Talk:Ellipse/Archive 1

Ellipse

Current talk page

1

Probably this picture could be used: [1] (Fig.207), see [2]. Boris Tsirelson 05:12, 2 May 2010 (UTC)

I added two pictures from that source. Incidentally, feel free to change/adapt/add stuff, CZ is a wiki, you know. --Paul Wormer 12:08, 2 May 2010 (UTC)

trammel, figs.

1. I have doubts about fig.2; it seems to me that the central line should overlap with the red and the green ellipsi, until their centers.
2. I like the deduction of the Trammel. Who was Trammel? What century, what country did he/she live?
3. How about to borrow the animation of the trammel device grom wikipedia; I mean, http://en.wikipedia.org/wiki/File:Bsgrinder.gif ?
4. As for the drawing with a string... I used the loop to draw ellipsi; then the whole ellipse can be drawn at once. (The lenth of the string should be extended with a piece of length equal to the distance betwin the focuses.)
Dmitrii Kouznetsov 18:30, 2 May 2010 (UTC)

Dmitri, I'll try to follow your advice on fig. 2 later today. I made it with Autocad and have no code. If you can make a better picture I would appreciate it. Look in an English dictionary for trammel, it is not a person. (I didn't know the word either). We are not allowed to use animated gifs, or using stuff from Wikipedia. Material from WP is only allowed if the real name of the author is known and we have written permission. With regard to the loop: I followed the description and the picture in the book. Boris suggested that I include the picture and I wrote the text to match it.

I have a question to the mathematicians: two things are missing. The quadric Ax²+Bxy+Cy²+Dx+Ey+F=0 and the definition by means of directrices (vertical lines at fixed distance from foci). Is that bad?--Paul Wormer 06:13, 3 May 2010 (UTC)

Ah, now I can answer, since Peter gave appropriate terms. I believe, these are medial gaps! (That is, less than major gaps, but larger than minor gaps.) But I must admit, this classification is not uncontroversial. Boris Tsirelson 14:48, 3 May 2010 (UTC)

A trammel point is also a woodworking guide that attaches to a router (tool) to cut elliptical holes. It's actually a little misnamed, as it physically consists of several pieces, such as an anchor point, guide arm, and attachment to the cutting tool base. Howard C. Berkowitz 14:54, 3 May 2010 (UTC)

Another way to construct an ellipse is illustrated here. --Daniel Mietchen 19:38, 3 May 2010 (UTC)

What a pity that it contains the two irrelevant spheres; the cone and the plane would be nice without them. Boris Tsirelson 19:48, 3 May 2010 (UTC)

I don't think the spheres are irrelevant, since they define the focal points. --Daniel Mietchen 20:07, 3 May 2010 (UTC)

The picture does not show this. It rather looks as if the two spheres were touching (and the plane cuts them?) The connection between the spheres and the foci would be better shown in a two-dimensional projection. (Eventually there should be conic and second order curve for a general discussion.)

The general quadric is not needed here because this is about geometrical properties (perhaps? the equation with shifted center?).

What I rather miss is the ellipse as affine transformation of a circle (x,y) -> (x,ky).

Peter Schmitt 22:22, 3 May 2010 (UTC)

On "gaps": major, medium, minor -- that is, of course, partially a matter of taste (and can be argued), just as "useful" is. Only "mistakes" should be non controversial, at least in mathematics. --Peter Schmitt 22:53, 3 May 2010 (UTC)

(unindent) dynamic figures
Paul, do you know if we can use java applets on CZ pages? Or should I ask in the forum? --Peter Schmitt 22:26, 3 May 2010 (UTC)

Current state is no, unfortunately. --Daniel Mietchen 23:01, 3 May 2010 (UTC)

Approvals etc

Let me express my opinion, probably quite controversial.

0. It is not the question, "is it bad?"; it is THE question, "should the article be approved?".

1. The approval mechanism is THE feature of CZ. We should not dream of Google juice when our articles are "unapproved, subject to disclaimer, not to be cited".

2. Unfortunately, in order to approve advanced math articles we need many (20..100) active math editors (then it will be reasonably probable to find at least two editors competent in the favorite matter of an author).

3. Fortunately, in order to approve undergraduate math articles it is enough to have just two active math editors, provided that... see (6) below.

4. Two necessary conditions for approval: (a) not misleading; in math context it just means, no errors; and (b) useful.

5. Desirable but NOT NECESSARY, and in fact not reachable: unimprovable. It is always possible to add something, or make a small improvement. "Useful" does not mean "as useful as at all possible". I understand that in a political context, to miss some aspect may be an intolerable bias. But in math context this is not an issue. Some aspect is missing? Well, work on it AFTER approval, if you can and want.

6. Thus, I call math editors to strive to approve articles (satisfying the two necessary conditions), not to find a reason to delay the approval.

A1. Regretfully, today we have at most two active math editors: Peter Schmitt‎ and Dmitrii Kouznetsov. (I would be happy to be wrong in this point.) I've asked both about possible approval of "Ellipse". One did not reply (yet), the other made some remarks.

A2. I can apply for the editor status, if I'll feel that this will help. That is, if at least one existing editor will support my attitude expressed above.

Boris Tsirelson 11:15, 3 May 2010 (UTC)

By the way (off-topic), my attitude to refereeing journal articles is similar: a referee should point out errors and minor, evident improvements. However, a substantial improvement is a business of authors. The referee can (if needed) write a subsequent article later (being already an author, not a referee). Boris Tsirelson 11:26, 3 May 2010 (UTC)

Boris, I'm somewhat puzzled by the fact that your general comments are on this particular page. The forum would be a better place for your thoughts.

I gave this article level 2, which means that it is (IMHO) not yet ready for approval. Further, I did not know that you're not a math editor, my question to the math editors was also directed to you. Do I understand from your above comments that in your opinion the directrix and quadric don't need to be added to this article?

Finally, one editor is enough to approve an article, so if you become an editor you can approve any math article you want.

