". Oleg Alexandrov (talk) 16:15, 16 August 2006 (UTC)
POLYLINE redirects to this page but is then not discussed. Can someone mention POLYLINE here, or make a distinct page?
- colinear also redirects here, but is not mentioned on the page. What gives? Sim 14:15, 24 March 2007 (UTC)
The article said "In three dimensions, a line must be described by parametric equations". This is wrong: a line in any dimension can be described by a linear equation. I changed "may" to "must" and added a couple of linear equations for a 3D line. This leaves the definitions section a bit rambly, in my opinion --- why describe a 2D line in slope-intercent, versus a 3D line in parametric and standard form? --- but I felt it was a step in the right direction since it is at least correct. It does have the advantage of getting the link to linear equation earlier in the page. Owsteele 14:39, 14 December 2006 (UTC)
Should the title be renamed to "Line (geometry)"? This would match "Point (geometry)" and "Square (geometry)". Jason Quinn 17:51, 4 April 2007 (UTC)
- The concept of line is used in branches of mathematics outside geometry (analysis, for example). --Cheeser1 00:02, 23 September 2007 (UTC)
y=mx+c to y=mx+b
c is usually used in calculus for the constant. It really could be anything though, there is no rules for choosing the particular letter you use, just guidlines and some loose standards that are always changing accoring to the math course your in or the time your taking it or personal preference. It makes no difference to the equation. Brentt (talk) 21:41, 9 December 2008 (UTC)
- I was always taught y=mx+c, I'm not sure what "m" stands for, but "c" is for "constant". If you want to use b in place of c, you should probably change m to a as well - y=ax+b would be a reasonable choice, although ax+by=1 would be a more common form using a and b. But, as you say, it really makes no difference. --Tango (talk) 22:07, 9 December 2008 (UTC)
Colinear or Collinear?
I was surprized to find that collinear (with two l's) seems to be more widely used than colinear. Are they both correct? Colinear makes more sense to me (as in co-linear), but I'm not a native speaker. --CyHawk (talk) 21:45, 3 February 2008 (UTC)
- I'm fairly sure that it's two Ls. The Oxford English Dictionary contains only an entry for collinear. I believe colinear is an unambiguous (and probably common) misspelling, but two Ls is correct. --Cheeser1 (talk) 20:26, 6 April 2008 (UTC)
Line with two origins
This is a cool example of a line with two origins and I was thinking of making a wikipedia article on it, but I wasn't sure if it deserves its own article or if it should be put in some other related article? LkNsngth (talk) 20:09, 6 April 2008 (UTC)
Lines in a Cartesian plane can be described algebraically by linear equations and linear functions. In two dimensions, the characteristic equation is often given by the slope-intercept form:
m is the slope of the line. c is the y-intercept of the line. x is the independent variable of the function y. In three dimensions, a line is described by parametric equations:
x, y, and z are all functions of the independent variable t. x0, y0, and z0 are the initial values of each respective variable. a, b, and c are related to the slope of the line, such that the vector (a, b, c) is a parallel to the line.
Formal definitions This intuitive concept of a line can be formalized in various ways. If geometry is developed axiomatically (as in Euclid's Elements and later in David Hilbert's Foundations of Geometry), then lines are not defined at all, but characterized axiomatically by their properties. While Euclid did define a line as "length without breadth", he did not use this rather obscure definition in his later development.
In Euclidean space Rn (and analogously in all other vector spaces), we define a line L as a subset of the form
where a and b are given vectors in Rn with b non-zero. The vector b describes the direction of the line, and a is a point on the line. Different choices of a and b can yield the same line.
Properties In a two-dimensional space, such as the plane, two different lines must either be parallel lines or must intersect at one point. In higher-dimensional spaces however, two lines may do neither, and two such lines are called skew lines.
In R2, every line L is described by a linear equation of the form
with fixed real coefficients a, b and c such that a and b are not both zero (see Linear equation for other forms). Important properties of these lines are their slope, x-intercept and y-intercept. The eccentricity of a straight line is infinity.
More abstractly, one usually thinks of the real line as the prototype of a line, and assumes that the points on a line stand in a one-to-one correspondence with the real numbers. However, one could also use the hyperreal numbers for this purpose, or even the long line of topology.
The "straightness" of a line, interpreted as the property that it minimizes distances between its points, can be generalized and leads to the concept of geodesics on differentiable manifolds.
Ray In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is the set of points C on the line containing points A and B such that A is not strictly between C and B. In geometry, a ray starts at one point, then goes on forever in one direction. —Preceding unsigned comment added by Gon56 (talk • contribs) 07:26, 14 June 2008 (UTC)
Shortest Distance Between Points
The declaration that the line is the shortest distance between two points is limited to Euclidean Space and further the sitation (number 3) states that it was "sort-of" proved by Euclid and "assumed" by Pytheagoras. Neither of these are proofs and neither should they be stated as such. —Preceding unsigned comment added by 220.127.116.11 (talk) 11:50, 19 November 2008 (UTC)
Furthermore , such a definition should be explained to have no tangent with the real, physical world, being a construct that only works on paper or monitor screens. A straight line might be possible as an entry vector for particles entering a black hole , if those exist, anywhere else in the Universe a line would be a curve , since space is bent and folds with each and any gravity source around. Even more important ,the lack of such explanation results in confusion for young people as they can't integrate the preconceptions induced by faulty teaching with the real world .(just think how you first reacted when encountering the reality of measuring the distance between two cities on different continents) —Preceding unsigned comment added by Pef333 (talk • contribs) 04:04, 7 January 2010 (UTC)
Nature of lines
Hi Tango. Glad you were happy to compromise without disturbing the "straight curve" lead. The fact that lines are fundamental might distinguish them from less fundamental objects, but otherwise makes little headway towards saying what a line actually is. A lead needs to be concise, and therefore should not be wasting words on such peripheral issues as the degree of fundamentality of "line" (which is highly debatable in any case, see below) when it should be trying to get to the essence of the concept as soon as possible.
