Talk:Curve

Addition to 'See also'

I added French curve to the 'See also' section. Perhaps trivial and obsolete, but certainly curve-related. —Preceding unsigned comment added by Moris JM (talk • contribs) 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 C^k manifold (i.e. a manifold whose charts are k times continuously differentiable), then a C^k differentiable curve in X is a curve c : I --> X which is C^k (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)

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)

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)

Curved lines in applied geometry

Mechanics, engineering and applied geometry are somehow related although, here, isn't it that the surface lines are only called curved lines for example, I dont understand why the addition of "curved" to a line is needed here even for nonstraight plane objects since it is obviously a line, anyway. --Mathstrght (talk) 09:10, 4 November 2022 (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)

This sounds true to me, but I would use the term radius of curvature. We have $\rho ={\frac {1}{\kappa }}$ where $\rho$ is radius of curvature and $\kappa$ is the curvature. Since $\kappa =0$ for a straight line, $\rho$ is undefined. FionaLovesCats (talk) 23:54, 21 April 2022 (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?  --Lambiam^Talk 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)

Comparative geometry of curved lines

Is something that is very easily applied with projections of curved lines into straight plane or planes. --Mathstrght (talk) 09:08, 4 November 2022 (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. --KSmrq^T 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.

As I asked another editor, and I now ask you, why you oppose this entry? Curves is the fastest growing fitness company in the world with 10,000 locations. Last year, for the first time since the U.S. Government started keeping tract, the rise in women's obesity went unchanged. It did not increase, nor did it decrease. However, the significance, as mentioned, it did not increase. But in the male category, there was still an increase. The only major contribution that could possible have made the difference in the female population, though the research is not yet complete, is the rise of female attendance in physical fitness. Curves has about 4+ million members and all the copycats combined are around one million, and have many many former members who had results. Curves is such a phenomena and is known to be such that Avon has partnered with them and has placed them in every Avon Magazine for the past year; and there is now a new partnering campaign with General Mills where they are making and selling Curves cereal, and Curves bars (snack). If you look a the search engines, there are more items for the fitness club than any other and there are more searches for Curves than there are for the math concept or any other. I fail to see why you all remain close minded and will not allow this minor compromise, which has been done by many other articles. I think your comment shows your ignorance as Curves has set major records in the Guinness book of world records, in Business Atlas, in the New York Times, etc. To tell the world that you are the fast growing franchise in the history of franchises is not just some gym. The company has gained the attention of the world business market, all the competition, the President of America's fitness council, the industry, and the medical world because of the milestones, the waves, contributions, and the changes it has brought/made in the fitness and medical world through exercise. I'm not trying to sell the company, but I am trying to show you how ignorant your comment was and how significant this company is and continue to be in the world. In the female population, Curves is a common household name and most likely everyone on here has a mom, sister, girlfriend, wife, niece, or aunt in the program or considering it. Now, I challenge you to defend why you refuse to allow a perfectly acceptable practice, otheruses4, within the wikipedia community and defend your close mindedness. Kind regards. --Maniwar (talk) 13:51, 1 August 2007 (UTC)

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)

Space curves

The section "Algebraic curve" says

Algebraic curves can also be space curves, or curves in even higher dimension, obtained as the intersection (common solution set) of more than one polynomial equation in more than two variables.

Would it be correct to make this more specific by saying the following?:

Algebraic curves can also be space curves, or curves in even higher dimension n, obtained as the intersection (common solution set) of n–1 polynomial equations in n variables.

Loraof (talk) 15:12, 15 October 2015 (UTC)

This is not correct. However, I agree that we need to be specific. I'll edit the page for fixing this. D.Lazard (talk) 11:42, 16 October 2015 (UTC)

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Sections on differentiable curves

Because of several non-equivalent definitions of a curve, this article must be structured as a broad-concept article. I have done the job, except for the sections on differentiable curves.