--Paul Wormer 12:19, 3 May 2010 (UTC)

Really, by one editor?! Did I miss a major change of the rules? Or did I misunderstood the rules from the beginning? Boris Tsirelson 14:33, 3 May 2010 (UTC)

Why here and not on a forum? Because we have no math forum (WP has, by the way). And the "general" forum is too verbose for a mathematician... Sorry Paul, here by "mathematician" I mean everyone contributing to math articles. Boris Tsirelson 15:11, 3 May 2010 (UTC)

In principle, there is a math forum — it's just rarely been used (like all the other Workgroup-specific forums), and as far as I can see, it does not render TeX. --Daniel Mietchen 15:31, 3 May 2010 (UTC)

About level 2: not only me, also Peter considers it as basically ready. But of course, if you want to work on it further, I do not want to disturb you with a premature approval. Boris Tsirelson 15:17, 3 May 2010 (UTC)

By the way (off-topic), my attitude to refereeing journal articles is similar: a referee should point out errors and minor, evident improvements. However, a substantial improvement is a business of authors. The referee can (if needed) write a subsequent article later (being already an author, not a referee). Boris Tsirelson 11:26, 3 May 2010 (UTC)

(edit conflict: written parallel to Paul's comment)

On the whole, I agree with you, Boris. With (4) I would also include: (c) and there are no MAJOR gaps (meaning: all top "level topics" are at least mentioned; drastic example: prime number without: there are infinitely many, and factorization of natural numbers). Unless you include this already with (b), usefulness.

As for your item (5), I have to admit that I thought about some additions to Paul's text (instead of thinking about approval), but when I read your message this night (even before reading this), it occured to me that, in this case, it might be better to approve first and then make changes.

I, and -- if I remember it correctly, Paul too -- are not so much concerned with approvals, one reason being that (re)approval of approved but extended aricles is not easy to do under the current rules with a lack of non-involved editors.

Another difficulty is that even minor changes (clarifications of wordings, etc.) are likely to be classified as "adding content" instead of mere "copy editing". (If you want to see an example for this, then you will find it on the talk page of Complex number. I hope that these rules will be revised some time in the future.) Having three editors would change this radically because then (even) all three may have contributed content AND were allowed to approve the article.

Peter Schmitt 12:45, 3 May 2010 (UTC)

Peter, I am glad to see that our views are rather close. About (c), I agree. About a number of inconveniences caused by approval, I believe that we should be patient to them, since, I believe, approval is very important for survival of CZ. Boris Tsirelson 14:40, 3 May 2010 (UTC)

Till now I felt that my idea of approval is too different from that of CZ; thus I was reluctant. But now I feel I should apply for editor status (and then we'll be three math editors, wow). Boris Tsirelson 15:22, 3 May 2010 (UTC)

Let's wait for Dmitrii with approving. Maybe he knows how to draw nice conic intersections, and has some more comments.--Paul Wormer 16:23, 3 May 2010 (UTC)

Of course. And by the way, I am now in the list of (active) math editors. Boris Tsirelson 18:36, 3 May 2010 (UTC)

Boris, I have registered only a few weeks earlier than you. I, too, do not really know what the "CZ idea of approval" is. I think that this is still up to evolution, and will certainly be a much discussed topic (together with the approval process) when, finally, a Charter will be ratified ...

My personal idea is (briefly): "a correct and reasonably complete survey/summary of the topic, well structured and reasonably well written. For "developed" it is essentially the same: correctness, of course, but I would accept some more omissions, and more flaws in the presentation.

Welcome, and thank you for joining as Editor. --Peter Schmitt 22:43, 3 May 2010 (UTC)

Boris, and other Math editors:

You noticed that you had been added to the list of approved math editors' even before I got a chance to send you a note advising you of that decision. Your request came to me from the Constabulary earlier today, and after I reviewed it the case seems pretty clear-cut. Happy editing! And I join you in hoping this will result in increased approvals of articles from this workgroup! The procedure for approving Editors from among existing active Authors is pretty informal, so if those of you who are currently editors wish to recommend others, please contact me or any of the other EPA's with your recommendations.

Roger Lohmann 19:54, 3 May 2010 (UTC), Editorial Personnel Administrator

Thank you. Boris Tsirelson 04:19, 4 May 2010 (UTC)

Affine transformation

"What I rather miss is the ellipse as affine transformation of a circle (x,y) -> (x,ky)". Peter, remember the Wiki idea. Why don't you add it to the present article? I don't see appear soon two (algebraic and geometric) articles on the ellipse, so, for the time being, all pieces of info should go in this article, including the y scaling.

In my opinion the algebraic equation (before rotation and translation) is pretty important, too. It so happens that I read a lecture by Weyl the other day in which he stated: "As you know the ellipse is ax² + 2bxy + cy² = 1 with a > 0, b > 0, ac−b² >0". Anybody reading this should be able to find it in Citizendium, because that's what an encyclopedia is for.

Boris, congratulations, quite an honor ;-). Yes one editor alone can approve an article, see, e.g., Macromolecular chemistry.

--Paul Wormer 08:12, 4 May 2010 (UTC)

Thank you for the information, it helps. Boris Tsirelson 12:32, 4 May 2010 (UTC)

For current rules on approval see [[CZ:Approval process#Who may approve: For current rules on approval see CZ:Approval process#Who may approve

Paul, you need not remind me that I may edit the article. It was not a request that you do it, but only in answer to your question.

One reason to refrain from editing can be the intention to save the right to approve an article (now less imporatant because there is a second participating editor).

Another reason may be the attempt to preserve a uniform style because additions by another author may make the page less coherent.

In any case, I'll do a copyedit of the article (soon). --Peter Schmitt 12:50, 4 May 2010 (UTC)

English

English is difficult. Peter changed "reflection in a line" to "reflection about a line". My first reflex was that that couldn't be right, but it is. I checked a book on group theory by an American author and found that he writes "reflection about a line" and "reflection in a plane". Prepositions, they are a nightmare. --Paul Wormer 08:30, 7 May 2010 (UTC)

PS Peter, did you also check the math? Errors in the math are more disturbing than errors in the language.--Paul Wormer 09:09, 7 May 2010 (UTC)

I have not yet finished. Yes, of course, a mathematical error would be much more disturbing. Most of the changes were not meant to correct your English but as an attempt to express some point more clearly. Following Boris' suggestion I want to leave the article essentially as it is now, and do not want to make major changes where I would, perhaps, use a different approach. --Peter Schmitt 21:29, 7 May 2010 (UTC)

I knew "reflection about" before, but now I saw a book where "reflection in" is used. Don't know what is "better" or more "common". --Peter Schmitt 12:22, 12 May 2010 (UTC)

"in" seems much more natural to me. Peter Jackson 14:39, 12 May 2010 (UTC)

I'd definitely use "about a line" and "in a plane". To me, reflection "in a line" reads as swapping the points end-to-end about some unmoved center point. I'm an English teacher & native speaker, know a bit about some areas of math but haven't looked at geometry since high school decades ago. Sandy Harris 03:45, 15 May 2010 (UTC)

This does get messy. Thinking further, I find that going up a dimension I'd want to say reflection "in 3-space " "through" a plane. I do not claim this is correct mathematical terminology, just what I'd say.

