Talk:Osculating circle

message: merge: see Talk:Contact (mathematics) --W!B: 03:32, 27 March 2006 (UTC)
replies: text: In that article, "osculating" seems to mean having three points of intersection.  That is not what it means in this article.  I was under the impression that this usage is standard in differential geometry. Michael Hardy 22:17, 27 March 2006 (UTC)
thread_title: Merge

message: When reading
: Osculate literally means to kiss; the term is used because osculation is a more gentle form of contact than simple tangency.
I am not sure the word "gentle" is the best.  To be fair, I am having trouble coming up with a good improvement.  "accurate", "tight", "precise", "lingering", "following", "parallel", "consistent", "refined", none seem to better capture the concept here.  But I don't like "gentle" either.  
Maybe someone with a moment of inspiration Roget could be proud of can come through and help out.  Baccyak4H 18:10, 7 September 2006 (UTC)
replies: text: "Gentle" makes sense because if you go from a tangent line to the curve, your acceleration abruptly changes, whwereas if you go from the osculating circle to the curve, it does not. Michael Hardy 18:32, 7 September 2006 (UTC)
thread_title: Minor wording question

message: In one of the figures, this expression is used, although it is never defined nor described in the article.  My best guess is that it means the circle shares one additional order of derivative with the curve than a typical osculating circle.  I would think this clarification should be pursued in the article in lieu of the caption being altered to avoid this reference.  Thoughts? Baccyak4H (Yak!) 20:31, 10 September 2007 (UTC)
thread_title:  Four-point contact? 

message: In the section 'Mathematical description', the curve is first called gamma and later C.  —Preceding unsigned comment added by 212.201.44.249 (talk) 09:01, 6 October 2008 (UTC)
thread_title:  Variable names 

message: Are there generalizations to higher dimensions?
If yes maybe someone can add them.  —Preceding unsigned comment added by 160.45.42.253 (talk) 14:06, 2 December 2010 (UTC)
thread_title:  higher dimensions 

message: The defined normal vector is in contradiction (sometimes it's opposed) to the one in the Wikipedia page "Differential geometry of curves", also in contradiction to the one used (at least) by we the physicists, and also in contradiction to the one that appears in the Frenet frame (intrinsic coordinates). In the Wikipedia "Differential geom..." a curve is plotted by a vector function, let's call it r(t) to simplify (where r is a two-dimensional cartesian vector), then the tangent unit vector is T(t) = r'(t) / |r'(t)| (where | | is length of a vector), which can also be called velocity unit vector, V(t) = T(t), and finally the normal unit vector N(t) can be defined by a Gram-Schmidt orthogonalization if the acceleration r"(t) is not perfectly perpendicular to the velocity r'(t), but this reduces to using the normal acceleration, simply N(t) = normal(r"(t)) / |normal(r"(t))| = <normal acceleration> / <length of normal acceleration>. So the normal unit vector, like the normal acceleration, is always centripetal (unlike the yellow vector in the Lissajous picture), according to this page of differential geometry and according to many books and according to what physicists agree and according to the Frenet frame. Grausvictor (talk) 19:42, 30 January 2017 (UTC)
thread_title: About the [Dubious-Discuss] in the definition of the Normal vector and the yellow vector in the Lissajous picture

message: I've been bouncing back and forth between these pages all day. I want to point out that the definition of Curvature as given on the Curvature page relies on the osculating circle. However, that page doesn't develop that concept and instead relies on what it calls equivalent equations.  Meanwhile, this page uses the concepts from the page Curvature to define the osculating circle.
I know that Wikipedia is not a textbook and is designed for working mathematicians and such, but this is enough a pedagogical inconsistency that it should be corrected. I would naively suggest that some geometric or analytic method (I was suspecting an analytic method) be used on this page.
Fantasticawesome (talk) 03:17, 2 June 2020 (UTC)
thread_title:  Please consider contributing an analytic definition