Talk:Differential geometry of curves

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Field: Geometry

So far, this page just collects up material from various places. It clearly needs editorial work, and the addition of further topics.

Charles Matthews 08:49, 9 May 2004 (UTC)

Request for picture

It would be nice if someone could add a picture illustrating tangent vector, normal vector, binormal vector and osculating plane for a three dimensional curve. MathMartin 15:32, 21 May 2004 (UTC)

I added an illustration, although it doesn't show the osculating plane. Mark.Howison 21:06, 6 September 2007 (UTC)

Notation

I'm fairly sure the definition given for the Frenet vectors is wrong. E.g, the second vector, the normal, is given as c"/|c"|, which is proportional to the acceleration, and can be tangent to the curve.

However, I'm not completely certain of the correct definition for n dimensions. Could someone check and correct?

Yes they are probably wrong, I am checking the definitions.MathMartin 12:57, 22 May 2004 (UTC)

Also, is there any particular reason for the inner product notation. Writing the formulae as dot products, and suppressing most of the arguments, would make the page look less cluttered, and hence make the essential core of the formulae much clearer. Carandol 21:12, 21 May 2004 (UTC)

I prefer the dot product because it is harder to confuse with scalar multiplication.
The inner product notation is less familiar to most people, and takes up more space.
What do you mean by suppressing most of the arguments ?
Not explicitly stating the dependance on t every time. I think its easier to interpret
$\mathbf{e}_i \cdot \mathbf{e}_i' = |C'|\, \chi_i \quad i
than
$\langle \mathbf{e}_i(t) , \frac{d\mathbf{e}_{i+1}}{dt}(t) \rangle = |\frac{dC}{dt}(t)|\, \chi_i(t) \quad i
because the second formulation contains inessential details. Carandol 05:14, 23 May 2004 (UTC)
I know the page is too cluttered at the moment, but I am still working on the page trying to find a good, coherent way to present the material. I will declutter the page in a few days. What is the deal with using v and v dot in the proof ? The dot notation (for derivatives) is not used on the page and v (as a vector) is not defined. I would like to remove the proof completely as it seems unneccessary. MathMartin 12:57, 22 May 2004 (UTC)
The proof probably is unnecessary. It should be enough to state the result, but I didn't want to change the structure of the page. However, the previous proof was incorrect because it claime the differential of c'/|c'| was c"/|c'|.

If we were stating the proof it would be to use an arbitary vector v, without inconvenient scalar factors to differentiate. Carandol 05:14, 23 May 2004 (UTC)

I would like the page to focus on the main theorem of differential curve theory and the methods used in differential geometry of curves. The important definitions should be on this page too. The more abstract curve stuff on general topological spaces is on the curves page but exactly what definitions belong to what page is still open, there is some unnecessary redundancy. I have to speak with User:Tosha as soon as this page has reached a sufficient maturity.

If you would like some more focus on physics (which seems likely as you speak of speed, acceleration and dot product) I would prefer to put it on a different page, perhaps the examples page, or a usage of differential geometry in physics page.

Not focus on physics, appeal to physical intuition Carandol 05:14, 23 May 2004 (UTC)

In any case it would be nice if you could leave the structure and focus of the page unchanged for a few more days until I am finished. I am of course very grateful for spellchecks and the remark on the frenet vectors.MathMartin 12:57, 22 May 2004 (UTC)

I will put all my replies here, because the deep indentations are too much for my brain :).

I want to keep the following notation

$\langle \mathbf{e}_i(t) , \mathbf{e}_i'(t) \rangle = \|c'\|\, \chi_i(t) \quad i

$\mathbf{e}_i \cdot \mathbf{e}_i' = |C'|\, \chi_i \quad i

The article should deal with differential geometry on an arbitrary Riemannian manifold (I have not made this clear) and therefore I think

$\langle \mathbf{a} , \mathbf{b} \rangle \mbox{ and } \| \mathbf{b} \|$

is more apropriate and less ambiguous than the slightly shorter

$\mathbf{a} \cdot \mathbf{b} \mbox{ and } | \mathbf{b} |$

But you are correct that people used to euclidean space might only know the last notation.

I would prefer to keep the reference to the variable t because the frenet vector ei depends on t. The moving frame part of the frenet frame should be made clear because it is the central concept to differential geometry of curves.

I agree that your notation is simpler and might highlight the essential details better but it assumes the reader knows of the unwritten conventions (like dropping the t which in a different context means ei does not depend on t).