Is there some more formal usage among mathematicians, reflection "relative to a line" or some such? If so, should that be introduced here? Sandy Harris 06:47, 15 May 2010 (UTC)

Thank you, Sandy. I'll leave "about" then (I know it from an author who cares about (American) language). Space is not needed here, but I'll remember your remarks. At the moment I cannot think of a "relative to" (but there could be a use for it). "Reflection" is a common term, and the use of the preposition is derived from common usage, I think.

On a similar vein: Would you "translate over a vector"? --Peter Schmitt 08:56, 15 May 2010 (UTC)

I'd translate by a distance. Once vectors get involved, I would not be sure what to say and would need to ask a mathematician. Sandy Harris 11:43, 16 May 2010 (UTC)

Quadratic equation

So far, the proof only shows that the conditions are sufficient, while the statement of the conditions for an ellipse is ambiguous and does not make clear whether they are meant to be necessary as well. --Peter Schmitt 12:33, 8 May 2010 (UTC)

Please go ahead, make the conditions necessary and sufficient and remove ambiguities. I conceived of this section myself, without an example. Not being a mathematician I can believe that its formulation can be improved greatly.--Paul Wormer 12:55, 8 May 2010 (UTC)

Paul, as I already said, my approach to the whole topic would be entirely different. Thus I restrict myself to copyediting the article, and staying close to your style. Your approach is probably right for readers who have a similar relation to mathematics as you (and your notation following practice in physics). They might not like my way of telling the story.

I changed your bb symbols to bf because these are almost exclusively used for some standard sets and objects, and if you write vectors in bold, then it is only natural to use bold for matrices.

And I replaced the equiv-symbol because it usually is used to show equivalence with respect to some equivalence relation (and not for definitions or equality.) I think it isn't used in physics, either. In mathematics, if definitions are indicated at all, often either ${\stackrel {\rm {def}}{=}}$ or := is used. The latter has the advantage that it can be used in two directions: := or =:

--Peter Schmitt 23:13, 8 May 2010 (UTC)

Toward Approval

This version of this article has been nominated for single editor approval by User:Peter Schmitt. I am concerned that some of these edits are content related edits. I would appreciate some more expert input to help me make that determination before I consider locking this article using the single editor approval. Of course, if two more editors endorse the article, the concern would become moot. D. Matt Innis 04:24, 10 July 2010 (UTC)

While (my opinion) all my edits should be considered as copyedits I have to leave it to you (or others) to judge whether they are acceptable under the current rules. But since I am the only editor who has made any edits, at most one additional editor (Boris) is needed, not two. --Peter Schmitt 10:28, 10 July 2010 (UTC)

Absolutely. D. Matt Innis 12:18, 10 July 2010 (UTC)

About the diff:

• "Use ... and one obtains" --> "Using ... one obtains" : no change of the meaning;
• $\equiv$ --> $:=$ (and two similar cases): no change of the meaning, just a better notation;
• 𝔸 --> Q (in formulas): no change of the meaning, just a better notation for the same matrix;
• "it was used that" --> "this uses" : no change of the meaning;
• $\mathbb {R}$ --> $\mathbf {R}$ : no change of the meaning, just a better notation;
• "blue line" --> "blue-red line" : no change of the meaning, just more careful.

Boris Tsirelson 17:50, 10 July 2010 (UTC)

Thanks, Boris. My real concern was mathbb{A}-->mathbf{Q}. As a constable, I can't make that call myself. D. Matt Innis 18:44, 10 July 2010 (UTC)

Since Matt is not satisfied by your statements, and though Hayford is, and in order to finish this process, I have rewritten the section you changed (and added the condition that the plane does not pass through the vertex). This satisfies Matt and he accepts the nomination if you join in:

Difference between my versions, Boris change, my undoing change.

(This does not mean, however, that I shall stop to criticize approval rules that make no sense.)

--Peter Schmitt 12:08, 20 July 2010 (UTC)

(undent) Thanks, Peter, for the diffs, that does make it so much easier. So now I'll stand back and wait for Boris to put his name in the first slot of the template and change the date and versioin number. That will be the signal for me to take another look. D. Matt Innis 13:23, 21 July 2010 (UTC)

And, Peter, you could remove the approval template for now, since the article is somewhat in (re-)construction now; when needed I'll fill the template again. Boris Tsirelson 15:25, 21 July 2010 (UTC)

I do not think that this is necessary. We are actively working on it. This is quite normal, I think.

I have finished the "major" copyediting, but I will not check it now (it is late in the night). If you find any problems, or have suggestions I will take care of them, but not immediately. I will be away until some time on the weekend. --Peter Schmitt 00:59, 22 July 2010 (UTC)

Second degree equation

What is meant by "conditions": necessary? sufficient? both? Not knowing this, how can we prove them? Boris Tsirelson 20:54, 20 July 2010 (UTC)

"By translation of the origin over t   the linear terms in f(r) have been eliminated, only two quadratic terms (in x′ and y′), one bilinear..." — the reader could guess what are x′ and y′, but we'd better write that x′=x-t₁ and y′=y-t₂. Boris Tsirelson 06:49, 21 July 2010 (UTC)

---

The formulation

$f_{t}:=f(t_{1},t_{2})\neq 0\quad \mathrm {with} \quad f_{t}>0\Rightarrow A+C<0\quad \mathrm {and} \quad f_{t}<0\Rightarrow A+C>0$

is rather cumbersome in comparison with the equivalent formulation

$(A+C)f_{t}<0$

Boris Tsirelson 18:11, 22 July 2010 (UTC)

Done, but I kept the "cumbersome" formulation, too -- it probably is clearer for non-mathematicians. --Peter Schmitt 00:16, 24 July 2010 (UTC)

---

Necessary/sufficient again: the arguments in the article prove that the conditions are sufficient (for the equation to give an ellipse), but I do not see how they (even after a small modification) can prove necessity. If the determinant vanishes then these arguments fail, but it does not mean that an ellipse cannot result. It is possible to analyse this case; it leads to several possibilities (parabola, two lines, ...) and they all are not ellipses; but I doubt that we have to do it in the article. Rather we could write that sufficiency is proved here, while necessity holds but is not proved here.

