Talk:Curve

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Older discussion at Talk:Curve/archive1

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)

Contents 1 last changes "focussing on differential geometry" 2 universal curve 3 Mechanical curve 4 I would like to put my two cents in 5 Same curve or not?

[edit] 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)

Please, can we have a proper discussion of issues here, on this page?

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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] Same curve or not?

Consider the following two plane curves:

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? --Lambiam^Talk 20:33, 27 December 2006 (UTC)