Just as the page currently assumes the reader knows about inner products, which are too large a subject to explain on this page. We could explicitly state we're dropping the t.
The biggest problem with the current format is the double brackets, the <a(t),b(t)>. Even though they're different in shape, it still takes the unpractised eye a perceptible moment to match them up correctly. How about a simple <a,b> with an explicit statement of convention? That would have most of the clarity advantages of my earlier suggestion, but would work equally well in non-Euclidean spaces. Carandol
Ok, you convinced me. The double brackets are bad. So we put a note somewhere (?) in the article that we drop t but we keep the inner product notation.MathMartin 17:23, 23 May 2004 (UTC)
Use the full notation, with (t) once, for the first appropriate equation, then say immediately afterwards something like "To keep the notation simple, rather than explicitly stating the dependance on t every single time, we will simply assume all quantities are functions of t, unless we explicitly state otherwise. This lets us write the last equation more simply as ... " Carandol

I want the article to be accessible to people who do not know the subject so I think it is better to have a more verbose and less ambiguous notation than the other ways round.

But we have to compromise somewhere, or the essential details get lost amidst the the thickets of explanation. Vectors themselves are a compact way of stating lists of coordinates with many implicit properties, assumed without a full explanation. The only question is precisely what level of compromise is appropriate for this topic, and you are putting forth a pretty good case. Carandol

What we could do is make a seperate page for curves in 2 and 3 dimensional euclidean space for people who want to look up certain formulas. The notation on the new page could be briefer and use dot notation.

I think the current Frenet-Seres page does much of that. Perhaps it could be expanded upon, or used as a basis for other pages covering different aspects of this topic in 2 & 3 dimensions at the same level, in compact dot notation. Carandol
I do not like the current Frenet-Serret page. Too much proof and no overall concept or explanation. I would like to see a page about plane and cubic curves together with the frenet serret formulas and the curve theorem. I already tried to talk a bit about the tangent, normal and binormal vectors in my page (as I think those concept are important for many people) but it does not really fit in well. The formulas to calculate those vectors simplify quite a bit in 3-dimensions. We only have to take care the notation stays compatible in some sense.MathMartin 17:23, 23 May 2004 (UTC)
It could do with some improvement, once we establish what it should cover, and a suitable notation. I've written a little on the same subject at Wikibooks, http://wikibooks.org/wiki/Calculus:Vectors#Tangent_and_Normal_Vectors. Those two sections do make more assumptions than appropriate for an encyclopedia, since they are in a book, and they don't cover as much as you're suggesting, because of the target for that book, but still: would that general approach be useful for this encylopedia? Carandol 18:37, 23 May 2004 (UTC)
I do not know :) My point is I am currently studying differential geometry so I want to write about differential geometry. I understand that most people want to do vector calculus in R^3 and do not need the Schmidt-orthogonalization algorithm to construct a Frenet frame. They only need tangent, normal and binormal vector and call this frame TNB Frame. As we are in R^3 we can simplify the equations (for example define B := T x N). When they use wikipedia to look up definition they would probably be confused by my page and even more confused by the curve page. But although I understand the need for a 2,3 dimensional differential geometry of curves page I will probably not contribute to it. So it is completely yours as far as I am concerned.
What I am trying to say is my differential geometry page is probably getting too big so it makes sense to split it into 2 pages. One focussing on the general theory and one on 2,3 space. It may be apropriate to use a slighly different notation on 2,3 space page. But nonetheless the pages (including the curves page) should be compatible in some sense so people can easily look up the more abstract definitions. So I would like to have a curve(topological space) <- differential geometry (Riemannian manifold) <- 2,3 Euclidean space, where the abstraction increases towards the curve page.
The next topic I am going to write on is surfaces in a riemannian manifold.
MathMartin 19:14, 23 May 2004 (UTC)

On a sidenote should the curve be denoted by

$\mathbf{c}(t) \mbox{ or } \mathbf{C}(t) \mbox{ or } \mathbf{\gamma}(t)$

I have seen all three notations in wikipedia but I think it should be uniform.

γ is more abstract, but C makes think of a constant. On balance I'd say γ Carandol
I agree, γ is best. But before we change the notation we should ask User:Tosha if he will change his notation too.

