Talk:Steiner conic

The correct statement of Steiners' theorem

This is not the essential statement of Steiner:

Steiner's Theorem: In a pappian projective plane, If U and V are any two distinct points of a conic, and P is a variable point of the conic, the lines PU and PV are projectively, but not perspectively, related.

This is the essential statement made by Steiner:

Given two pencils $B(U),B(V)$ of lines at two points $U,V$ (all lines containing $U$ and $V$ resp.) and a projective but not perspective mapping $\pi$ of $B(U)$ onto $B(V)$. Then the intersection points of corresponding lines form a non-degenerate projective conic section.(figure 1)

See: Jacob Steiner’s Vorlesungen über synthetische Geometrie, B. G. Teubner, Leipzig 1867 (bei Google Books: [1]), p.96 — Preceding unsigned comment added by Ag2gaeh (talk • contribs) 19:24, 3 April 2015 (UTC)

This had been my understanding as well, however, when I started to look for English references all I could find were the statements by Coxeter and Meserve. Since they were in complete agreement, I felt that I had no choice but to write it up that way. I know that Steiner did not write the abstract version given above and that this has to be gleaned from what he did write, but I've looked at seit 96 and I can't see that the statement is supported by what is there (perhaps it is my poor command of German that is at fault). If someone could lead me through the argument I would be very appreciative. Bill Cherowitzo (talk) 22:20, 3 April 2015 (UTC)

Perhaps You looked into the 1. Volume. But the essential statement is on page 96, bottom, of the 2. Volume. The statement You called Steiner's theorem is the easy part. The difficult part, which is due to Steiner is to prove that the point set generated by a projective mapping is a conic section. For a complete proof in English: see Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.(PDF; 891 kB), S. 38.--Ag2gaeh (talk) 07:09, 4 April 2015 (UTC)

I am sorry that you did not understand my post. I was agreeing with you about what should be called Steiner's theorem. However, my opinion (and yours) on the matter is of little consequence. We are editors and not contributors to this encyclopedia. We report on what is in the reliable secondary sources concerning the material and do not espouse our own viewpoints. I found your edit summary in reverting my work to be both baffling and insulting. I clearly stated above that I was doing an editor's job, reporting what was in the reliable secondary sources and not interjecting my personal opinions. You removed citations to established well known sources and replaced them with your citation which by WP:Scholarship is not considered reliable. I see no recourse but to return the page to the way it was (but keeping the new dual conic material). At such time as you (or I) find a reliable secondary source that supports the point of view, that we think is correct, of what should be called Steiner's theorem, I will be happy to help in modifying the page to incorporate that information. We do not, as you have done, simply remove material that we disagree with. Bill Cherowitzo (talk) 21:44, 8 April 2015 (UTC)

I regret that You found my change after the discussion above "baffling and insulting". But may I remember: You changed the sence of Steiner's theorem with no chance for a discussion in advance. So, I think, a third person should decide, which statement is due to Steiner.--Ag2gaeh (talk) 11:22, 9 April 2015 (UTC)

Not the change, just the two line edit summary. Bill Cherowitzo (talk) 18:40, 11 April 2015 (UTC)

See http://www.math.poly.edu/~alvarez/teaching/projective-geometry/chapter_five/node4.html Theorem 4.1.--Ag2gaeh (talk) 07:17, 10 April 2015 (UTC)

I would not consider this a reliable source either. Bill Cherowitzo (talk) 18:40, 11 April 2015 (UTC)

Further remarks:

The book "Jacob Steiner's Vorlesungen,..." was written by H. Schröter, which, I think, is a reliable secondary source.

Question: Who, if not Steiner, proved that the projective generation of a conic works ?

No one is arguing that the results are not due to Steiner. The question is, what does the mathematical literature refer to when they use the expression "Steiner's Theorem"? Steiner did not use the term, nor did Herr Schröter ... so referring to either of them does not settle the issue. Other mathematicians applied the expression to some result of Steiner and we need to find out which result is being referred to in the existing literature. Bill Cherowitzo (talk) 18:40, 11 April 2015 (UTC)

What some people call "Steiner theorem" may be called "Steiner property of a conic". The essential and difficult part is the proof that the projective generation works.--Ag2gaeh (talk) 07:09, 11 April 2015 (UTC)

I agree. As a mathematical writer, I would use the expression just that way, but as a Wikipedia editor I am not free to call things as I would like them to be called ... I must report on what other people have said. Bill Cherowitzo (talk) 18:40, 11 April 2015 (UTC)

The second part of the book "Jacob Steiner's Vorlesungen,..." starts at page 225 of the Google-PDF-file. The essential statement (p. 96 of the book) can be found on PDF-page 339. --Ag2gaeh (talk) 09:20, 11 April 2015 (UTC)

Thanks for that. The web page had some glitches which hid the second part of the book. Bill Cherowitzo (talk) 18:40, 11 April 2015 (UTC)

Some more references:

H. Lüneburg: Die euklidische Ebene und ihre Verwandten, p. 104: "Die nächsten beiden Sätze, die beide von J. Steiner stammen und gemeinsam unter dem Schlagwort Steiner'sche Erzeugung der Kegelschnitte rangieren, ..." = "The next two theorems are due to J. Steiner and are known as The Steiner generation of conics... ".

