Talk:Ellipse/Archive 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...

"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?

"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.

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

"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.

"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.

:<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).

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.

"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>...".

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

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?

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).

---

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, <math> |\vec{r} - e \vec{a}| + |\vec{r} + e \vec{a}| = a-ex + a+ex = 2a <math>. Boris Tsirelson 12:04, 25 July 2010 (UTC)