# User talk:Abel Cavaşi

- СПУТНИКССС Р 22:04, 20 February 2006 (UTC)

Welcome!

Hello, Abel Cavaşi, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers:

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## On minor edits

Hello, and welcome to Wikipedia! Please note that a number of your recent edits have been reverted because they were mathematically incorrect, and moreover labeled as minor edits. Please see Help:Minor_edit for details about what a minor edit actually is. Cheers, Silly rabbit (talk) 05:46, 25 December 2007 (UTC)

The helix, at least as it is conventionally understood, has constant curvature and torsion. It is not merely the ratio which is constant. An easy way to see this is to realize that a doubly infinite helix is acted upon transitively by a 1-parameter family of Euclidean transformations (the corkscrew motion which carries the Euclidean space along the helix). Since curvature and torsion are Euclidean invariants of the curve they must therefore be constant, since we can get from any one point to any other by means of a corkscrew transformation. Silly rabbit (talk) 14:05, 25 December 2007 (UTC)
Indeed, if κ and τ are constant, then obviously their ratio is constant. I did not know about Lancret's theorem. Still, the articles were correct as stated before, with the stronger statement, that the curvature and torsion are constant. Silly rabbit (talk) 18:55, 25 December 2007 (UTC)
Lancret's theorem (apparently) holds for helices of variable radius. If you check the article Helix, and Frenet-Serret formulas, the helices under consideration have constant radius. Silly rabbit (talk) 19:10, 25 December 2007 (UTC)
Yes, it is necessary and sufficient that the curvature and torsion be constant for a space curve to be a helix. Period. End of discussion. If you believe otherwise, then there is nothing more to say. There may be a theorem which states something else, but that would need to be checked. Anyway, I do know what a helix is, and you appear to be inserting misleading information into articles without providing references. I call you on it, and now you have the audacity to insult me. Please note that I wrote most of the current article on the Frenet-Serret formulas, and I think I have a fairly good reputation on Wikipedia as an expert in geometry. Silly rabbit (talk) 01:38, 26 December 2007 (UTC)

I'm saying that I don't believe Lancret's theorem as stated. The mathworld reference is not enough, since they seem to provide no further references on the theorem. Furthermore, the theorem cannot possibly be true as stated, by the fundamental theorem of space curves. For any pair of functions κ>0 and τ there is a curve with curvature κ and torsion τ. Any two curves with the same curvature and torsion are congruent under a proper Euclidean motion. Now, consider the pair of function κ = et, τ = et. There is a curve with curvature κ and torsion τ. Lancret's theorem asserts that the curve is a helix. However, it cannot possibly be by the argument I have already given. The curvature and torsion of a helix are constant because they must remain invariant under a one-parameter family of Euclidean transformations preserving the curve: the corkscrew motion. Now, why don't you go and find another reference about Lancret's theorem. I suspect it may involve affine invariants of helices, and so be out of place in an article such as Frenet-Serret formulas which deals with the Euclidean invariants.

Now, I am puzzled that you would say that the torsion and curvature of a helix are not constant. This is a well known result to be found in essentially any calculus book. See also the second volume of Spivak's Comprehensive Introduction to Differential Geometry. The converse, that any curve with constant curvature and torsion is a helix, is also a standard fact and there are various ways of seeing it. For instance, given a constant κ and τ, there is a helix with these as the curvature and torsion; specifically the parametric curve

${\displaystyle {\frac {1}{\kappa ^{2}+\tau ^{2}}}\langle \kappa \cos t,\kappa \sin t,\tau t\rangle .}$

So, I have now provided a more-or-less complete proof that a curve is a helix if and only if it has constant curvature and torsion. I have also given you very thorough reasons for rejecting Lancret's theorem. If you believe that I am still mistaken, then please provide some corroborating references. Otherwise, I don't have anything further to say about it. Silly rabbit (talk) 16:06, 26 December 2007 (UTC)

The definition of helix you are using does not agree with the one given in the Wikipedia article Helix. This is what I meant when I said that on Wikipedia, helices are of constant radius. I note that the Mathworld definition agrees with Wikipedia's, as does that in most standard books. Of course, if you use a different definition, then you are going to have different results. But you do need to be forthcoming about what you mean. Silly rabbit (talk) 21:36, 26 December 2007 (UTC)

## AfD nomination of Overacceleration

An editor has nominated Overacceleration, an article on which you have worked or that you created, for deletion. We appreciate your contributions, but the nominator doesn't believe that the article satisfies Wikipedia's criteria for inclusion and has explained why in his/her nomination (see also "What Wikipedia is not").