For now, the phrase "Since, by assumption, the determinant det(Q) = AC−B2 ≠ 0" is not justified; equivalence is lost. Boris Tsirelson 17:33, 22 July 2010 (UTC)

Really, it is not hard to prove equivalence between the condition

$AC-B^{2}>0$ and $(A+C)f_{t}<0$

and the condition

there exist a shift and a rotation that transform $f(\cdot )$ into $\mathrm {const} \cdot ({\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-1)$ for some positive $\mathrm {const} ,a,b$.

But it does not solve the problem. The second condition implies the (ultimate) third condition below, but the converse implication probably is harder to prove; here is the third condition:

the transformed function vanishes at (x,y) if and only if ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$.

Indeed, what if, say, the square of the transformed function is equal to the cube of $\mathrm {const} \cdot ({\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}-1)$? I know it cannot happen, but this is not self-evident Boris Tsirelson 06:20, 23 July 2010 (UTC)

In the copyedit process I tried to put the "necessary" in without changes. I'll critically reread it to identify where too much is assumed. --Peter Schmitt 00:38, 24 July 2010 (UTC)

Here is a simpler example of the logical subtlety with the necessity. The equation $x^{2}+2xy+y^{2}=0$ cannot be transformed to the equation $y=0$ by a shift, a rotation and multiplication by a coefficient. And nevertheless it represents a straight line! Now, if "our" conditions are violated then indeed the given equation cannot be transformed to the equation ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$ by a shift, a rotation and multiplication by a coefficient. So what?? Boris Tsirelson 19:08, 24 July 2010 (UTC)

About "second-order curve" I understand your intention; but be careful with special cases; the equation $x^{2}+y^{2}=0$ is equivalent to $x^{2}+2y^{2}=0$.Boris Tsirelson 19:19, 24 July 2010 (UTC)

Another example: the equation $(x^{2}+y^{2})^{2}=1$ is equivalent to $(x^{2}+y^{2})^{2}+x^{2}+y^{2}=2$. Thus the phrase "since the equation of a curve is unique up to a proportionality factor" would better be "since the second-order equation of a curve is unique up to a proportionality factor". Boris Tsirelson 19:29, 24 July 2010 (UTC)

Now it is better. Boris Tsirelson 19:35, 24 July 2010 (UTC)

(after 2 edit conflicts)
Yes, I noticed it, too. (And adding "second-order" would not help, as your example shows. Could be non-degenerate, perhaps.)
Everything would be simpler (and logically more consistent) if we were talking about the classification of second-order curves. But, in this article, I want to retain Paul's "basic" (or elementary) approach. It did not work well to try and force the "necessary" on his derivation which was rather not a proof but a calculation showing what conditions suffice. --Peter Schmitt 19:56, 24 July 2010 (UTC)

Sorry for edit conflicts... But do not think that I press you to prove necessity. Indeed, I wrote: "Rather we could write that sufficiency is proved here, while necessity holds but is not proved here." This is not a textbook, and we are not obliged to prove anything. I only want not to claim to be proved here what really is not. Boris Tsirelson 20:03, 24 July 2010 (UTC)

No need to excuse for the edit conflicts -- this is bound to happen. No, I do/did not feel "pressed" to prove it. It was my idea because I wanted to add necessity to Paul's statement and then, naturally, to add it to the proof, too. But it is better to leave it as it is now (det and trace are not proved, either.) --Peter Schmitt 20:45, 24 July 2010 (UTC)

Proof

"With the help of the inverse Q⁻¹ the equation can be rewritten to..." — not quite equation, rather, equality. Boris Tsirelson 18:14, 22 July 2010 (UTC)

"Ir is known that the determinant of a matrix is invariant" — should be "It is known". Boris Tsirelson 18:17, 22 July 2010 (UTC)

"hence

$\det(\mathbf {Q} )=AC-B^{2}=\alpha _{1}\alpha _{2}>0.$

holds if" — remove the fullstop after the display. Boris Tsirelson 18:19, 22 July 2010 (UTC)

Done, but since I am not sure that "equality" is better here, I choose "rewrite the equation for f" --Peter Schmitt 00:07, 24 July 2010 (UTC)

Eccentricity

"For the following two inner products (indicated by a centered dot) we find,

${\vec {r}}_{1}\cdot (e{\vec {a}})=0\quad {\hbox{and}}\quad {\vec {r}}_{1}\cdot {\vec {r}}_{!}=b^{2}.$" — should be $\dots \cdot {\vec {r}}_{1}$, not $\dots \cdot {\vec {r}}_{!}$. Boris Tsirelson 12:54, 21 July 2010 (UTC)

Thank you for spotting this -- it is difficult to see (both on the page and on the edit page).

As you noticed I am currently trying to increase readability and facilitat following the calculations. I hope this works. I avoided such changes as long as I wanted to be able to approve the article.

(I hope I did not introduce too many typos and mistakes.) --Peter Schmitt 13:05, 21 July 2010 (UTC)

"it is difficult to see" — maybe just because you use higher resolution than me... Boris Tsirelson 15:19, 21 July 2010 (UTC)

Well, then I better wait for you to finish the "active part of the trajectory", and then reread. Boris Tsirelson 15:22, 21 July 2010 (UTC)

Algebraic form

Proof: some displays are now too long for one line. Boris Tsirelson 12:57, 21 July 2010 (UTC)

Really? Not on my screen. I'll change it (though this clearly should count as a copyedit :-) --Peter Schmitt 13:01, 21 July 2010 (UTC)

Probably you use a high resolution, and read rather small characters.