Leisurely introductions are fine pedagogically, but the length necessary to get anywhere with them makes them better suited to the main body. As an example of this sort of thing see my attempt at exposing the motivation for, and underlying machinery of, toposes, admittedly a dry read but there is no way a mathematically capable reader new to toposes can extract that information from the preceding material in the article oneself---but no way should that go in the lead! Then compare it with John Baez's attempt at a similar thing, which is great fun but throws the baby out with the bathwater by failing to adequately convey what's going on under the hood (bonnet). The automotive analogy is a good one: John tries to convey what it feels like to drive one in various terrains while I try to explain the principle of the engine inside. Both help, but engineers and mathematicians are more likely to want the latter.
That said, your objection to my long list of characterizing properties in the lead was well taken, as was your suggestion that this more detailed material should be in the main body. After sleeping on what I'd written I'd come to the same conclusion myself and was going to move it to the body and write simply "straight line" in the lead when I noticed that you'd beaten me to it and had simply reverted my edits, which was fine by me by that time.
In the meantime I'd come to the realization that my wordy characterization had arisen from too hasty an attempt to replace "fundamental object" with something more specific, at a time when I didn't really have a suitable replacement ready and so just threw the kitchen sink at it. I now think that most of the other characteristics belong elsewhere than in the line article, namely in the more general classes of which lines are a particular subclass. This is what inheritance is all about in object-oriented programming, and the inheritance concept seems to provide an equally good organizing principle for encyclopedias.
Whether lines actually are more or less fundamental than curves is a nice question. A line as a subspace of the Cartesian plane can be defined without loss of generality as the set of zeros of a two-variable affine form, namely the solutions of ax + by + c = 0. Curves in the same setting cannot be defined in that way without significant loss of generality as one only obtains algebraic curves with that approach, no sine waves, space filling curves, etc. etc. But if you do limit yourself to algebraic curves then a line as the linear case of an algebraic curve is less fundamental by virtue of being an instance of a more general and therefore more fundamental notion.
Moving beyond Euclidean space, intensionally defined curves, those structured with suitable data appropriate to curves expressed without reference to a higher dimensional embedding space, are arguably a fundamental concept in their own right. A particularly simple example is a curve as a structure endowed with two metrics for respectively arc and chord length, the sort of entity one might run across in a CAD system like Autocad. This is a self-contained yet simple concept of "curve" admitting an equally simple notion of "straight," namely that the two metrics agree! Curves of such a kind are fundamental in the same sense that rings, lattices, etc. are fundamental, making their special cases slightly less fundamental (but only slightly less when the definition is as simple as mere coincidence of the two metrics).
If anything makes lines fundamental it would surely be that they are conceptually simple, being the path referred to in Newton's first law of motion (Newton assumed space was always flat), and encountered early on as one of the simplest instances of a geometric object, only points being simpler (unless a point is defined as the intersection of two nonparallel lines!). But this brings us to the difficult question of what it even means to be "fundamental." I guess this is a big part of what bothers me when I see it in the first sentence of an article, the other equally big part being that even if we all agreed on what it meant it still says very little: few if any concepts list "fundamental" among their defining characteristics. --Vaughan Pratt (talk) 23:40, 7 December 2008 (UTC)
- It is important for a lede to establish the notability of a subject in addition to introducing it. We need to say why it is worthwhile to consider lines, their fundamental nature is part of the reason. Whether or not lines are fundamental depends on your approach to geometry. If you approach it from the point of view of coordinate geometry then the fundamental objects are points, defined as pairs (or larger n-tuples) of real (or complex, or whatever else) numbers (which are themselves defined in terms of rational numbers, which are defined in terms of integers, which are defined in terms of natural numbers, which are defined in terms of 0 and a successor function which are left undefined). Everything else is then defined in terms of those points (both lines and more general curves are simply collections of points). If you approach it in the same way as Euclid, then you have both lines and points as undefined concepts and a load of axioms about them (eg. "two points are joined by a unique line") it doesn't matter what points and lines are, what matters is how they interact. So, lines are fundamental to pure Euclidean geometry, they are not fundamental to coordinate geometry. This article should discuss lines in both contexts (and others), so I think it is appropriate to describe them as fundamental in the lede (perhaps "It is a fundamental concept in certain forms of geometry." although that seems unnecessarily verbose to me). --Tango (talk) 23:55, 7 December 2008 (UTC)
- Agreed about notability---I shouldn't have neglected it in my version and am happy to see it back. On the matter of Euclid, if being a primitive of Euclid's axioms is what qualifies "line" as fundamental for you then you should edit the leads of circle and angle accordingly, since Euclid's axioms have those along with points and lines as equally primitive notions. Bear in mind however that when Tarski formalized Euclidean geometry in first order logic to make it more rigorous he dropped "line" as a primitive notion, raising the question of whether Euclid's more informal understanding was misguided in making lines primitive. -Vaughan Pratt (talk) 02:17, 8 December 2008 (UTC)
- Well, Euclid's axioms were full of flaws, try Hilbert's instead. They have as undefined concepts "points", "lines" and "planes" and also the relationships "lies on", "between" and "congruent" (this is from Appendix B of Faber, the reference I used in the article). Circles and angles are not mentioned. I'm not familiar with Tarski's approach, but I'll read up on it when I get back from lunch. --Tango (talk) 12:34, 8 December 2008 (UTC)
Tango has argued for retention of the second sentence of the lead, "It is a fundamental object in geometry." I am just as strongly against it. However I don't want to get into an edit war with Tango because these often turn out badly. What do others feel about what this sentence contributes to the lead?