IMO, Differentiable curve should redirect to Differential geometry of curves (this latter title is confusing, as I do not understand why this article is not named "Differentiable curve" and what is the difference between "Differentiable geometry of curves" and "Study of differentiable curves"). The part of Curve devoted to differentiable curves should be reduced to a summary of Differential geometry of curves, whose size should be of the same order as that of the two other sections. If some content would be lost by this reduction, it should be added to Differential geometry of curves.

Also, the two articles must be made coherent: presently, a eight-shaped curve is a differentiable curve in one of the articles, and not in the other.

I could probably do these modifications, but it would be much better if they would be done by an expert. D.Lazard (talk) 16:36, 1 May 2019 (UTC)

Differentiatial geometry and calculations for curved lines

Also it is segments that maybe used in differential geometry calculus and these are actually used. --Mathstrght (talk) 09:06, 4 November 2022 (UTC)

What is a curve?

In articles about curves, one cn find several contradictory definitions of a (topological or differentiable) curve, that are contradictory. For example

• Roughly speaking a differentiable curve is a curve that is defined as being locally the image of an injective differentiable function $\gamma \colon I\rightarrow X$ from an interval I of the real numbers into a differentiable manifold X, often $\mathbb {R} ^{n}.$ More precisely, a differentiable curve is a subset C of X where every point of C has a neighborhood U such that $C\cap U$ is diffeomorphic to an interval of the real numbers. In other words, a differentiable curve is a differentiable manifold of dimension one. (in Curve#Differentiable curve.) This implicitly asserts that an injective differentiable function is a diffeomorphism on its image. For example, the cusp $\gamma (t)=(t^{2},t^{3})$ is a curve for the first sentence, not for the second one.)
• a vector-valued function $\gamma :I\to \mathbb {R} ^{n}$ of class $C^{r}$ (i.e., the component-functions of $\gamma$ are $r$-times continuously differentiable) is called a parametric $C^{r}$-curve or a $C^{r}$-parametrization. Note that $\gamma [I]\subseteq \mathbb {R} ^{n}$ is called the image of the parametric curve. (In Differential geometry of curves#Definitions). This means that a parabola is not a curve, but the image of a curve, and that $\gamma (t)=(t,t^{2})$ and $\gamma (t)=((t+1),(t+1)^{2})$ are different curves. Also, an eight-shaped curve is a differentiable curve for this definition, but not for either definition given in curve.

In summary, we have three different definitions of a differentiable curve: one that includes cusps and eight-shaped curves, one which exclude eight-shaped curves but not cusps, and one that excludes both cusps and eight-shaped curves. Moreover, often in contradiction with the common usage, this is the parametrization that is called a curve, not its image. It seems that it is a WP:OR tentative for formalizing the common usage of using "the curve $\gamma$" as an abbreviation for "the curve defined by $\gamma$".

Suggestion. I suggest to use the following definitions in all related articles:

• A topological curve is the image of an interval by a continuous function $\gamma .$ If the function is differentiable, then this is a differentiable curve.
• In the preceding definition, $\gamma$ is a parametrization of the curve.
• The phrase "the curve $\gamma$" is a commonly used abbreviation for "the curve defined by $\gamma$".
• A parametric curve is the pair formed by a curve and a defining parametrization. ("Parametrized curve" would be more proper, but is less common.)
• A regular curve is a curve that can be parametrized by a function that is a diffeomorphism of an interval or a circle onto its image. Regular curves are (connected) differentiable manifolds of dimension one.

As implementing this suggestion may need to edit several article, some comments would be useful before editing these articles. So, I'll add notifications at WT:WPM, Talk:Differential geometry of curves and Talk:Plane curve. D.Lazard (talk) 13:59, 9 May 2019 (UTC)

I'm a bit squeamish about some of this. For one, I've never heard "topological curve" used as such. A continuous function from the interval to a space is generally just called a path (or an arc if it's injective). Presumably you're doing this by analogy with topological manifold, but does this actually get used? Even if so, I can't imagine it's all that common. Also, for injectivity (smooth or not), if that's really needed, it's often fine and easier to just state separately. It really depends on context, but using manifold language may be overkill in certain places. Might have more to say later, but that was the main thing that struck me right away. –Deacon Vorbis (carbon • videos) 15:48, 9 May 2019 (UTC)