About "count as a copyedit" now I can trust only a declaration of Matt (and only if the declaration is recent enough). Boris Tsirelson 15:16, 21 July 2010 (UTC)

---

About

$\left(|{\vec {r}}+e{\vec {a}}|-|{\vec {r}}-e{\vec {a}}|\right)={2e{\vec {r}}\cdot {\vec {a}} \over a}$

I wonder, why the brackets in the left-hand side? Boris Tsirelson 18:05, 22 July 2010 (UTC)

They were overlooked when I divided by 2a ... --Peter Schmitt 23:48, 23 July 2010 (UTC)

Circle and ellipse

A circle, is it a special case of ellipse? It depends on the definition, of course; but what is the widely accepted definition? Regretfully sources often are unclear in this point. However, let us see what they say about the eccentricity. The Mathword says "Define a new constant 0<=e<1 called the eccentricity (where e=0 is the case of a circle)". But "Introductory college mathematics" by Harley Flanders and Justin J. Price says (on page 245) that 0<e<1. Thus maybe we are free to choose a definition; but then we must conform to it. If we insist that a circle is an ellipse then the semi-axes are always well-defined as lengths, but not always as line segments. The phrase "both ellipse axes are symmetry axes" becomes vulnerable. Also the phrase "A cone can be generated by revolving around the axis a line that intersects the axis of rotation under an angle α (between 0 and 90 degree)."; strictly between, or weakly between? "If the angle between the plane is greater than α (but smaller than a right angle) then it intersects the cone in an ellipse."; and if not smaller than a right angle (thus, equal to it), then what? Still an ellipse... "(Otherwise, the intersection is either a parabola or a hyperbola.)" - also vulnerable. Maybe it is better to say once that a circle may be included or excluded, but we assume it to be excluded unless otherwise stated explicitly. The "eccentricity" section ends with an eccentric statement: 0<=e<=1; well, maybe the circle is an ellipse, but a line segment probably should not be. Boris Tsirelson 17:57, 22 July 2010 (UTC)

You are right, of course, that essentially it is a question of the definition whether a circle is an ellipse. But I think that you will not find a mathematician who would not say: "A circle is (a special case of) an ellipse." Just like a square is a rectangle, a parallelogram, a rhomb.

In common language, however, I have heard the view that a square is not a rectangle.

It is indeed an oversight that the special role of a circle in some of the statements is not mentioned, though I am not sure ie cccccf it is serious in all the cases you mention. I'll think how best to deal with this. --Peter Schmitt 00:33, 24 July 2010 (UTC)

The definition used for "ellipse" in this article clearly includes both extreme cases (circle and line segment). --Peter Schmitt 09:26, 24 July 2010 (UTC)

Polar representation relative to focus

"of the vector (cf. figure 5) with endpoint on the ellipse" — but where is the other (first) endpoint? On the second focus, in fact. Boris Tsirelson 18:22, 22 July 2010 (UTC)

This vector was used earlier, but -- yes -- it should be repeated here. --Peter Schmitt 23:54, 23 July 2010 (UTC)

---

---

A progress, I see; and, Peter, please let me know when it will be my turn to reread the article. Boris Tsirelson 17:52, 24 July 2010 (UTC)

Boris, if I have not added some more blunder, I am finished. --Peter Schmitt 19:57, 24 July 2010 (UTC)

The new version

"(In a circle, any diameter can be chosen as the major axis or as the minor axis." — close the bracket...

Done

"Usually, in common language, these extreme cases are not referred to as an ellipse because "circle" (or "point") and "line segment" describe them better, but in mathematics they are included because they satisfy the definition." — Well, if we are quite serious about the definition, we could also say about the case $d<F_{1}F_{2}$: the empty set, is it also an ellipse, or not?

I have evaded this problem by (implicitly) stating $d\geq F_{1}F_{2}$. --Peter Schmitt 14:37, 26 July 2010 (UTC)

"A plane that intersects the axis in an angle greater than α then it intersects the cone in an ellipse." — Either add "if" (and remove "that") in the beginning, or remove "then it" in the middle.

Done

"Hence, (in fact the Pythagoras theorem applied to POF₂)," — should be P₁OF₂ .

Sorry, I overlooked this

"so that the eccentricity is given by
:<math>
e = \sqrt{ \frac{a^2-b^2}{a^2}}\quad\hbox{with}\quad 0 \le e \le 1.
</math>e "

— remove the "e" after the display.

done

"Such an equation always describes a conic." — (1) Does the reader know what is a conic? (2) Is a conic defined so that "always" is appropriate here (with all possible degenerate cases)?

"$(A+C)f_{t}<0$

or, equivalently, $\quad f_{t}:=f(t_{1},t_{2})\neq 0\quad \mathrm {with} \quad f_{t}>0\Rightarrow A+C<0\quad \mathrm {and} \quad f_{t}<0\Rightarrow A+C>0$"

— it is better to define $f_{t}$ near its first occurrence, not second.

Done

:<math>\begin{matrix}
                         At_1+Bt_2 &= -D \\
                         Bt_1+Ct_2 &= -E 
\end{matrix}</math>.

— remove the dot (fullstop) after the display (it appears on a strange place, like a superscript to E, at least on my browser).

done

Switching back to a quadratic equation
:<math>
\left(\mathbf{r}''\right)^\mathrm{T} \boldsymbol{\alpha} \mathbf{r}'' + f_t = \alpha_1 (x'')^2 + \alpha_2 (y'')^2 + f_t = 0 .
</math>
we see...

— remove the fullstop at the end of the display.

done.

"and we can apply the assumption

${\begin{aligned}\alpha _{1},\alpha _{2}>0&\quad {\hbox{i.e.}}\quad 0>\alpha _{1}+\alpha _{2}=A+C\quad \Rightarrow \quad f_{t}<0\\\alpha _{1},\alpha _{2}<0&\quad {\hbox{i.e.}}\quad 0<\alpha _{1}+\alpha _{2}=A+C\quad \Rightarrow \quad f_{t}>0\\\end{aligned}}$"

— swap "0<..." and "0>...".

I was careless, it seems ...

"This shows that the conditions given are sufficient ." — remove the space before the fullstop.

Done.

by solving the equation given for the vector '''t'''.
http://en.citizendium.org/wiki/Creative_Commons_CC-by-sa_3.0

— what could it mean?