H. Lenz: Vorlesungen über projektive Geometrie, Akad. Verl. Leipzig, 1965, S.56.

W. Blaschke:Projektive Geometrie, p. 57.

B.M. Saler (1984):Synthetic Geometry, p.136 (The Steiner definition of a conic).--Ag2gaeh (talk) 17:03, 12 April 2015 (UTC)

Perhaps the title should be changed to Steiner generation of conics (s. lead of the original version) ?--Ag2gaeh (talk) 08:33, 12 April 2015 (UTC)

The intention of the original article was, to describe the special construction/generation of a conic due to Steiner. I called it, as in the German version " Steiner's theorem or Steiner generation of a conic ". You deleted the second, more pecise, part. In Coxeter's book on page 82, the essential statement/theorem due to Steiner is called Steiner's construction, which You do not mention. Your not so important "Steiner's theorem" was not the subject of the article. You are free to add material. But it would be nice, if You kept the essential statement. What You call (rather neutral) the projective generation of conics is called in German books unanimously "Steiner-Erzeugung der Kegelschnitte" and in Coxeter's book "Steiner's construction". I introduced the shorter titles Satz von Steiner (Geometrie) and Steiner's theorem (geometry). If You think the title is misleading we should discuss a new one.--Ag2gaeh (talk) 13:28, 12 April 2015 (UTC)

I agree, a change of the title (and the corresponding reorganization of the page) would deal with this issue. I have a slight preference for "Projective generation of a conic" as that seems to be the term most often found in the English literature when the results are mentioned (for some reason that I don't understand, these results do not often get attributed to anyone, unlike in the German literature). Let me propose a compromise; title the article "Steiner's projective generation of a conic" and create redirects to it from "Steiner's generation of a conic" and from "Projective generation of a conic". My reasoning is that I would like readers to be able to find the article without having to know beforehand that Steiner is the author of the results. Bill Cherowitzo (talk) 18:38, 12 April 2015 (UTC)

Compromises are good, in general. But "Steiner's projective generation of a conic" seems to me a bad compromise. In German we would say, it is like "ein weisser Schimmel". A Schimmel is already a white horse, so there is no need for the word "weiss" (white). I think the new title should be either "Steiner's generation of a conic" or "Projective generation of a conic". Perhaps some other geometers should help to decide. In the English literature I found, the special generation was always linked to Steiner. But English speaking people should decide this apparently difficult case. --Ag2gaeh (talk) 20:44, 12 April 2015 (UTC)

If You agree, I generate a new article Steiner conic (s. Coxeter 1993, p. 80) with the content of my sandbox. The article Steiner's theorem should be deleted.--Ag2gaeh (talk) 07:48, 18 April 2015 (UTC)

That would be ok by me. I would most likely edit the new article again to make the English a little more colloquial (minor editing) and point out that you need to say something about characteristic two in the dual conic section since the tangent lines of a non-degenerate conic form a degenerate dual conic in that case. Bill Cherowitzo (talk) 19:09, 19 April 2015 (UTC)

I added remarks for Ch=2 case to my sandbox content and moved it to Steiner conic. Please improve it. I redirected Steiner's theorem to Steiner conic. But I think, it should be deleted.--Ag2gaeh (talk) 15:37, 20 April 2015 (UTC)

Just a remark to char=2: In general not any line through the knot is a tangent (s. Hartmann, p. 25). So, the dual of a conic is i.g. only a subset of points of a dual line.--Ag2gaeh (talk) 20:54, 20 April 2015 (UTC)

Oops. Sorry, I was thinking of the finite case. I'll fix that. Bill Cherowitzo (talk) 01:55, 21 April 2015 (UTC)

wrong procedure

Sorry, for doing the wrong procedure. The talk page of Steiner's theorem should be moved to this talk page. I ask for help. --Ag2gaeh (talk) 15:51, 20 April 2015 (UTC)

It's fine, I can move the page for you if you want as soon as this page is deleted. Pishcal — ♣ 15:59, 20 April 2015 (UTC)

Image

{{Mathematical diagram requested}}

Steiner conic is illustrated in Halsted's book:

• Synthetic Projective Geometry, page 20.

A similar image is requested to enhance this article. Rgdboer (talk) 21:25, 21 May 2020 (UTC)

Done. --Ag2gaeh (talk) 07:26, 23 May 2020 (UTC)

The clear illustration is given as Example 1: Projectivity given by two perspectivities. Thanks to Ag2gaeh for this image. Rgdboer (talk) 22:58, 25 May 2020 (UTC)

Notation

The projectivity $x\barwedge X$ is the composition of two perspectivities:

$x\ {\overset {A}{\doublebarwedge }}\ b\ {\overset {B}{\doublebarwedge }}\ X$ where b is a point on the line at the right of the diagram.

For an expression of the Steiner conic one then has

$\bigcup \{xX\cdot yY:x\barwedge X\ \ \land \ \ y\barwedge Y\}.$

Articles on projective geometry can be enhanced by use of standard symbols for perspectivity and projectivity. — Rgdboer (talk) 20:58, 21 June 2020 (UTC) +correction — Rgdboer (talk) 21:02, 21 June 2020 (UTC)

I deliberately omitted such formal notations and think an image and some common words are more informative to people, who are not familiar to such kind of language. Such symbols may be convenient for lengthy statements or proofs.--Ag2gaeh (talk) 07:27, 22 June 2020 (UTC)

I support Ag2gaeh's reasoning for not using these symbols. While useful in specialized publications, they do not enhance the readability of an article for someone who is not already familiar with the topic. There are enough complaints about the impenetrability of math articles, we do not need to compound the problem.--Bill Cherowitzo (talk) 17:26, 22 June 2020 (UTC)