Talk:Differentiable curve

message: 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)

message: 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)
replies: text: I added an illustration, although it doesn't show the osculating plane. Mark.Howison 21:06, 6 September 2007 (UTC)
thread_title:  Request for picture 

message: 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)
replies: text: 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)
thread_title: Answer for the last question

message: Can someone please add some references to this article?
Thanks! Liransh 19:34, 6 July 2007 (UTC)
replies: text: Started: Added a new section called Additional Reading with first reference (Kreiszig). TomyDuby (talk) 01:12, 25 June 2008 (UTC)
thread_title:  References 

message: 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)
replies: text: 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)
replies: text:  Dear Arcfrk, Thanks for your clear explanation. I will use it for updating this section. TomyDuby (talk) 11:47, 17 June 2008 (UTC)
thread_title:  Length and natural parametrization  

message: 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)
thread_title:  Main theorem of curve theory 

message: This translation from Ukrainian WP appears to contain significant new content, but as I'm not a mathematician I'm not best placed to help. Opinions from actual mathematicians please? OnionRing (talk) 20:06, 26 May 2016 (UTC)
replies: text: Significant new content, really? Where? Boris Tsirelson (talk) 11:01, 27 May 2016 (UTC)
replies: text: Section "The curvature of the curve"? I had nominated it for speedy db-a10, but then wondered if I was missing some content that could usefully be merged in. If not, then please feel free to speedy it a10 again. Thanks, OnionRing (talk) 11:21, 27 May 2016 (UTC)
replies: text: We have curvature already, which has been forked off of this article, and includes everything that is there (and more).  I have WP:PRODed the new article.  Sławomir Biały (talk) 11:38, 27 May 2016 (UTC)
thread_title:  Proposed merge from [[Користувач:Перевод]] 

message: The article was marked for copy editing based on excessive use of first-person pronouns (I). I believe this is incorrect as the usage of I is in notation and that this call for copy editing is not necessary.  — Preceding unsigned comment added by Ibatts19 (talk • contribs) 16:57, 24 February 2019 (UTC)
thread_title:  Marked for copy editing 

message: I have open a discussion in Talk:Curve#Sections on differentiable curves, which may concern watchers of this article. D.Lazard (talk) 16:43, 1 May 2019 (UTC)
thread_title:  Sections on differentiable curves in [[Curve]]