I don't know. This was obviously dropped (unnoticed) by the cursor ...

Gardeners construction: "string around the triangle F₁F₂C" — should be FGC.

Oops -- should have looked at the graphics ...

About the periapsis and apoapsis: is it evident that the distance to a focus is maximal (minimal) at S₁, S₂? Not immediately, I believe; rather, it follows from Eq.(3).

Added a comment, though we are not forced to prove everything ...

---

Now, a less boring matter. Subsection "Proof" of Section "Algebraic form" proves that every point of the ellipse satisfies the equation. What about the converse? It can be proved with some additional effort. Namely:

Knowing that $x^{2}+y^{2}=r^{2}$, ${\frac {{\vec {r}}\cdot {\vec {a}}}{a}}=x$ and $e^{2}={\frac {a^{2}-b^{2}}{a^{2}}}$ and doing the calculations backwards we pass from ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$ to $|{\vec {r}}-e{\vec {a}}|^{2}=(a-ex)^{2}$ and $|{\vec {r}}+e{\vec {a}}|^{2}=(a+ex)^{2}$. Does it follow that $|{\vec {r}}-e{\vec {a}}|=a-ex$ and $|{\vec {r}}+e{\vec {a}}|=a+ex$? Yes, it does, since the $a-ex\geq 0$ and $a+ex\geq 0$; these inequalities are ensured by two facts: $0\leq e\leq 1$ and ${\frac {x^{2}}{a^{2}}}\leq 1$. Finally, $|{\vec {r}}-e{\vec {a}}|+|{\vec {r}}+e{\vec {a}}|=a-ex+a+ex=2a$. Boris Tsirelson 12:04, 25 July 2010 (UTC)

Implemented, though written differently. --Peter Schmitt 13:41, 26 July 2010 (UTC)

---

The work is close to finish. Looking forward to the final version, I sign the approval. As for me, Peter is welcome to join. (As for Matt, — I do not know.) Boris Tsirelson 19:35, 25 July 2010 (UTC)

I have fixed all the minor things. Sorry for the typos. (I did not mind to do it, but you could have saved some of your time by doing the typos instead of describing them. Matt accepts copyedits of typos.) Thank you for the very careful job. Unfortunately I sometimes insert a letter at the position of the screen cursor when getting too near to the touch pad.

I'll think about what to do with the remaining remarks. --Peter Schmitt 00:05, 26 July 2010 (UTC)

"Matt accepts copyedits of typos." — I'd prefer a more formal policy defining what is acceptable in such situation. Given that a constable need not understand math, nor even the wiki markup, it is problematic. A typo in an English word would be OK. But what about deleting "e" at the end of a display? Isn't it a change of a formula?! Boris Tsirelson 07:34, 26 July 2010 (UTC)

I understand why you did not do it, Boris. It was just a remark to document that (without any gain in quality) it considerably increases the effort (and patience) needed to get an article approved. --Peter Schmitt 13:45, 26 July 2010 (UTC)

We may invent a trick (for the next case, I mean): I copy the article to my sandbox and copyedit it there; then you copy my sandbox to the article, check diff, and continue from this point. A bit silly, but can help. Boris Tsirelson 20:13, 26 July 2010 (UTC)

Just because I can't read math doesn't mean I can't read English ;) If you really wanted to trick the process, just give him your account and password! But, I know you wouldn't do either. Where's the integrity in that? D. Matt Innis 01:28, 27 July 2010 (UTC)

What is the problem of integrity? I did not mean any crime. (Too silly, to plan a crime on this place.) As far as I understand, you agree that it is OK if I suggest changes to Peter, he agrees and implements them, and then I approve. The principal point is that nothing can be approved by me unilaterally, without explicit acceptance by Peter. Am I right here? Now, what does it change, whether I write on the talk page "remove 'e' after the display" and he does, or alternatively, if I remove the 'e' in my sandbox and he implements? Note, I wrote "then you copy my sandbox to the article, check diff, and continue from this point" rather than "you copy it blindly". It is a better technical realisation of the same social interaction process. Do you agree? Boris Tsirelson 07:42, 27 July 2010 (UTC)

The integrity is here: in all cases, Peter is assumed to be responsible for every edit made by him, no matter who, when and how suggested him the edit. Boris Tsirelson 08:20, 27 July 2010 (UTC)

Matt, you know that we follow the rules because we want this approved. We also know that you have to follow the rules. We only complain and argue the rules because we want to point out and "prove" that not all rules make sense but are purely formal obstacles and deserve to be reconsidered. Take this article as a concrete example: It will be approved because Boris and I cooperate until we both are satisfied. It would still be necessary that we both agree on the result if we both could edit the page directly. But it would be much easier and need much less effort to reach this result. (It may even be that a minor improvement is not made because Boris does not think it worth the effort to point it out to me.) Not all Citizens may have the patience and take the time to do it the formal way. (The situation is even more difficult if an Editor wants to approve an article where no cooperating author is available.) --Peter Schmitt 09:34, 27 July 2010 (UTC)

Boris and Peter, please don't consider my comment above as accusatory; I actually wrote it with a smile on my face, but looking back, I see that was not clear. I know that neither of you are interested in cheating the system. I actually think it would be a good compromise. The problem, though, is in the constable's evaluation when they check to see if the single editor approval included any content edits. The page history won't show the copy edits as the nominating editor and could be used to sneak a content edit in when there weren't three editors.

I see nothing wrong with leaving a diff on the talk page that shows the edits that were in the nominating editor's sandbox. This needs to show in a prominent place under a heading such as ==Nominating editor's sandbox edits==. There can be several links if necessary as the article progresses and the constable can then include them in on the Approval page in case someone suspects something was fishy.

I agree this would make single editor approvals much more efficient. Sorry for the confusion and thank you for you continued patience.

D. Matt Innis 12:55, 27 July 2010 (UTC)

A constable on duty, when joking, should not forget the ":-)" sign!

(unindent) But the matter is still not so clear. Here are some principal questions.

• Which edits may be made by the (single) nominating editor? Some possible answers: nothing at all; spelling corrections (in English words) only; the same, and also punctuation errors; everything that does not change the meaning, which must be evident to any constable; everything that does not change the meaning, according to an expert opinion.
• Which edits may be suggested by the (single) nominating editor to someone else (an author or another editor)? Some possible answers: (the list above); everything at all.