[0, 1]² as a subset of R² is both the image of a Peano curve and of a Hilbert curve. I would not call these the same. What about defining a curve as an equivalence class of parametrizations, where two parametrizations γ and γ' are equivalent if there exists an order-preserving isomorphism φ between their domains such that γ = γ'∘φ?  --Lambiam 20:57, 17 May 2019 (UTC)

The parametrization γ defined by γ(t) = (t², 0) if t < 0, (0, t²) otherwise, is a differentiable function, but I wouldn't call the corresponding curve differentiable at the origin.  --Lambiam 21:05, 17 May 2019 (UTC)

To editor Lambiam: I have clarified that "topological curve" is a term that is introduced here for practical purpose. D.Lazard (talk) 13:45, 21 May 2019 (UTC)

I'm afraid it is still not always clear when "curve" refers to the image (a subset of some space), and when it refers to a function whose domain is an interval and whose range is such an image. Note that Differentiable curve redirects to Differential geometry of curves, although the latter article does not actually present a definition of the notion "differentiable curve"; the term does not even occur. Similarly, Parametric curve redirects to Parametric equation, but that article also does not actually present a definition of the notion "parametric curve" – although the term is used, but somewhat ambiguously. Furthermore, some curves are defined as a locus, which may or may not be specified in the form of an equation, but does not generally determine a parametric curve (take e.g. the definition of an ellipse as the set of points having a given sum of the distances to two given points). Above, where you wrote, "If the function is differentiable, then this is a differentiable curve", I interpreted "the function" to refer to the function mentioned in the preceding sentence, and "this" to the entity called there a "topological curve", no other plausible antecedent being in sight.  --Lambiam 14:22, 21 May 2019 (UTC)

Thinking more about it, I (now) feel that the best course of action is to present a historical account, starting with how the Ancient Greek geometers approached this (a locus defined in words), then the approach through parametric equations arising with the invention of analytic geometry, and finally as the image of a function on an interval. A comparison – without suggesting that any of these is intrinsically better than any other – will reveal that these definitions of "curve" are not interchangeable (e.g., a hyperbola can be defined as a locus, but only one branch at a time as the image of a function on an interval; conversely, (the image of) a bounded space-filling curve is never the solution set of an algebraic equation – one has to cheat and use a non-algebraic equation like $\mathrm {max} (|x|,|y|,1)=1$). To conclude, there could be a section about different notions of equality for curves.  --Lambiam 14:56, 21 May 2019 (UTC)

There is already an historical section, and I understand your suggestion of a "historical account" as a suggestion for rewriting this section (the first one) for focusing on the evolution of the concept of curve. I would agree with this. It is in this spirit that I have added to the lead the remark that Euclid's definition is not really different from the modern one as the image of a continuous function.

About the various definitions: I have recently rewritten the lead and added the section "topological curve" for clarifying that there are various definitions of a curve, which are strongly related but not equivalent. I have tried to include all common definitions of a curve as a guide for the reader for understanding to which specific article he should go. This can certainly be improved, and I may have forgotten some important kinds of curves, but I think that this is the right starting point, and the remaining problems can be solved by a specific discussion, except for differentiable curves.

For differentiable curves, the problem is deeper: we have, in Wikipedia, at least three non-equivalent definitions. A reader who want to know whether a specific curve is differentiable cannot find the answer in Wikipedia. I have given the example of eight-shaped curves and cusps. You have found another example. So, we desperately need an accurate definition of a differentiable curve. Without that, we cannot have a correct version of the corresponding section of Curve, and of the detailed article Differentiable curve. D.Lazard (talk) 01:36, 22 May 2019 (UTC)

