Testwiki:Featured article candidates/Problem of Apollonius

The following is an archived discussion of a featured article nomination. Please do not modify it. Subsequent comments should be made on the article's talk page or in Wikipedia talk:Featured article candidates. No further edits should be made to this page.

The article was promoted by User:SandyGeorgia 17:33, 13 October 2008 [1].

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Problem of Apollonius

Nominator(s): Willow (talk)

I'm nominating this article for featured article because I believe that it exemplifies the best of Wikipedia's work. This is a legendary problem on which dozens of mathematicians have worked, from Apollonius of Perga to the present. The problem can be solved by a great variety of methods, illustrating for students the richness and relatedness of mathematics, and highlighting some episodes in its history, such as the resurgence of geometry in the early 19^th century. The problem is also relevant for modern navigation systems such as the GPS.

The article has undergone a Peer Review and became a Good Article over a month ago, with few changes since, indicating that it is stable. It has 43 kB of readable prose, one table, 23 static images and one animation that clarifies a mathematical transformation. It also has 72 inline-cited references (some are doubled up into a single citation), 11 books for further reading, and 7 external links to reputable websites. But above all, I hope that everyone enjoys the article! :) Willow (talk) 11:59, 22 September 2008 (UTC)

P.S. Note to reviewers: please do not change &-n-d-a-s-h-;'s to –, and other such HTML coding; I do that on purpose to help with proofreading. Thanks!

Comments

• Current ref 34 (Eric W. Weisstein "Four Coins propblem) is lacking a last access date. Also, shouldn't it be Weisstein, Eric W. to fit with the rest of your references?

Otherwise sources look good, links checked out with the link checker tool. Note that I couldn't evaluate the non-English sources ... even the Latin. Ecclesiastical medieval latin did not prepare me for mathematical Latin! (grins) Ealdgyth - Talk 12:05, 22 September 2008 (UTC)

Thank you so much, Ealdgyth, for your scrupulous attention to detail! :) I think I've fixed up that reference. And now that I know your preferences, I'll give you only medieval Latin to evaluate: Sensus noster iam marcescit, et in nobis refrigescit iam fervor ingenii; si quaeratur, quis hoc fecit, respondemus, "Nos affecit labor frequens FACii." ;) Willow (talk) 14:07, 22 September 2008 (UTC)

(flees from the Latin) Argh! College memories... Argh! Thanks Willow, looks like you're done! Ealdgyth - Talk 14:11, 22 September 2008 (UTC)

• Support - Looks solid. I knew that Willow was working on it and she's definitely made a change in the article. Good luck! —Sunday | Speak 20:28, 22 September 2008 (UTC)

Thank you, m'lord! :) A rose by any other name... :) Willow (talk) 14:29, 23 September 2008 (UTC)

• What gorgeous images ! But they need to be within sections, not above them; I hesitate to move them in myself, as that will create some layout problems (and there are a couple that are creating white spaces already). Also WP:MOS#Quotations on pull quotes and quotation mark/graphics. SandyGeorgia (Talk) 00:02, 23 September 2008 (UTC)

The pull-quote from Apollonius might be stricken soon, based on other critiques, so I won't fix that just yet, although I would be interested to know whether Template:Tl templates are categorically forbidden in Featured Articles? I kind of like the big splashy quotation marks, but others such as the royal pentomphaloid of blessed memory dislike them intensely, and I can imagine they might seem, well, a little unprofessional. :( In any event, I'll bow to the consensus, whatever that is.

The pull-quote has been stricken. Willow (talk) 15:39, 23 September 2008 (UTC)

The image layout is a bit of a quandary for me. As I'm sure you're aware, it used to be a layout requirement that left-aligned images came before the subsection heading. Now that wording has been softened to: "Do not place left-aligned images directly below subsection-level headings, as this can disconnect the heading from the text it precedes. This can often be avoided by shifting left-aligned images down a paragraph or two."

I can try to do that in some places, such as the beginning of the "Solution methods" section (Figure 3) as well as Figures 9 and 11; however, I don't see any solution for Figures 6–8? I'm sure you don't want me to interpolate unnecessary text just to satisfy an image layout issue. Willow (talk) 14:29, 23 September 2008 (UTC)

I might've found an agreeable compromise on the image layout issues. Does everyone like the present layout? Willow (talk) 15:39, 23 September 2008 (UTC)

Note that this is a subject of discussion at WT:MOS (See Prince of Canada et al). - Dan Dank55 (send/receive) 21:35, 23 September 2008 (UTC)

I'm sorry, but I'd much rather make geometrical animations and write medieval Latin poetry all day for Ealdgyth than wade through (or into) such conflicts, which depress me and tie my stomach up in knots. :P Can you just tell me what to do once it's all over? I really don't care one way or the other, as long as I can explain my topic without too much interference from the MoS. Thanks, Dan! Willow (talk) 23:02, 23 September 2008 (UTC)

Second that thought :-) It looks very nice, but there's still one left-aligned image left under a third-level heading (in "Number of solutions"). Can that be moved right? I think the left-right alternating business is purely aesthetic and much less important than accessibility for readers who use screen readers, but honestly, I'm not sure how a screen reader would do with all the math formulas anyway. MoS suffers from WP:TLDRitis, but if you want to consult Graham about how his screen reader does with the article, you could post something brief to the talk page of WP:ACCESS. SandyGeorgia (Talk) 03:01, 24 September 2008 (UTC)