As for me, "everything at all" should be the answer to the second question. This is what I meant when writing "in all cases, Peter is assumed to be responsible for every edit made by him, no matter who, when and how suggested him the edit." If you do not agree then I am puzzled: what do you think about the whole process documented on this (long) page? If I may suggest only whatever I may make myself then why suggest at all? And in every case, my idea was: whatever I may suggest I may suggest in every technical form equally well, be it a note on the talk page or an edit on my sandbox. An answer from Matt is welcome; an official answer from the constabulary is even more welcome. Boris Tsirelson 14:57, 27 July 2010 (UTC)

Alas, you have found a loop-hole.

Of course we are heading toward; can Boris write an entire article in his sandbox; have Peter move it to mainspace; then Boris approve it?

This is not a constable or a constabulary decision. This is Editorial Council material, so for the time being, I'd keep documenting on this talk page. I suggest bringing it to the forum where I would be glad to comment.

D. Matt Innis 17:54, 27 July 2010 (UTC)

I see. Yes, you are right; this is the ultimate formulation of the question. However, the loophole is not really found by me. I recall that Peter wrote about this issue repeatedly, probably on the Approvals Manager talk page, in a slightly different form. Boris Tsirelson 18:55, 27 July 2010 (UTC)

Ah, well then I apologize to Peter for apparently not understanding or not fully following through with his thinking. The question of course becomes, do we no longer allow a three single editor approval, or do we allow a two editor approval? Interesting... and obviously something that needs discussion. D. Matt Innis 19:08, 27 July 2010 (UTC)

Yes, yes! I recall this point of Peter. Also, my thoughts culminated in these words (now buried on talk archive 8 of Matt):

CZ can be reliable only if many active editors are present. Then a wrong statement will raise a protest (of at least one editor) with high probability. Otherwise, nothing helps: if say only two editors work then there is a chance that they are a conspiracy.

The same in other words: a clever design can produce a reliable system of many non-reliable elements. But they must be many; otherwise no design can help. Boris Tsirelson 19:43, 27 July 2010 (UTC)

Yes, I remember those exact (last) words and agreed (to myself), but to actually think it ALL the way through with a very probable example is like getting hit bewtween the eyes with a board... I now see said the blind man :-) D. Matt Innis 19:55, 27 July 2010 (UTC)

Toward Approval 2

I see that there is a new date and Editor in the ToApprove template. It appears as a single editor approval and the editor has not edited the article (the one edit was removed). Unless, I see or hear anything otherwise, this article will be locked on July 28, 2010. If there are additional edits that need to be included, make sure to update the metadata template with the new version before July 28.

Thanks for all the hard work that went into this article. Citizendium is a better place thanks to you guys! D. Matt Innis 20:27, 25 July 2010 (UTC)

Error in Wikipedia

Curiously, the corresponding WP article is erroneous; compare WP ellipse with CZ ellipse. They write $F(B^{2}-4AC)>0$ instead of $4AC-B^{2}>0$, $(A+C)f_{t}<0$. A note is just left there by me, with a link to CZ. By the way, an approved CZ article is treated by WP as "reliable source"; unapproved one is not. Boris Tsirelson 20:27, 25 July 2010 (UTC)

Actually, they have no definite rule on the subject. It would have to be argued/fought out any time someone wanted to cite a CZ approved article and someone else objected. Certainly some people there reject them. Peter Jackson 08:41, 26 July 2010 (UTC)

I see. Moreover, I observe it on this case! Now they write $B^{2}-4AC<0$ but refuse to write $(A+C)f_{t}<0$ because CZ does not give reliable source for this claim (and "The threshold for inclusion in Wikipedia is verifiability, not truth"). They do not say "wait a day or two, it will be approved in CZ, and then...". Well, if you want to help WP (but also our readers, probably), you are welcome to add a reference, if you find one. (I have no such books on my shelf.) Boris Tsirelson 20:04, 26 July 2010 (UTC)

Suggestion

I'm not about to change the actual text, since I am not a mathematician and it is up for approval, but I have a suggestion. I'd remove the section of the introduction starting "There are two extreme cases". It seems to me that would make the intro flow better. Then I'd add a new section, after "conic section", along these lines.

==Special cases==

There are some special cases, effectively the limits of the ellipse. Usually, in common language, these extreme cases are not referred to as an ellipse because "circle" (or "point") and "line segment" describe them better, but in mathematics they are included because they satisfy the definition.

If the distance between the foci is zero, the ellipse becomes a circle. The two foci and the center are all identical, the eccentricity is zero, and any diameter can be considered as the major axis or as the minor axis. In algebraic terms, this is the case of a = b, and we call a = b = r, the radius of the circle. We also have d = 2a = 2r for the diameter of the circle. In conic section terms, it is the case of a plane perpendicular to the axis of the cone.

If a = b = 0, the ellipse degenerates to a point. In conic section terms, this is a plane intersecting the vertex of the cone. The eccentricity of the ellipse is undefined in this case since calculating it would require division by zero. In geometric terms, if a plane intersects the cone only at the vertex, then the intersection is the same whatever the slope of the plane.

Wrong. The intersection can be a pair of lines, or a double line. --Peter Schmitt 14:47, 26 July 2010 (UTC)

If the distance between the foci equals d. Then a = 0 and the ellipse degenerates to the line segment bounded by the foci. In conic section terms, this corresponds to a plane tangent to the cone; they intersect on a line.

Wrong. A line is not a line segment. --Peter Schmitt 14:47, 26 July 2010 (UTC)

If the distance between the foci is greater than d, then the ellipse equation has no solutions, the locus of points on the ellipse is the empty set. Geometrically, this corresponds to a plane that does not intersect the cone.

This can only happen if the cone is a line.

What is the eccentricity in the last two cases? Sandy Harris 01:41, 26 July 2010 (UTC)

It is 1 for a line seqment. --Peter Schmitt 14:47, 26 July 2010 (UTC)

Since this refers to eccentricity, we need to either put it after the eccentricity section or add a one-line definition of eccentricity in the introduction. I like the latter approach better. Sandy Harris 02:06, 26 July 2010 (UTC)

Peter, what do you think?