I have not looked into this, but it is possible – in fact, not unlikely – that the literature also contains materially different definitions for the concept of "differentiable curve". For a simple curve, I think the only sane definition is that there is a (unique) tangent for every interior point of the curve. In particular, for a curve defined as the set of points (x, y) that satisfy an equation of the form y = f(x), i.e. the graph of a function, this should boil down to the differentiability of function f provide that we allow the derivative to assume values ±∞. For a general topological curve, sanity similarly requires that we retrieve the definition for a simple curve when applied to a curve that happens to be simple. Parametrizations are an entirely different species; does the literature actually use the terminology of "differentiable curve" for these?  --Lambiam 20:57, 22 May 2019 (UTC)

A few complaints

Hello curve enthusiasts! Here's what I think should change here.

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In the section Curve#Differentiable_arc it is stated that "Arcs of lines are called segments or rays, depending whether they are bounded or not". I think that the whole line should also be considered an arc to itself. I propose that it instead say "Arcs of lines are segments, rays, or lines depending how they are bounded". This issue comes down to whether we require a strict subset relation or not in the definition of arc, and I think we should be explicit about that either way.

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In the section Curve#Length_of_a_curve a formula is written

$\operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\sup \!\left(\left\{\sum _{i=1}^{n}d(\gamma (t_{i}),\gamma (t_{i-1}))~{\Bigg |}~n\in \mathbb {N} ~{\text{and}}~a=t_{0}<t_{1}<\ldots <t_{n}=b\right\}\right).$

I think we should lose the parentheses because it's common to see $\sup A$ where $A$ is some set, and we don't lose any information by doing this. I would prefer

$\operatorname {Length} (\gamma )~{\stackrel {\text{def}}{=}}~\sup \!\left\{\sum _{i=1}^{n}d(\gamma (t_{i}),\gamma (t_{i-1}))~{\Bigg |}~n\in \mathbb {N} ~{\text{and}}~a=t_{0}<t_{1}<\ldots <t_{n}=b\right\}.$

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Within that same section is the formula

${\operatorname {Speed} _{\gamma }}(t)~{\stackrel {\text{def}}{=}}~\limsup _{[a,b]\ni s\to t}{\frac {d(\gamma (s),\gamma (t))}{|s-t|}}.$

I think that $[a,b]\ni s\to t$ is unnecessarily confusing. I would use the \substack LaTeX command like in this article to stack $s\to t$ and $s\in [a,b]$ but that doesn't seem to be supported by Wikipedia. At least I get errors when I try :(. I would even just consider

${\operatorname {Speed} _{\gamma }}(t)~{\stackrel {\text{def}}{=}}~\limsup _{s\to t}{\frac {d(\gamma (s),\gamma (t))}{|s-t|}}.$

as the domain of $\gamma$ has already been specified. FionaLovesCats (talk) 01:03, 22 April 2022 (UTC)

These seem good suggestions. Be WP:BOLD and edit the article yourself. D.Lazard (talk) 10:01, 22 April 2022 (UTC)

Straight lines, functions, bend spaces with curves

I think curved line is very rear to use but curve simply never in mathematics, this is some jargon that is not acceptable. --Mathstrght (talk) 09:04, 4 November 2022 (UTC)

"Curved line" listed at Redirects for discussion

An editor has identified a potential problem with the redirect Curved line and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 November 16#Curved line until a consensus is reached, and readers of this page are welcome to contribute to the discussion. Steel1943 (talk) 17:29, 16 November 2022 (UTC)

Direction

Shouldn't the definiton of curve mention the notion of direction? A continuous unidimensional line which gradually varies direction. JMGN (talk) 16:24, 12 April 2024 (UTC)

If the direction changes, one has a curved line; that is a curve. So this definition is circular. Moreover if the direction is defined, this means that one has a differentiable curve, and there are many curves that are not differentiable. Moreover, if the direction varies gradually, one has a continuously differentiable curve.

In short, the concept of the direction of a curve cannot be defined without having first defined a curve, and, in any case, your definition is much too restrictive. D.Lazard (talk) 16:44, 12 April 2024 (UTC)