Talk:Line (geometry)

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Contents 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 13.1 Discussion 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?

[edit] Collinear vs Coplanar?

If 3 points are collinear, are the necessarily coplanar? 76.111.81.183 —Preceding signed but undated comment was added at 22:52, 22 September 2007 (UTC)

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)

[edit] Too advanced?

This page is good, but delves right into the advanced stuff. 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)

[edit] 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)

[edit] 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 (UTC)

[edit] Two links to Bulgarian (Български)

This is because the content of this article is in two bg articles.

[edit] Question

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 70.59.199.11 (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 70.59.199.11 (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)

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)

[edit] Notation

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 67.171.149.4 20:07, 16 March 2006 (UTC)

[edit] 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)

[edit] POLYLINE

POLYLINE redirects to this page but is then not discussed. Can someone mention POLYLINE here, or make a distinct page?

Beau Wilkinson

colinear also redirects here, but is not mentioned on the page. What gives? Sim 14:15, 24 March 2007 (UTC)

[edit] 3D Lines

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)

[edit] Title

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)

[edit] y=mx+c to y=mx+b

I changed it because that is how the formula actually is written, according to the math courses I have taken. —The preceding unsigned comment was added by 75.4.13.98 (talk) 23:46, 14 May 2007 (UTC).

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)

[edit] Requested move

[edit] 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)

[edit] 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)

[edit] lines history

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:

where:

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:

where:

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)

[edit] 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 82.68.215.206 (talk) 11:50, 19 November 2008 (UTC)

[edit] 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)

[edit] Second sentence

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)

I'm struggling to see in what way it's a geometry if it doesn't have points... What's the definition of an automorphism without reference to points? --Tango (talk) 11:37, 9 December 2008 (UTC)

OR comments removed--Ramsey2006 (talk) 17:04, 16 February 2009 (UTC)

[edit] half line?

The ray section makes it sound like a ray is half as long as a line, when they are in fact the same length (half of infinity is infinity).M00npirate (talk) 01:44, 27 January 2009 (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)