Articles on circles, angles, etc. content themselves with characterizing the concept and its applications without trying to position them in the hierarchy of fundamentality. My feeling is that lines should be described in the same spirit, and that those responsible for the article on them should take a neutral point of view on whether lines deserve to be singled out from other concepts as "fundamental." Otherwise we're going to get into interminable arguments as to whether circles, angles, etc. should also be accorded this special status of "fundamental." I much prefer the terminable kind. --Vaughan Pratt (talk) 06:35, 8 December 2008 (UTC)
- I'm not strongly opposed to that sentence but I don't think it adds much or any value to the article. Maybe it is intending to allude to the last sentence of the first paragraph of the Euclidean geometry section, the one that says that lines are not so much defined as postulated about, but if so it's doing so in a clumsy and opaque way. —David Eppstein (talk) 07:17, 8 December 2008 (UTC)
Hmm. I see Vaughan's point, but I can't say I feel very strongly about it one way or the other. Certainly no one is going to defend the assertion "you can learn geometry just fine without ever bothering about lines", so in that sense they're fundamental. But I don't see the need to say so just here. Are there likely to be readers who are confused on this point? --Trovatore (talk) 09:16, 8 December 2008 (UTC)
- I think the sentence doesn't add anything important to the article. It's not even clear what it is supposed to mean. I have moved the "fundamental concept" information a bit further down into the sentence about Euclidean geometry, where it is unquestionably true. The style can probably still be improved. Is this a reasonable compromise, otherwise? --Hans Adler (talk) 10:13, 8 December 2008 (UTC)
- I don't care strongly one way or another, but I doubt that the use of the term "fundamental concept" imparts useful information to the reader – whether layperson or expert, and I prefer to keep ledes simple when possible. In reference to Euclid-style axiomatic geometry I further prefer the terms "primitive notion" or "fundamental object" (the latter being used at Point (geometry)) over "fundamental concept". I also happen to think it is better to put this in the section Line (geometry)#Euclidean geometry, and to simply state in the lede that "The geometric concept of a line stems from Euclidean geometry, but is also found in many other geometries." By the way, we have no article defining the notion of a geometry as a mathematical structure (encompassing structures such as Hyperbolic geometry, Elliptic geometry, Spherical geometry, Dowling geometry, Klein geometry, Calibrated geometry, Incidence geometry, Finite projective geometry, Cartan geometry, Absolute geometry, Point-free geometry, Partial geometry, Inversive ring geometry, and Zariski geometry), so we can't properly wikilink the word "geometries". --Lambiam 12:28, 8 December 2008 (UTC)
- I don't feel particularly strongly about it, if people think it should be removed then fine. However, I think lines are a very important concept in geometry and there should be something in the lede saying that in order to show the notability of the subject. "Fundamental" is a pretty vague term, I know, but it gets the idea across. We can then explain in more detail how they are fundamental in later section. (I have nothing against "fundamental object" over "fundamental concept", in fact I think I may have written it like that in one version - "primitive notion" is rather too technical for the lede in my opinion.) --Tango (talk) 13:10, 8 December 2008 (UTC)
- Hans Adler's change (associating "fundamental" with Euclid) did the trick nicely, and hopefully meets everyone's requirements. I did a little further tweaking that hopefully achieves people's goals for this lead even better. It reads really smoothly now, thanks everyone for your help! --Vaughan Pratt (talk) 05:22, 9 December 2008 (UTC)
I'll have to admit that I find this particular controversey rather humorous, since my OR has involved geometries where lines are indeed the fundamental notion and which are rather pointless in the sense that the automorphism group of the geometry does not even preserve the points, so that points are not even definable. --Ramsey2006 (talk) 17:13, 8 December 2008 (UTC)
- Ok, so you have a group G and a nonexistent point P (or many) such that G doesn't preserve P. Not sure I'm getting the picture yet. --Vaughan Pratt (talk) 05:22, 9 December 2008 (UTC)
- "Half line" is a standard name for it. It is half of a line, it just has the same length as a whole line. Similarly, half of all integers are even, despite there being just as many integers as there are even integers. --Tango (talk) 01:52, 27 January 2009 (UTC)
Disambiguation of the term "Lineaments"
Shouldn't the article give some identities related to lines? I mean stuff like equations for the distance between two lines (as in skew lines) or a point and a line (might merge perpendicular distance here), the angle between two lines, determining if two lines are parallel or perpendicular, etc.? If you look at e.g. the triangle and plane articles, they have a lot more equations in them. -- Coffee2theorems (talk) 12:19, 29 October 2009 (UTC) a line's degrees is —Preceding unsigned comment added by 18.104.22.168 (talk) 23:22, 2 March 2010 (UTC)
The page Collinear points redirects to here, yet there was no mention of collinearity in the article. I've added a brief section that needs formatting and possibly expanding. Dbfirs 08:19, 4 April 2010 (UTC)
We seem to have a circular definition here because the definition of curve is "deviation from a straight line". Can we not find a better definition of a straight line, such as the extremely well-known "shortest path between two points" (in Euclidean space)? We could also mention the (possibly later addition by Heron or Diophantus) "definition" of a line in Euclid's Elements as breadthless length that lies equally with respect to the points on itself. How about using Wiktionary's "An infinitely extending one-dimensional figure that has no curvature; one that has length but not breadth or thickness"? Dbfirs 07:04, 29 June 2010 (UTC)
- You are right about the circular definition, but it was due to a questionable definition of curve. Namely, it is questionable to define a curve based on the definition of a line. This is what I suggest:
- A curve is an infinitely long series of points that extends without end and without gaps
- A line is a straight curve, that is an infinitely long series of points that extends in opposite directions without end and without gaps
- Notice that the concept of "direction" is based on the concept of straight line, so the reference to the concept of Curvature or "straightness" (null curvature) is crucial to obtain a non-circular definition.
