# Talk:Curve

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Field: Geometry

## Untitled

Older discussion at Talk:Curve/archive1

I added French curve to the 'See also' section. Perhaps trivial and obsolete, but certainly curve-related. —Preceding unsigned comment added by Moris JM (talkcontribs) 17:49, 7 February 2010 (UTC)

## Explanation of rewrite

I rewrote the article to focus on the curve definition used in differential geometry. I think it is much more accessible now and with the importance of differential geometry for physics most people will come looking for this definition. I moved the topological definition of curve to the end of the article.

The last edit by 145.254.193.73 was also me, forgot to log in :)

MathMartin 14:29, 5 May 2004 (UTC)

I am still working on the article. My main goal is to make the article more accesible and useful by focussing on the differential geometric aspects of curves and the common definitions like regular, jordan curve etc.

I removed the following from the article because it talks to much about manifolds and too little about curves. Furthermore some of the text is now duplicated in the definition section.

If X is a differentiable manifold, then we can define the notion of differentiable curve. If X is a Ck manifold (i.e. a manifold whose charts are k times continuously differentiable), then a Ck differentiable curve in X is a curve c : I --> X which is Ck (i.e. k times continuously differentiable). If X is a smooth manifold (i.e k = ∞, charts are infinitely differentiable), and c is a smooth map, then c is called a smooth curve. If X is an analytic manifold (i.e. k = ω, charts are expressible as power series), and c is an analytic map, then c is called an analytic curve.

There are other things people might be looking for. I expanded the section on algebraic curves, which is certainly one of them. Gene Ward Smith 18:32, 25 April 2006 (UTC)

What do we gain by using manifolds instead of Rn ?

MathMartin 15:44, 5 May 2004 (UTC)~

If we had the jet bundle article there would be a quick comeback on that.

One thing to bear in mind, is that in the longer term articles do aim to be comprehensive. That is not the same thing as having an expository strategy, and following it.

BTW, this page is getting long, and some archiving is called for, especially if it is going to be actively edited.

Charles Matthews 16:00, 5 May 2004 (UTC)

If we had the jet bundle article there would be a quick comeback on that.

Did not get this. What do you mean ?

One thing to bear in mind, is that in the longer term articles do aim to be comprehensive. That is not the same thing as having an expository strategy, and following it.

What do you mean by this ? Should I revert the definitions to the more abstract stuff ? What is a longer term article ? I do not think a have removed stuff from the page. I just reordered it (topological curve and algebraic curve at the end) and focussed on the differential geometric stuff. At least thats my intention.

MathMartin 16:22, 5 May 2004 (UTC)~

Well, jet bundle is a Requested Article which will get done one day. Jets are equivalence classes of curves in manifolds (cf your removal from the page); and are a basic concept.

So, all I'm saying is that future developments should be borne in mind, here. This is always going to be a major page. There is more than one way up the mountain, and I'm not objecting to your path.

Charles Matthews 16:45, 5 May 2004 (UTC)

## last changes "focussing on differential geometry"

I can not stand these last changes, there is Diff geom subsection, if you feel something should be added do it there or make new page, call it curves in euclidean space or so.

Tosha 02:13, 7 May 2004 (UTC)


Charles Matthews 06:53, 7 May 2004 (UTC)

My main point, aside from what I said before is: I do not think it is good to use the most abstract definition (like defining length on metric spaces instead of euclidean spaces). I think the page should have mostly one level of abstraction (at the moment the beginning of the page talks about topological spaces, then we use metric spaces, then we use differentiable manifolds). The page should use one setting (e.g. euclidean space) to define interesting curve definitions. If necessary on can always say "this definition can be abstracted to topological spaces ..".

If you do not agree Tosha, I will probably start my own page on euclidean curves, but then we will duplicate much material. And I would probably link to your curves page whenever I wanted to point out the more abstract definitions.

So it makes more sense for me to put the curve stuff in one central page. But I cannot "stand" the page as it is right now. The differential geometric subsection is not enough for me. I think most people come looking for the differential geometric definitions and not the more abstract definitions, so those should be central to the page.

MathMartin 08:57, 7 May 2004 (UTC)

There can be different ideas on exposition. If this is a disagreement about the order of topics, mainly: could MathMartin and Tosha just give their ideal orders, and discuss that. If it's about level of treatment in the differential geometry, in the end probably there will be multiple discussions; but it is better if they all start 'in the same place'.

