Talk:Osculating circle

Merge

merge: see Talk:Contact (mathematics) --W!B: 03:32, 27 March 2006 (UTC)

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)

Minor wording question

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)

"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)

I agree it makes sense that way. I only suspect there is a much more precise word, a better one. But as I cannot come up with it, it shall stand. If I come up with any ideas, I'll post here first. I encourage others to do the same. Baccyak4H 03:48, 8 September 2006 (UTC)

Four-point contact?

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)

Variable names

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)

higher dimensions

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)

About the [Dubious-Discuss] in the definition of the Normal vector and the yellow vector in the Lissajous picture

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)

The examples given here seem to follow standard mathematical conventions, as is in any introductory multivariable calculus text. The only dubious thing is that the formula given for a plane normal vector is incorrect if the curve changes concavity.Hamptonio (talk) 12:54, 1 September 2017 (UTC)

The section starts by defining γ(s) as a regular parametric plane curve, i.e. a smooth curve in a Euclidean plane. For these curves, there is nothing dubious about the marked formula as the normal is, by definition, one of two possible unit vectors perpendicular to the tangent (see https://en.wikipedia.org/wiki/Normal_(geometry)). As such, the tangent vector T(s) = γ'(s) = (x'1,x'2) has two corresponding unit normals with, again by definition, the formula N(s) = ±(-x2',x1')/|T(s)| = ±(-x2',x1')/|γ'(s)|. And by convention when mathematical texts refer to "the normal", they refer to the positive solution for N(s). The formula as presented is entirely correct. Pomax (talk) 7:20, 25 November 2018 (UTC)

Please consider contributing an analytic definition

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)