- — Paolo.dL (talk) 13:45, 12 December 2010 (UTC)
- I guess that topologists did not find a better way to define a curve than the statement "curve not required to be straight" (or something similar). Also, "series of point" is not perceived to be a correct terminology. I edited trying to avoid circular definition.
- — Paolo.dL (talk) 10:37, 13 December 2010 (UTC)
Four definitions: 1 informal, 2 different, obsolete, and obscure, 3 incomplete, and 4 circular
The new intro contains four definitions. In my opinion, only the first is ok, but it is informal. The others are useless. Moreover, the second is not consitent with the first.
- The first is informal, but it can be understood.
- The second, by Euclyd, is different from the first as it does not require null curvature, so it allows for "curved lines". Euclyd's definition is almost obsolete. In current terminology, lines are typically required to be straight (as explained in definition 1), and "curved lines" are most often called "curves". Moreover, Euclyd's definition is absolutely useless and obscure when it tries to define the concept of "straightness" ("The straight line is that which is equally extended between its points"!). The inconsistency with respect to the first definition is not explained in the introduction.
- The third is not given! Actually, it seems to be a group of similar but not identical definitions for which only the names of the authors are listed!
- The fourth, which is described as the most commonly used, is naively circular, as it is based on the definition of a Cartesian coordinate system, which in turn cannot be defined without reference to the concept of straight line. Namely, such a coordinate system is formed by Cartesian axes, which in turn are (directed) straight lines. In other words, we can't define a Cartesian coordinate system before defining a straight line.
I can't believe that this is the state of the art. In my opinion an expert editor is urgently needed, to rewrite the introduction.
- Some comments on Paolo.dL's comments:
- First "definition" is not really a definition, but only an intuitive description.
- I agree that the second is obsolete. I have explicitly stated that by writing "until seventeenth century". I agree that the concept of "straightness" is obscure. A note should be added that this means, I guess, that a line is the shortest path between two points (in the cited book, this definition is followed by more than one page of explanations, which, for me, are as obscure as the definition). Note that in the cited book, I am not able to recognize what is Euclid's translation and what is comments by the translator.
- The third is too technical to be given here. The list of authors should better be completed by links to pages describing their systems of axioms. I have not yet searched if these pages exist in Wikipedia. In any case, as fourth definition is equivalent to the third one, one of them is sufficient here.
- I do not agree that fourth definition is circular: In coordinate geometry, a point (in the plane) is identified with a pair of real numbers (its coordinates) and the Cartesian axes are defined by the lines which satisfy the linear equations x=0 or y=0. It should be added somewhere (but probably not in the introduction) that this definition depends on the field of the coordinates which is usually the field of the real numbers, but may also be the complex field (complex line) or a finite field (finite geometries, presently important in cryptography).
- I may also add that I do not agree to define straightness by null curvature: Curvature is a notion which needs a metrics and is not a primitive property. Moreover, in non Euclidean geometries, lines are defined as geodesics (shortest path) and may have curvature (It is well known that it is the case for the geometrical space of the general relativity). D.Lazard (talk) 15:37, 13 December 2010 (UTC)
- I have just read the corresponding article in Citizendium (see the banner in the top of the talk page). It contains many good things which should be imported, although merging seems difficult. Especially the definition of the line by "betweenness", which appears to me as the translation in modern language of Euclid's "equally extend between points". D.Lazard (talk) 16:21, 13 December 2010 (UTC)
- By the way, the intro in that article defines the line informally as a straight curve (that "does not bend"). Although I used this definition in one of my previous edits, now I think it would be better to have a definition independent of that of curve. Besides that, I like the other parts of the informal definition in the intro, although I agree that they are formally invalid.
- The definition based on "betweenness" is fascinating, I agree, but it is based on the concept of "shortest distance", or "Euclidean distance", and how can you define a shortest distance without a straight line? By defining a Euclidean norm? That's not enough, as the numbers you plug into it are coordinates, so you also need to define a Cartesian coordinate system, and again this cannot be done without straight lines.