Charles Matthews 09:32, 7 May 2004 (UTC)

I do not think this is about order of topics, but more about focus of article (differential geometric curves vs. more abstract curves)

Ok here is my order of topics:

• Introduction:already in the article, albeit a bit short
• Definition: This should list the most important and accessible definitions relating to curves. The definitions people mainly come looking for on this page. Should be very brief.
• Notes: Explain the definitions a bit, give some background why the definitions are useful
• Examples: See definitions in action
• History: Curves are quite old, so we should have some historical information (curves as conic sections, usage in physics)
• other topics deserving their own section: Length, Equivalence classes, arc length ...
• more abstract curve definitions: Curves on topological spaces, metric spaces, algebraic curves ..

MathMartin 10:07, 7 May 2004 (UTC)

I understand your point, but this is an article about curve, if you want you can make one say curve in Eucledan space, make a remark in the beggining of curve that this article is a bit advanced and send less advenced readers to the new one. I think it would be a good idea and copying material from page to page is not a problem.

In my opinion the article was much more interesting before your changes.

On history, there was nearly no information in the history subsection, exapt that straight line was not curve before but now it is. I could not imagine a person who would get anything out of it so I removed it. I think the history subsection should be included only if the history is interesting, not just born grow.

Tosha 15:02, 7 May 2004 (UTC)

I do not understand in what sense the previous article was more interesting ? I did not see any clear focus in the article. If you already know about curves you probably can find some interesting information in the old article but it was definitely lacking a coherent presentation and was not accessible.

I admit my history section was a bit thin. What I was trying to point out is how the concept of a curve changed from a static one (conic section) to a dynamic one (curve of a point mass).

Charles Matthews does not think it is a good idea to have two articles on curves. But I do not think we can reach any conclusion other than doing two articles on curves. If I have time I will create a new curve article using the name "Curves (in Euclidean Space)" or something. Perhaps someone neutral can later merge the pages.

MathMartin 15:48, 7 May 2004 (UTC)

Well, OK, assuming these are understood positions, now. I could try to find a compromise edit. I don't myself have such strong feelings - is the Frenet stuff, which used to be in all the textbooks, important? Or is it quite boring, as Frank Adams once told me? I think it could be argued either way.

The point about curves in Euclidean space not being completely separate: better to have a summary in this article, and See main article ... there. This should ensure better consistency, and also gives a chance for two expositions, at different paces.

Charles Matthews 16:07, 7 May 2004 (UTC)

the new editition "focussing on differential geometry" looses style, it becomes borring topic in calculus. I do not object (and never did) someone will need such presentation but this one should also survive.

I just want to note that it is unlikely that anybody will open this page to find out what curve is, and most likely that someone will look for specific information about curve here, that is the reason it should contain general definitions (not just "do it your self"). I do not see why not to make separate article on smooth curves, I think the subject is very different, all these regular-free curves could be covered and it is too much for one article.

One more thing, we should not look hard for compromize, it will simply make the article worse

To make it short: Let's split.

Tosha 23:48, 7 May 2004 (UTC)


I agree with Tosha. Martin, I think you're bringing too much of a bias to what is considered important. You just automatically assume that (of course??) anyone who comes here must be interested in differential geometry. This is not necessarily true. It is true that differential geometry is a hot topic and important in physics, but it is hardly the only manifestation of the "curve concept", and to assume that the needs and desires of readers who come here are matched in line with your own is a bit "diffeo-centric" (pardon the term). There are many topologists who study curves outside the setting of Reimannian metrics, e.g. purely for their topological properties or metric space properties. There are others who are fascinated with nowhere-differentiable curves, Peano curves, and other "pathological" examples. There are fractal curves, which are more and more important. People in number theory and algebraic geometry will most likely think of elliptic curves and varieties when they first hear the term "curve", and since there is a definite geometric interpretation and flavour to these objects, they also qualify as "curves". This article should give a general overview of the curve concept (which can have intuitive explanation and history, but will need to be somewhat abstract by nature), a summary of different types of "curves" in different areas of math, and then each of these can probably be fleshed out in a separate article. This happens often with really general topics. It's also a fine line between being too abstract and not general enough, usually this is worked out by giving some motivation and examples from concrete cases before general definition. Also, as more subtopic pages become available, links within other articles can be made to point to the SUBTOPIC so as not to lose a casual reader in an abstract definition (this is done with limit, where the author has a choice of several different pages to "direct" the reader to.) Revolver (YES...I know I said I was on leave to write my paper, but I've popped in very occasionally anonymously...guess I must be "wikipediholic"...)