It would be nice to get the opinions of some other people too. MathMartin 14:29, 23 May 2004 (UTC)

Certainly Carandol 16:56, 23 May 2004 (UTC)

${\gamma}$ is the best, the only problem it looks bit ugly in the line (sometimes on some browsers), but hopefully in a few years with MathML everything will be nice. c would be the second choice. The page looks good, I think it is ok to move most of diff.geom subsection from curve, although I would leave basic definition. I think length and jordan curve should stay in curve, but it is ok if you would repeat it here. Infact I would limit this page to smooth case only, it would make it more compact, the abstract definition of length and some other things would not be needed anymore. It would make it simpler and more usefull for phisics. If you want I may try to do that, and you can alwas reverse it back.
Tosha 18:05, 24 May 2004 (UTC)

It makes sense to only use smooth curves on differential geometry of curves page. I think the length definition should remain (only in terms of c'(t) and not the abstract definition) even if duplicated on your page. You can remove the definition for jordan curve from my page. You can do the edits and I will check later.
Is there any convention for using inline symbols (like γ etc.). On my browser (Mozilla) it looks good and most math pages use lots of inline. So I think we should change to γ even if it renders badly on some browsers.
MathMartin 20:35, 24 May 2004 (UTC)

Ok

Tosha 22:15, 24 May 2004 (UTC)


I'd just like to note that on my browser gamma prime ($\mathbf{\gamma}'$)is hard to distinguish from just gamma ($\mathbf{\gamma}$). Its just the way it gets rendered, but someone who doesnt realize it should be a differentiation might not realize it. 207.112.58.65 (talk) 18:40, 27 January 2008 (UTC)

Frenet-Serret formulas

I think this sub-section written much better than the article Frenet-Serret formulas, maybe it is better to move this there? Tosha

I do not know if this is a good idea. The Differential geometry of curves article is already quite long and perhaps the Frenet-Serret formulas deserves a page of its own. But the current page should be heavily edited and the two articles should be made more compatible. I think the long proof should be deleted and some examples could be added (User:Charles Matthews has already added an introduction).MathMartin 22:08, 7 Sep 2004 (UTC)
I just had a look at Frenet-Serret formulas article and I am not in the mood for editing it. But although the article is in bad shape I think the article should stay in a somewhat different form, like an intuitive example (explanation) of the Frenet-Serret formulas in 3 dimensions. I talked with User:Carandol about this some time ago (see thread above). So if you intend to delete most of the page drop him a short note perhaps he is interested in a rewrite.MathMartin 22:25, 7 Sep 2004 (UTC)

I agree with this (the proof should be deleted) but I would also delete whole page Frenet-Serret formulas and move the section from here to Frenet-Serret formulas. so diff geom of curves would get smaller and better and Frenet-Serret formulas will get better. I'll talk to User:Carandol Tosha

Sounds ok, I just think it is a good thing to keep the different notation in Frenet-Serret formulas article (perhaps in the form of an example for R3), so that a reader coming from engineering background or from [Wikibook Calculus] is able to get a intuitive grasp on the content. If User:Carandol is not interested in doing it, I might add the example later.MathMartin 09:16, 8 Sep 2004 (UTC)

Having derived a neat form for radius of curvature, but using only 3D space:

$\frac {\left|\mathbf{c'}\right|^3} {\left|\mathbf{c'}\times\mathbf{c''}\right|}$

where my $\mathbf{c}=\mathbf{c}(t)$ corresponds to your $\gamma(t)$

I was delighted to find that I could derive my formula from your definitions. Seeing that you have put a helpful example of what one of your formulae simplifies to in 3D, viz binormal vactor, may I similarly add my formula? (And I can see that using $\gamma$ is a good idea!) Very helpful page, thank you. I was able to catch up on the whole multi-dimensional (ie >3D) way of doing things. --Ian.patient 09:58, 23 May 2005 (UTC)

Hi Ian and welcome to wikipedia. Generally articles here are written by several people so no one owns a page. But in this case most of the article has been contributed by me so I think I can accept your compliment :). You are free to edit the page and one of the wikipedia maxims is to be bold in your edits so if you think you can improve the page go ahead.
As can be seen from the discussion above there is some need for a discussion of curves in 3 dimensional space, so you might want to consider starting a new article on this topic. Differential geometry of curves is already quite long and I think the main article should mostly deal with the general n-dimensional case. There are also Curves in differential geometry and Frenet-Serret formulas which deal with more concrete examples. MathMartin 12:25, 23 May 2005 (UTC)

Tengent Vector

You should probably start this section by stating that the tangent vector is γ‘(t) (as appears in the top of the DGoC entry), before discussing it's direction (unit vector) and length.

A circle is a curve that is perpendicular to its tangent vector? Maybe I am missing something - but when is a curve EVER perpendicular to its tangent? I think there is something screwy there, needs to be clarified.