- Unfortunately, that article does not give the axiomatic definition by Hilbert, which is probably the only one which would formally work. Indeed, the so called "modern definition" is based on the definition of a space, and how can you define a Euclidean space without a standard basis, and how can you define that without a Cartesian Coordinate system? I wonder why the authors in that article do not seem to realize that this "modern definition" is as circular as that based on "betweenness"?
- My opinion is that a careful distinction is needed between informal description/definition and formal/mathematical definition. Hopefully they should be close together, but I am not sure it is possible, at least without higher mathematics than possible in this article. E. Artin, in the book I have cited, shows that, when starting from axioms like Hilbert's one may construct a structure of field on the points of the line, and, conversely that, constructing the geometry from the structure of vector space allows to prove Hilbert's like axioms (I am not sure that Artin's axioms are exactly the same as Hilbert's). Constructing geometry from field theory is roughly as follows: Starting from a base field R, usually the field of real numbers, one constructs Rn the set of n-uples of elements of elements of n, which has a structure of vector space of dimension n. Affine lines in this space may easily be defined in it by linear equations between the elements of the n-uples, called coordinates. Anything works well, and rather easily, with this construction. The only difficulty is of philosophical kind: the space in which we live is not a vector space. In particular, it has no given origin nor given coordinates system. The fact that there is no fixed coordinates system may be obviate by working with a vector field which becomes explicitly isomorphic to Rn as soon as a basis has been chosen. Similarly, the origin may be "forgotten" by considering an affine space, which is a set endowed with a vector space which operates bijectively on it (translations). The set of the vectors of a vector space is an affine space whose associated vector space is itself. Finally, the Euclidean space may be constructed by choosing a metrics, i.e. a quadratic form, on the affine space. The best presentation, I know, of all of this is Marcel Berger book of geometry (in French) but, if you cannot read French, the references in the article affine space may be convenient, especially Coxeter's book. There is absolutely no circularity in this algebraic construction of the geometry. As you can see, the definition of the lines and other geometric objects is rather simple in Rn, but this space does not correspond easily to the intuition that everybody has of the space of the geometry. On the other hand the notion of affine space is more intuitive, but needs higher mathematics. D.Lazard (talk) 00:37, 14 December 2010 (UTC)
I do understand that you can define sets of n-tuples that you can call "straight lines", outside a vector space. These, however, are just sets of n-tuples, not sets of "points", so they are not really "lines", nor "straight lines". For practical applications, they are useless. But as soon as you make it a vector space, by choosing a basis, there's no way to define a basis without a coordinate system. Even without a fixed origin, a basis needs coordinate axes. I did not study affine spaces, but I can't imagine a metric which could build an Euclidean space unless you start from another space in which the coordinate axes have a know curvature or shape, and orientation in space. And how can you know the shape, and how can you define angles, whitout knowing what a straight line is? So, I am not convinced that this definition is non-circular. It may be so complex that you don't realize it is circular, perhaps. Of course, I am not sure, but at least I can say that your explanation does not convince me. Paolo.dL (talk) 12:20, 14 December 2010 (UTC)
- It seems to me that you make a confusion between mathematical objects and the representation you have of them in your mind: Since the end of 19th century or beginning of 20th all the mathematics is build on top of set theory. That is, each mathematical object is an element of a set. The nature of the element itself (at least for the basic objects, like numbers and points) does not matter. The important thing is the structure of the set and the operations which are defined on the elements, which should be chosen to correctly reflect the properties of the reality which are modeled. Thus, as soon as the n-uples of real numbers have the properties you have in mind for a point, there is no problem to define a point as such a n-uple and the line as the set of n-uples which has some properties, namely to satisfy linear equations. This is exactly what has been done in the figure of the article, where the lines are implicitly identified to the set of solutions of their equations.
- On the other hand, a mathematical model never reflect all the properties of the reality, and mathematical theories may be hierarchized by the number of basic properties they introduce. This is the difference between affine and Euclidean geometry. Affine geometry consider only the geometrical properties which do not involve the distance (like "the 3 medians of a triangle are concurrent", the middle of a segment being definable without defining any distance). The curvature needs a distance to be defined, but it is not needed to recognize a curve from a line: a curve (defined, for example as in the topology section of curve), is a line or a segment if and only if it intersects every line in 0, 1 or infinitely many points. Curves, and, in particular ellipses, are defined in affine geometry, but Euclidean distance is needed to distinguish a circle from an ellipse.
- It is clear that these considerations are of a too high mathematical level to be introduced in this article. But it seems important that the article should not contradicts them, as they are now a consensus between professional mathematicians. It is in this spirit that I have rewritten the introduction of this article, insisting implicitly on the evolution of the mathematicians mind (by citing an old text). It is also the reason for which I have removed the mention of curvature in the introduction of curve, as not important as this place. D.Lazard (talk) 15:44, 14 December 2010 (UTC)
- My point was that a n-tuple does not imply a basis. An Euclidean space does, and this basis in turn implies a Cartesian coordinate system. And you did not convince me that you can define a Cartesian coordinate system without a primitive definition of straight lines, orthogonality, distances, etc. But this is not my most important point.
- I also wrote that you can call "straigth line" a set of n-tuples with some math properties, if you like, and I am perfectly aware that this would solve the problem. This, however, means that we should not care about defining a Cartesian coordinate system. Who cares if the axes are, according to some criterion, "curved"? The numbers in the standard basis are always the same, even when the axes are curved.