I think we can have a perfectly satisfactory, balanced curve page, mentioning at least all the major usages; and providing links to more detailed expositions. In fact, it is hard to see how anything else should work, in the long run.

Charles Matthews 09:46, 8 May 2004 (UTC)

I have now looked at the two mathematical encyclopedias I have. This article compares quite well; and the differential geometry/algebraic curve material is put in separate articles. Charles Matthews 13:19, 8 May 2004 (UTC)

Ok you convinced me. I probably have a bias towards differential geometry. I wont change the structure of the article and will try to put my stuff in the differential geometry section. If this section becomes big enought I will put it on a seperate page.

MathMartin 19:52, 10 May 2004 (UTC)

I've actually collected up a number of short articles that were already here, and made differential geometry of curves. It's a start; I'm aware that it requires edits to sort out.

Charles Matthews 20:47, 10 May 2004 (UTC)

I am a bit perplexed. My and Tosha did nothing but arguing about the curve page and meanwhile you have rewritten the page and created a new differential geometry curves page. This seems like a very good strategy to create/rewrite pages. I will just argue a bit and you do all the work :)

MathMartin 21:00, 10 May 2004 (UTC)

It's actually a revolutionary new management strategy. I'm thinking of called it 'where angels fear to tread', or something. Or perhaps it's a very old strategy, called 'getting your hands dirty'.

Charles Matthews 21:30, 10 May 2004 (UTC)

## universal curve

The redirect for uc seems too specific. Perhaps an abstract here would be a good idea? -MagnaMopus 18:57, 18 January 2006 (UTC)

## Mechanical curve

I've created Mechanical curve as redirect to this page, but if there's someplace better for it to redirect please edit it. From the page at Archimedes: "the first Greek mathematician to introduce mechanical curves (those traced by a moving point) as legitimate objects of study". Is that like Spirograph or something? Thanks! Ewlyahoocom 20:20, 11 May 2006 (UTC)

Yes, a spirograph is one example, but a simpler one might be an astroid, an involute, or really any continuous function if the moving point is taken to be an abstract and not necessarily a concrete object whose movement is being measured by the curved function. LokiClock (talk) 20:11, 25 June 2009 (UTC)

## I would like to put my two cents in

I say that a straight line is a curve with an undefined radius.--Luke Elms 20:25, 22 November 2006 (UTC)

## Same curve or not?

Consider the following two plane curves:

$\gamma_1 : [0, 2] \rightarrow \mathbb{R}^2, ~\gamma_1(t) = (t, t^2),$
$\gamma_2 : [0, 1] \rightarrow \mathbb{R}^2, ~\gamma_2(t) = (2t, 4t^2).$

Clearly, viewed as functions, these two are different. Are they also different when viewed as curves? Or are they different representations of the same curve? The present text equates the curve and the function, and thus appears to make them different. Most people (and, I suspect, most mathematicians) would consider them the same curve, just like 0.5 and 1/2 are different representations of the same number.

Is anyone aware of a (reliable, published) source that pays attention to this issue? If so, what do they have to say about it?  --LambiamTalk 20:33, 27 December 2006 (UTC)

This topic is discussed at the end of section Curve#Differential_geometry. 88.73.158.80 14:24, 3 May 2007 (UTC)

## Unwarranted extra disambig

Maniwar (talk · contribs) really, really, really wants curve(s) to be about Curves International, as was made clear in a prior discussion. As consensus made clear, it ain't gonna happen. Nor is there any good reason for an extra disambig notice above the usual one here; the disambig page already includes the necessary info. Reverting. --KSmrqT 02:23, 1 August 2007 (UTC)