On one hand I understand Sandy. On the other hand, a definition should be monolithic, not distributed across sections. Thus, if we want to go this direction, we could make the lead fuzzy, like that: ellipse can be defined as a locus..., as a conic section, algebraically, etc. Then, a section "Formal definition", like this: different definitions are equivalent up to degenerate cases; the main definition is: the locus of... Boris Tsirelson 07:26, 26 July 2010 (UTC)

I, too, understand Sandy. However, since the introduction is meant to give a brief summary of the basic information (some readers may not read more), we cannot follow the suggestion.

This article answers the question: What is an ellipse? And it answers it using the sum of the distances from the foci. Thus the circle has to be mentioned in the introduction (even non-mathematicians may think of this case) and not mentioning the other special case would leave a gap.

Starting from this definition some -- by far not all! -- basic properties (equation, a parametric form) of the ellipse are treated and derived.

Treating all properties of an ellipse in a single article would make it much too long. There can be (and, hopefully, will be some day) several articles using different approcaches.

The empty set can only occur as an intersection if the cone is degenerated to a line (its axis).

I'll think about a better integration of the special cases. (Perhaps both in the lead and in a separate section?)

--Peter Schmitt 09:02, 26 July 2010 (UTC)

The empty set should not be an ellipse, I think, since otherwise we'll have a lot of unnecessary troubles: what is its a, b, e, center etc etc. Who needs it? Boris Tsirelson 09:11, 26 July 2010 (UTC)

I agree (and the usual definitions of a curve exclude this, I think). I have handled this by inserting that d has to be at least the distance of the foci.

In reaction to Sandy's suggestion I have moved the special cases to the end of the introduction. Since all issues are treated in the text, I do not think that an additional section is needed. --Peter Schmitt 14:56, 26 July 2010 (UTC)

Is a = 0 correct in the distance between foci = d case?? I think it should it be b = 0. I copied that from the current article. Sandy Harris 08:47, 26 July 2010 (UTC)

Oh yes; "when the distance of the foci equals d. Then a = 0 and..." should be "b=0". Boris Tsirelson 09:07, 26 July 2010 (UTC)

Thank you, Sandy. You are right, of course. Even if two (or more) people read a text carefully there will always be left something that has been overlooked ... (Though, "always" is not possible :-) --Peter Schmitt 09:10, 26 July 2010 (UTC)

Another

As the article already includes several different approaches, why not the focus-directrix one? Paul asked about that earlier, but doesn't seem to have got a reply. Peter Jackson 08:44, 26 July 2010 (UTC)

If someone wants to write such section, why not. However I'd prefer it to go to Version 2; for now we want approve the already written article. Boris Tsirelson 09:01, 26 July 2010 (UTC)

The question was answered -- essentially: "as you like it!". I agree with Boris that it may or may not be included. It is not an essential part of this article. Personally, I prefer articles of moderate size. As I wrote above: There is much too much material on ellipses than fits into one article. There can be articles, each using another approach. There can be a separate article on the directrix, for instance. There can be an article summarizing properties of the ellipse (without deriving them), etc. There is much opportunity for future work. Adding more and more material would make a rather coherent article into an arbitrary collection of facts. (By the way, if I had started an article on "ellipse" I would probably have based it on the conic section aspect.) --Peter Schmitt 09:21, 26 July 2010 (UTC)

Let us organize an "Elliptic science" workgroup! :-) Boris Tsirelson 10:34, 26 July 2010 (UTC)

But do not count on me as a particularly eager contributor :-( --Peter Schmitt 19:26, 26 July 2010 (UTC)

Circumference? Area?

Finding the area or circumference of a circle is straightforward. We all memorised these in high school: c = pi*d = 2 *pi*r and A = pi*r^2. Should the article give the corresponding formulae for an ellipse? Sandy Harris 02:33, 26 July 2010 (UTC)

The area is simple, since it is a "squeezed disk". But circumference is quite complicated. Boris Tsirelson 07:20, 26 July 2010 (UTC)

The newer version

It is nice. But still, few remarks.

The beginning of the new proof: "We have" — I'd prefer "Proof. We have", since for now it is not clear that what was said is a claim to be proved now.

This would be a proof inside a proof -- I tried it differently.

The end of the new proof: $(a+{\frac {f}{a}}x)+(a-a{\frac {f}{a}})=2a$ should be $(a+{\frac {f}{a}}x)+(a-{\frac {f}{a}}x)=2a$.

Ooops.

Why not mention positivity of these $a-{\frac {f}{a}}x$, $a+{\frac {f}{a}}x$ as a consequence of the assumptions? It is an important part of the argument.

Added.

Again, about "Such an equation always describes a conic." — maybe just write "a conic section"? The term "conic" itself never appeared before, did it? Boris Tsirelson 16:36, 26 July 2010 (UTC)

Did I overlook this? --Peter Schmitt 19:23, 26 July 2010 (UTC)

---

Now it is OK with me. Boris Tsirelson 19:54, 26 July 2010 (UTC)

With a view on your WP comments I have added a phrase (to the general form) explicitly stating "non-degenerated". I updated the Metadata and added my name. --Peter Schmitt 21:14, 26 July 2010 (UTC)

Nice. However, with my poor English I am in doubt when to write "degenerated" and when "degenerate". Boris Tsirelson 07:56, 27 July 2010 (UTC)

I hesitated when writing it. Thinking again, "degenerate" fits better here because it is not the result of having "degenerated". --Peter Schmitt 08:54, 27 July 2010 (UTC)

Boris, if some hitherto undiscovered problem should surface you will have to postpone approval for two days since I'll not be available for one or two days. --Peter Schmitt 23:08, 27 July 2010 (UTC)

I see. Hope not. Boris Tsirelson 04:49, 28 July 2010 (UTC)

---

Matt, where are you? It is your move. Boris Tsirelson 08:50, 29 July 2010 (UTC)

I'll do it right now. Hayford Peirce 20:17, 29 July 2010 (UTC)

APPROVED Version 1.0

Discussion for Version 1.0 stopped here. Please continue further discussion under this break.

Congratulations, people, this article is *finally* Approved! Thanks for all the hard work! Hayford Peirce 20:35, 29 July 2010 (UTC)