- So, do not try to solve the problem just by saying I can't accept an abstraction. I can, but I think there's also an interesting problem of practical application. Let me try and represent a reader who doesn't care about bases and coordinate systems. You can say a line is straight if it is aligned with a ruler, but the reader might ask: "how could you decide that the ruler is straight? when no ruler existed, how could someone create the first ruler"? You can use the path followed by a photon, but then "how could you decide that the photon travels along a straight line?" (actually it doesn't). So, is there a better way to solve this practical problem? Perhaps not. I don't know, but I believe the reader should have an answer, and I am pretty sure an answer to this practical problem cannot be given using n-tuples. Paolo.dL (talk) 20:02, 14 December 2010 (UTC)
- All these questions are special instances of the same epistemological question: Most mathematical notions are models of some non mathematical reality; are these models adequate to represent this reality? It seems difficult to answer it in this article. The more so as many mathematicians do not agree with the sentence preceding the question. D.Lazard (talk) 11:08, 15 December 2010 (UTC)
- The answer to these questions is clear from the definition of the lines by linear equations. May be it should be better emphasized, even if the answer is clear on the figure. The answers follow from the fact that the field of the real numbers is infinitely long (Archimedean property of the reals) and continuous. However, the answers are different if one is doing geometry over another field (there are many papers in cryptography devoted to counting the (finite) number of points on elliptic curves over finite fields). D.Lazard (talk) 11:08, 15 December 2010 (UTC)
Is a curve a line?
When you refer to a line do you always mean a straight line? The article curve mentions that a curve can also be called a curved line, so lines aren't necessarily straight? --22.214.171.124 (talk) 09:24, 8 June 2011 (UTC)
- In maths, "line" almost invariably means "straight line". In everyday English, curves are often described as lines, though. A line is an example of a curve, though. --Tango (talk) 21:58, 8 June 2011 (UTC)
- By Euclid, "Line is a breadthless length", whatever that means. Conforming to this ancient definition which has prevailed for several millenia I would suggest that a "line" in mathematics should refer to any abstraction of a very long but not very thick object. Think about phrases such as "line integral", which definitely applies to "curved lines", and "line segment", which usually indicates that the line is straight. My opinion is that in technical terms a "line" without any prelimiters should refer to any one-dimensional manifold. Of course there are axiomatic systems where this definition is not very natural, but anyhow it should meet with any other description of the intuitive concept. Lapasotka (talk) 23:11, 8 June 2011 (UTC)
- I'd assume an "unqualified" line means straight line in Euclidean space, parametrized either by two points or a point and direction vector, although could also be a geodesic, the shortest path on a curved surface, like the equator or lines of longitude represents straight lines on a sphere, while other parallels are curves (circle paths). Tom Ruen (talk) 23:39, 8 June 2011 (UTC)
- When one resorts to "parametrizing" lines in a Euclidean space the description becomes void in an instant. In the end the real number system, which is used in the process, is itself custom tailored to encapture the idea of a "line" -- a breadthless length -- and the analytic geometry is then easy to build upon it. The ancient definition (Def I.2 in Euclid) of a line is more primitive than the concept of "straightness" (Def I.4) which states that "A straight line is a line which lies evenly with the points on itself". Here the straightness is, curiously enough, defined very similarly to the modern differential geometry, by means of symmetry. Only in analytic geometry it even makes any sense to define a "(straight) line" by means of algebraic equations, and even then the equations themselves do not capture the main idea. Think for example of az+bw+c=0, where z and w are complex variables. Lapasotka (talk) 02:06, 9 June 2011 (UTC)
- I guess that line was for ancient English mathematicians an abbreviation of straight line, and that is a modern evolution that this abbreviation becomes the norm. In fact things evolved differently in other languages: The French equivalents of straight line and curved line are ligne droite and ligne courbe and the modern abbreviations are droite and courbe which mean straight and curve; the word ligne has evolved differently, being used for the row of a matrix. Thus this discussion is not about mathematics but linguistic, being impossible to translate literally: The question of 126.96.36.199 would give the equivalent of "When you refer to a straight do you always mean a straight line?" D.Lazard (talk) 07:44, 9 June 2011 (UTC)
Definitions and Descriptions
I am somewhat sorry that I missed last year's discussions in the previous two sections (NOT!). It seems to me that the reason folks were going round and round without coming to any conclusion is a lack of clarity about what a definition means. In any axiomatic system (and geometry is certainly such, Euclidean or otherwise) there must exist primitive notions, objects or relationships that have no definition. Every definition in the system can ultimately be traced back to rest on the primitive objects and the relations between them which are given by the axioms of the system. When you have different axiom systems that describe the same subject, they do not have to have the same set of primitive notions. So it is possible that a primitive notion in one system can be a definition in a second system, because the second system has a different set of primitives. This is relevant to the previous discussions in the following way: The concept of a line is a primitive notion in most axiomatic treatments of Euclidean geometry - specifically, and very emphatically, in Hilbert's treatment and Euclid's treatment (although he didn't realize it). In coordinate geometry you can define a line by means of a linear equation because you have changed the axiom system and line is no longer a primitive object. An axiom system for coordinate geometry will generally have an axiom that says that the points of a line are in one-to-one correspondence with the real numbers (for example, G.D. Birkhoff's treatment, circa 1936) a statement that you would not find in Hilbert or Euclid. In Artin's Geometric Algebra which has already been mentioned, a new set of axioms for Euclidean geometry is given, and he can, with respect to this new set, define a line ... because it is no longer a primitive notion. These are not examples where there are two (or more) definitions for the same object because the definitions are with respect to different axiom systems and in essence you have changed the groundrules and are now comparing apples and oranges.