KSmrq, I think you really, really, really are closed minded and full of POV. Yes there was a discussion on the Curves talk page as I originally created the Curves page and had it redirect to Curves International. However, the math guru's decided they wanted it to go to Curve, the math concept and so I agreed with consensus and left it alone, though I feel it was a bit biased with editors of math articles voting. Fast forward to now, since I discovered that it is perfectly fine within the wikipedia community to utilize otheruses4 (see firefly, Paraffin, Apollo, Brass, Jerome, Syntax, Spice) as a few of the many examples which utilize this. The community allows the tag and it is not hindering nor vandalizing the community. However, what I see here is a monopoly of math experts not wanting to follow suit with a perfectly excepted format of wikipedia.
Curves International is only the largest fitness franchise in the world, larger than Bally's and Gold's gyms combined...so it's not just some gym. Since you, apparently, have read the discussion on the Curves talk page you will see I'm doing this as a compromise, if you will, rather than seeking another rfc. Most people when they search for Curves, are not looking for the math concept, but in fact are looking for the women's fitness center. This entry is in no way defacing the article, and edit wars (hopefully) could be avoided, so I think this is a good compromise.
There is discussion too at User talk:Maniwar. Oleg Alexandrov (talk) 15:13, 1 August 2007 (UTC)
Let's try to keep the conversation here and not be all over the place. A central place will be good, in my opinion. --Maniwar (talk) 15:47, 1 August 2007 (UTC)

## Curve (civil engineering)

What about curves in civil engineering??? Peter Horn 02:43, 19 July 2008 (UTC)

## The most general definition of curve

The most general definition of curve is that of a 1-dimensional Topological manifold. Why this definition is not given at all in the page?--pokipsy76 (talk) 10:21, 8 March 2009 (UTC)

Because that is not a definition of curve. JumpDiscont (talk) 05:27, 11 January 2011 (UTC)
NO, it is a perfectly correct definition and it is given in the page as "A curve is a topological space which is locally homeomorphic to a line". In fact, the term "1-dimensional Topological manifold" has been substituted by its definition to remains at the lowest possible mathematical level. Nevertheless, when Pokipsy76 did ask his question, this sentence was not yet in the page.
Note also that, usually, edits in talk pages are done at the end, and that there is a discussion on the definition of a curve at the end, which is more recent than Pokipsy76's question.
D.Lazard (talk) 10:35, 11 January 2011 (UTC)

## Jordan curve vs Jordan arc

The text treats “Jordan curve” and “Jordan arc” as synonyms. I believe this is wrong; a Jordan curve is a homeomorphic image of a circle, as stated, but a Jordan arc, as far as I know, is a homeomorphic image of a closed interval. In other words, a Jordan curve is closed, a Jordan arc is not. Isn't that how the terms are used in the literature? Hanche (talk) 02:26, 10 November 2009 (UTC)

## Incorrect details in Differential geometry section

The text starting "Another way to think about a curve..." and ending "by Frank Ayers, Jr." is incorrect. First let me note that I have Ayers book (I'm presuming it's his Schaum's book on DE's). On page 41 he gives the correct formulae for the equations of the tangent line and it's x and y intercepts for a general curve F(x, y) = 0. The equation given in the text for the tangent is incorrect and should have parentheses around the X-x part, i.e. it should read Y-y = (dy/dx)(X-x). The equation given for the y intercept is correct. The equation for the x intercept is incorrect and should be X = x-ydx/dy. The statement "...but this case we will use it to find the X and Y intercepts which are when x and y equal to 0" should read "...but this case we will use it to find the x and y intercepts which are when X and Y respectively are equal to 0". These are minor mistakes and easily fixed. A bigger problem is that the whole concept is wrong. It is not true to say that the sum of the x and y intercepts has to be equal to 2 for a curve. This is easily seen by considering the curve y=3 whose tangent line is itself and which has no x intercept and y-intercept 3. Another problem is that the two equations being combined are only simultaneously true when the tangent line passes through the origin in which case they again do not add up to 2. Recommend that the entire text be removed. Psmythirl (talk) 19:27, 6 January 2010 (UTC)

I don't understand this part or how it relates to the rest of the Differential geometry section. I think it uses X in a different sense to the rest of the section, but it doesn't make that explicit or define its terms. It seems to have more to do with the "Graph of a function" sense. But if it's mistaken as well as confusing, I agree it should go. (In fact, seeing as no one has offered a defence of it since your comment in January, I've decided to remove it for now.) Dependent Variable (talk) 11:12, 30 August 2010 (UTC)