No, If you take a strait line, it is paralel to its tangent and not perpendicular!

A circle is perpendicular to its tangent in any point on it and the differential equation it yields gives:

$0=\langle T,\gamma \rangle\Rightarrow 0=\sum_{i=1}^nx_ix'_i\Rightarrow 0=\frac12\sum_{i=1}^n\frac{d}{ds}(x_i)^2\Rightarrow \frac {r^2}2=\frac12\sum_{i=1}^nx_i^2$

for some $r\in\mathbb{R}$ hence

$\sum_{i=1}^nx_i^2=r^2$ which is a circle.

Niv.sarig 21:33, 5 April 2007 (UTC)

Note that it is not the curve that is perpendicular to its tangent (of course not), it is the vector drawn from the origin to the point where you take the tangent. I think this is what the question was about. -- josce —Preceding unsigned comment added by 152.77.24.38 (talk) 12:51, 11 December 2007 (UTC)

References

Started: Added a new section called Additional Reading with first reference (Kreiszig). TomyDuby (talk) 01:12, 25 June 2008 (UTC)

Length and natural parametrization

I have two problems with this section:

1. There is no definition in this article for smooth curve. Elsewhere, smooth function is defined as a function that has derivatives of all orders.

2. Surely a curve γ(t) does not have derivatives of all orders in order to be able to calculate its length. It is enough if it is once differentiable: he integral requires only γ'(t).

TomyDuby (talk) 02:06, 17 June 2008 (UTC)

On the first point: indeed, this article goes through the pains of defining various smoothness classes of parametrized curves, but ends up not using them other than by the way of cluttering the text (this is, however, a common problem in all treatments of the foundations of differential geometry). The word "smooth" is a jargon word that implies sufficient differentiability for the question at hand. I think that it may be best to remove the clutter and stay more intuitive throughout the article, after an explicit warning that the meaning of the term "smooth" depends on the context. As for the general definition of the arc length, note that this section refers to the main article on arc length. A parametrized curve with compact domain of the parameter is rectifiable (i.e. has a well-defined arc length) if and only if the coordinate functions have bounded variation. In the context of differential geometry, which is the scope of this article, one should stick with smooth curves (once continuously differentiable suffices in this case), and then for such curves the arc length is indeed expressible by the given integral. Arcfrk (talk) 05:29, 17 June 2008 (UTC)
Dear Arcfrk, Thanks for your clear explanation. I will use it for updating this section. TomyDuby (talk) 11:47, 17 June 2008 (UTC)

Main theorem of curve theory

It is written in the article: Given n functions

$\chi_i \in C^{n-i}([a,b]) \mbox{, } 1 \leq i \leq n$

with

$\chi_i(t) > 0 \mbox{, } 1 \leq i \leq n-1$

then there exists a unique (up to transformations using the Euclidean group) Cn+1-curve γ which is regular of order n and has the following properties

$\|\gamma'(t)\| = 1 \mbox{ } (t \in [a,b])$
$\chi_i(t) = \frac{ \langle \mathbf{e}_i'(t), \mathbf{e}_{i+1}(t) \rangle}{\| \mathbf{\gamma}'(t) \|}$

where the set

$\mathbf{e}_1(t), \ldots, \mathbf{e}_n(t)$

is the Frenet frame for the curve.

I am wondering: would it be possible to provide a proof or a clear reference for it? I think this is important, all the more so that this is a fundamental theorem. Also, it would be probably interesting to give a pointer to an equivalent (generalization) of the $\chi_i(t)$ functions and this characterization not only for curves, but for surfaces (which I do not know). Maybe the corresponding theorem can be detailed in the article about differential geometry of surfaces. 82.229.252.7 (talk) 16:13, 28 August 2011 (UTC) ambrym

Curve vs image of curve

A curve is defined as a function, and it is clarified that it is different than it's image.

But in the section "Reparametrization and equivalence relation" it says:

"Given the image of a curve one can define several different parameterizations of the curve"

the last word curve in that sentence, to which curve does it refer? it seems to refer to the image of the original curve instead.

Another problem I see is this one, suppose I define the unit circle in the usual way $C = \{ (x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1 \}$

Then C is a set (and thus not a curve), how does the concept of length of C relate to the length of a curve? Or the fact than one would like to call C regular because it has tangent lines, to the regular curve definition of this article. I think this relation (curve/image) should be clarified in some way. Maybe even change the definition of curve to it's image and call path to what is being called curve? (to have a name for the image)