- Although Euclid called "breadthless width" a definition of a line, we would not do so today. Rather, we would say that he was trying to describe (give a mental picture of) a primitive notion. This is a very dangerous thing to do since having no definition implies that there is no mental picture of the object. These mental pictures are only meaningful in the context of a specific model where the primitive notions have been identified with something concrete in the model. When you change the model you change the mental picture and this can be very confusing, especially to someone who is trying to understand the concept.
The only reason that I have not gone in and edited the lead here is that I do not know how to do it in such a way that would remain faithful to what I have just written (without being as preachy as I have been) and yet be at the level that this article is trying to achieve. Any suggestions would be welcome. Wcherowi (talk) 19:56, 16 September 2011 (UTC)
- I agree with the beginning of Wcherowi's considerations but not with the end of first paragraph nor the second one: Mathematical theories are usually models of the real (physical) word, like geometry, or tools for modeling it (number theory or analysis). Especially the line is an abstraction and an approximation of a visible structure of the real world. This is exactly what Euclid meant with his description of "breadthless width". The different axiomatizations of the geometry are different formalizations of the same intuitive concept. Thus the "mental picture" precedes the definitions and there is no reason to change of mental picture when changing the definitions. The mental pictures have to be changed only when previous mental pictures appear to be inaccurate, by the apparition of a paradox (Zenon's paradox, Russel's paradox, or Michelson's experiment which has lead to relativity). In our case, the various axiomatizations of Euclidean geometry have been proved to be equivalent. Thus it does not matter if line is a primitive or a defined object of the theory, as its properties are the same, and, thus it represents the same reality.
- Therefore, I agree with the present state of the introduction of the article, presenting first the intuitive notion, then explaining why it does not suffices an then presenting the main formal definitions, before to be more formal in the sections.D.Lazard (talk) 13:53, 17 September 2011 (UTC)
This may be an appropriate applied maths point of view, but a Formalist (and on Sundays I am one) sees no connection between a mathematical theory and whatever it is that you call "reality". It doesn't keep me up at night, but I would wonder what the physical referent of a Klein quadric in 5-dimensional projective space would be. I think it does matter whether something is a primitive or not. In the '50's Ma Bell (AT&T) had designed some telephone switching boxes which were models of the projective plane of order 5. In this model switches were lines. I would like to know how thinking about "breadthless widths" or any variant of that will help anyone understand the sentence I wrote before this one. I am not advocating dropping descriptions, but I am concerned about the limitations on our thinking processes that inappropriate descriptions can foster.
I fully agree with your outline of what the flow of an article should be. What I am grappling with, as an editor, is how to simplify something that I might understand from an advanced viewpoint without distorting it or providing a false impression. I believe this to be a very difficult task, but one that we need to master for good WP articles. Certainly one aspect of this task is to be very careful with the language that is used. So, when I see utter nonsense like defining something to be a primitive, which appears in the lead of this article, I tend to get upset and want to do something about it. Wcherowi (talk) 19:07, 17 September 2011 (UTC)
- I'm not a Formalist, even on Sundays, so I think the philosophical details would be better spelt out in the article on Formalism (mathematics), with just a brief mention in the article here. We can avoid any disagreements by reporting how others have defined a line, rather than arguing over what it "really is". Dbfirs 20:09, 17 September 2011 (UTC)
- I see what you mean now, and support your addition (see below). An anon editor has changed your "amongst" to "among" and changed "behaviour" to American spelling. I've reverted the latter, but left the former change (though I personally prefer your "amongst"). Dbfirs 16:47, 21 April 2013 (UTC)
Ray nonsense in the current article
I like to cite: "If the concept of "order" of points of a line is defined, a ray, or half-line, ...." This is totally misleading. You need no concept of order of points to define the ray. :-(
I state: There are 3 types of "lines":
1) of infinite extend in 2 directions : Do you want to call this line?
2) of infinite extend in 1 direction : Do you want to call this ray?
3) of infinite extend in 0 directions : Do you want to call this line segment?
Because in one dimension you have at most 2 directions, this classification is complete.