## Demoting to Start class

There a few issues with this article that lead me to the opinion that it still requires much work and is not of B class quality. First, there is an almost complete lack of references given for the material. The St Andrews and 2dcurves.com are good resources and make good external links, but they are mainly catalogs of individual curves, not a resource for the definition of a general curve or its properties. That leaves a single note and a single reference to support one section out of the entire article. Second, several of the sections are much to long considering the material is covered in another article. I brief introduction to the idea should be given only and details should merged with the other article. Finally, the article attempts to give a single definition of a curve when really there are several definitions depending on the context; a curve in topology is very different from a curve in algebraic geometry. I will try to address some of these issues myself, but the article should not be marked B class in its present state.--RDBury (talk) 19:17, 26 January 2010 (UTC)

It seems I have missed something vital here. What does B class mean? Where do I look for more info? Hanche (talk) 02:34, 5 February 2010 (UTC)
See the math rating banner at the top of the page, it has links to more information.--RDBury (talk) 07:14, 24 February 2010 (UTC)

## Definition of a curve

I reverted the edits by Paolo.dLfor the following reasons:

• The new version looks like a definition, while it is only an informal description. This is misleading.
• The term of series is improper, set would be proper.
• The new version omits the important fact that a line has no thickness.

Nevertheless there is an issue with these two pages curve and line. I'll try to solve them in a better way by editing them.

The old version, that you reverted, does not give a definition, but just refers to the definition of line (which is a special kind of curve), which in turn refers to the definition of curve. It's a circular definition. So, you are right that my definition is informal, but that's better than a fake definition. Besides the definition, there was additional text in my edit that you deleted without explanation. So, I am going to restore my edit, using the word "sequence" instead of "series", and explaining that the curve has no thickness, and you'll change whatever you don't like when you'll find a more formal and non-circular definition. Paolo.dL (talk) 10:06, 13 December 2010 (UTC)
The reference to the concept of Curvature or "straightness" (null curvature) is crucial to obtain a clear definition, and that was the part you deleted without explanation.
I see that topologists start from the definition of line. OK, then I'll change the informal definition of line to avoid circularity. Informal definitions are often needed in introductions. Notice, however, that the concept of "direction" is based on the concept of straight line, so the informal definition is not perfect.
Paolo.dL (talk) 13:45, 12 December 2010 (UTC)
Quite a tricky thing to define well without resorting to advanced mathematics. Neither def really seems to be good. The American Heritage® Science Dictionary has a fairly nice def: "A line or surface that bends in a smooth, continuous way without sharp angles."[1] I also think mentioning its 1 dimensional might help.--Salix (talk): 11:41, 13 December 2010 (UTC)

I feel uncomfortable with the "curved line" definition. I don't think it is an appropriate definition. Lines and curves are different geometrical entities. if the line is the extension of the shortest path between any two distinct points in the plane, then the curve encompasses all other paths. This "curved line" definition is a huge misconception, I believe.--74.192.202.208 (talk) 08:18, 2 December 2011 (UTC)

In older terminology, up to about World War I, the word "line" was used to mean what we would call a curve in today's language. The phrases "straight line" and "right line" were used for what we would call a "line", while "curved line" was a "line" that wasn't straight. See wikisource:1911 Encyclopædia Britannica/Line for an example of the old terminology in use. Such shifts in language are actually very common and it makes reading mathematics from more than a hundred hears ago a bit of a challenge.--RDBury (talk) 15:35, 2 December 2011 (UTC)

## path-image, alternate definitions, dubious

The section "Conventions and terminology", after defining a path to be a continuous map from R into a manifold, says that the word "curve" is used in diff.geom. & vect.calc. for a path, but that it is used in topologoy for the image of a path. This contradicts the definition given in the immediately preceding section, "Topology". What gives?

Also, the lead defines a "curve" as a space which is locally homeomorphic to a line, i.e., a one dimensional manifold. Is this wrong? I mean, it works for a simple curve that doesn't ever cross over itself, but in general the image of a curve may include (infinitely many) points that are not locally homeomorphic to a line. Is there a definition without this problem (without referencing paths, as they brings in the issue of multiple paths having identical images). Also, is a single point supposed to qualify as a topological curve (since it is the image of the path whose map function happens to be just a trivial constant)?

I think the different approaches and terminology should be given more priority in the article (ie. explained earlier and making the distinctions more clearly). Cesiumfrog (talk) 01:01, 3 May 2011 (UTC)