Why were we not able to state these simple facts clearly in the article? :-(
- While I don't think that this section on rays is particularly well written, your suggestions don't really lead to a better article. You are using two expressions, "direction" and "infinite extent" which, as far as I can see, can not be precisely defined without using the primitive concept of order. If you would like to make, say direction, a primitive term, you would need to supply the axioms which relate this term to the other concepts of Euclidean geometry. While there is historical precedent for doing this, it never really caught on. Failure to deal with these concepts precisely will lead to the same kinds of mistakes that Euclid made and were clearly pointed out by Hilbert and others. Bill Cherowitzo (talk) 23:24, 22 June 2012 (UTC)
- The wording is up to you! :-( I wanted to point out the logic / classification (in mathematics) behind of these "types of lines"! :-/ You may also define line as the straight connection between any two differnt points, if you also allow a point at infinity and -infinity. But this also is only a question of representation, not of the fact itself! Achim1999 (talk) 17:49, 24 June 2012 (UTC)
- This is not a matter of wording, it is a matter of logic. You are trying to replace carefully chosen terminology by vague undefined concepts (straight connection being one of them). Bill Cherowitzo (talk) 04:28, 25 June 2012 (UTC)
- NO!. It is not a matter of logic, it is a matter of representation because of the philosophic understanding of the pure (undisputed?) facts! Thus you may call it also a matter of mathematical philosophy, if you like. Achim1999 (talk) 09:26, 25 June 2012 (UTC)
- In German, which generated many historical mathematician/geometricians, there are for the object "gerade Linie", the words "Gerade", "Strecke" and "Strahl" in mathematics to precisly distinguish the general "(gerade) Linie" , in English "(straight) line", form the just wanted 3 types. Look into your mathematical dictionary for English words. :-) Achim1999 (talk) 17:57, 24 June 2012 (UTC)
- NO!. It is not a matter of logic, it is a matter of representation because of the philosophic understanding of the pure (undisputed?) facts! Thus you may call it also a matter of mathematical philosophy, if you like. Achim1999 (talk) 09:26, 25 June 2012 (UTC)
- I fail to see the point you are making. The german terms are translated as line, ray and segment when put into context. Bill Cherowitzo (talk) 04:28, 25 June 2012 (UTC)
- Yes, we agree on the simple geometric facts and also on the english words. We agree not of their representation (in the article to the reader) because of different philosophy / understanding behind them! You need special "ray"-concept, me not! Achim1999 (talk) 09:30, 25 June 2012 (UTC)
- Given distinct points P and Q in the Euclidean plane, they determine a unique line. Since this line exists you claim it has a natural ordering of its points. So, which of P or Q comes first in this natural ordering? Also, there is nothing metrical about the definition of a ray ... you do not need to measure anything – only order things. The article is not misleading, I think you are not appreciating the complexity of the issues involved. Bill Cherowitzo (talk) 04:28, 25 June 2012 (UTC)
- To decide whether P or Q is first, is not my/the problem. For eiter a line or a line segment you can't! And more you need not in this geometry context! Thus the whole ordering idea should be totaly avoided here, because it is unnecessary for this notation and complicates things without any win! Lastly the ray. Because it is asymmetric you may define a unique order! But I need no order, you want it here in the article! :-( Achim1999 (talk) 09:19, 25 June 2012 (UTC)
The ray section contains a nonsense, but not those asserted by Achim1999. It lies only in the conditional statement of the first sentence. In fact a "betweenness" relation on three points is always defined on (Euclidean) lines: Given 3 points on a line, one is between the two others. This relation (or an equivalent one) is either among the axioms of the geometry or, in coordinate geometry, a consequence of the total ordering of the reals. Thus a correct first sentence for this section would be:
"Given two points A and B, the ray with initial point A and passing through B is the set of the points C of the line containing A and B such A is not between B and C."
- The whole idea means, the author, and obviously many other surfacally-thinking guys here, have not understand that we are in simple geometry and need no order relations/notion here, but a unique classification of 3 cases are the basic of "point-line definition". Sadly, there is very low level of "want-to-be insights" of a specific geometric foundation on wikipedia. :-( Achim1999 (talk) 09:19, 25 June 2012 (UTC)
- I agree with D.Lazard and applaud his attempt to improve the section. The minor changes I made were mostly to improve the English. However, I did get rid of the ill-defined phrase, "finite in one direction and infinite in the other" as this can be made precise by using the betweenness relation (and betweenness is how you define order in geometry). This required the return of the correct diagram of a ray, since three points are needed to talk about betweeness.
- I believe that Euclid also thought that geometry was simple and saw no need for the notion of order (betweenness). Euclid was wrong! This defect in his treatment leads to "proofs" that depend on how the diagrams are drawn, and hidden assumptions whose lack would invalidate the logic of the presentations. You are asking us to ignore what we have learned over the past few centuries and return to the intuitive but faulty logic employed by Euclid ... I see no gain in doing so. Bill Cherowitzo (talk) 16:37, 25 June 2012 (UTC)
- You miss again the point. *sick* It is a question of (mathematical) philosophy, not of logic. Look at Hilbert's Axiomatization of Euclidean Geometry if you want to switch this topic. :-( He did it the first time completely and correctly! But this doesn't excuse your bad understanding of the foundations what can be done/defined with two (different) points in Geometry. THIS we (only/mainly) need in the article, not repeating Hilbert's work to get enlighted of the "same" origin for line, ray and line-segment! :-/ Achim1999 (talk) 20:19, 25 June 2012 (UTC)
Definitions and descriptions II
Having looked at the reader feedback comments on this article, it is clear to me that some segment of readers are not understanding the lead section. I am referring to the constant call for a "definition" of a line. What is said in the lead, and repeated in the first paragraphs of the Euclidean geometry section state clearly, at least to me, that there will not be a "definition". Perhaps this message is too subtlely delivered. I will attempt to address this problem head on with a section that I'll call Definitions versus descriptions in which some (but not all) of the discussion in the earlier section of this talk page will appear. This new section may at first appear to be too redundant of other material in the article, but I am hoping that other editors will help smooth out that problem. Bill Cherowitzo (talk) 16:39, 19 November 2012 (UTC)