Thank you, Sandy, for your kind encouragement! :) I'll definitely follow your advice on asking Graham for help. The lone holdout image you mention, Figure 11, is a special case. After some brooding and ignoring all rules, I moved it from the proper right-aligned to left-aligned. The dilemma was that a right-aligned Figure 11 stacks on Figure 12, pushing it down and causing unwanted whitespace just before the poem by Frederick Soddy, as you can see here. Figure 11 can't move upwards because it must come after the Table of the preceding subsection, and it's not really practical to switch the order of the subsections, since Figure 11's subsection refers to the one preceding it. I kind of like the present look, but I also don't want to shut anyone out of understanding the article due to an easily fixed layout glitch. Hoping that this is an OK solution, Willow (talk) 09:22, 24 September 2008 (UTC)

• Support and comments A really nice article, but two concerns: jimfbleak (talk) 06:44, 23 September 2008 (UTC)

• Forced image sizes. These override my thumb preference settings, and, although the images are excellent, make for an ugly layout. If this had not been at FAC, I would have removed most or all of the image sizes. I was going to withhold support on this issue, but I'll be away a for a few days, so supporting now in expectation of a fix.

• I can't decipher to whom this support belongs, no sig, and it will take me a long time to step back through the diffs; please add an Template:T1 template. SandyGeorgia (Talk) 01:45, 28 September 2008 (UTC)

I know forced image sizes aren't customary, but the default image sizes were too small to read the subscripts on the figure labels, such as the "2" in d₂. Is there a way of setting a minimum size? Whether a layout is ugly might be a little subjective (for instance, I find the defaults worse than the present article, naturally) but it seems that illegibility is something we can agree on. Willow (talk) 14:29, 23 September 2008 (UTC)

The image size preferences have been removed except for those in the table. Those were maintained for consistent spacing of the table rows. Willow (talk) 15:39, 23 September 2008 (UTC)

Odd I'm not a FA regular, but MOS:IMAGE does say If an image displays satisfactorily at the default size, it is recommended that no explicit size be specified. so too small labels would be a reason to give explicit sizes. I would say default size works well for most here but figs 6, 9, 10, 13 and the unnumbered fig near fig 3 seem a bit too small to me. (should that fig be given a number for consistancy?) --Salix alba 16:47, 23 September 2008 (UTC)

• opening sentence - ...is to construct circles that are tangent to ("touch") three given circles - I really don't like "touch"; tangent is linked and explained later in the text. An explanation in the key sentence disrupts the flow and seems inappropriate. Someone who can't pick up the idea of tangency pretty quickly won't get far in this article anyway. jimfbleak (talk) 06:44, 23 September 2008 (UTC)

I sympathize about the "touch" interfering with the flow and I've tinkered with other ways of saying it; but it's hard to satisfy all the desiderata at once. I feel that the lead has to be able to stand by itself; I don't think we can expect that the reader will read the rest of the article or even follow the links. This is especially true for people who might be reading a geometry article on a cell-phone with a slow connection. (An admittedly unlikely scenario but, hey, geometers can dream, too. ;) I also feel that the lead has to be accessible to people who have never learned the mathematical definition of "tangent to". I could remove the "touch" and interpolate a parenthetical sentence, e.g., "(Two circles are tangent if they touch at a single point.)" Would that help, or would it interrupt the flow even more strongly? Willow (talk) 14:29, 23 September 2008 (UTC)

I removed the parenthetical "touch" and added an explanatory clause, which I think is not too disruptive to the flow of the writing. Willow (talk) 15:39, 23 September 2008 (UTC)

Supportive comments by Jakob.scholbach (talk) 20:10, 23 September 2008 (UTC). Congrats, Willow, great article. (I read til Inversion methods for now.)

Thank you very much, Jakob! :) Willow (talk) 21:10, 23 September 2008 (UTC)

Is that a Support or a Comment? SandyGeorgia (Talk) 01:45, 28 September 2008 (UTC)

I wouldn't bother forcing the distinction. Third bullet in the instructions plus the subsequent sentence clearly suggest the artificiality of this distinction. What does count is the addressing of critical comments. Tony (talk) 05:01, 2 October 2008 (UTC)

I believe that Jakob means to say that he's leaning towards Support because he likes the article and will do so once his relatively minor concerns are taken care of. Which I'm trying to do...the main outstanding thing left is to re-write the lead paragraph to encompass more of the technical details of the article without losing readers because of accessibility. I'm brooding over that and will likely dive in on Monday. Willow (talk) 14:28, 28 September 2008 (UTC)

Exactly. Except for the lead section issue, I would already strike the "ive comments" above, but I will wait until that point is covered more adequatly. Jakob.scholbach (talk) 18:12, 28 September 2008 (UTC)

• Lead: "Apollonius' problem can be generalized in several ways. The three given circles can be of any size and at any distance from each other" - the wording is a bit unfortunate, I feel. After all in the original problem the positions are arbitrary right off the start, right?

You're quite right; that test was moved from somewhere else and not updated appropriately. I fixed the wording and made it a little more colourful at the same time. Willow (talk) 21:10, 23 September 2008 (UTC)

• "Statement section": Given your apparent talent with images, I think that section would benefit of an image of a tangent vs. a secant.

Sure, that's not hard. :) Do you think that tangent/secant lines/lines, line/circles or circles/circles would be more helpful? Willow (talk) 21:10, 23 September 2008 (UTC)

Hm, I just thought of a circle, and a tangent line and a secant line.

• Same section: I don't understand why you talk about two lines being tangent or not.

Perhaps I was misguided in this, but it seemed necessary for completeness to discuss the meaning of tangency for all of our geometrical objects, not just the obvious ones. That's also why I discussed how points could be tangent, and whether something was tangent to itself. Does that seem reasonable? Did I understand your meaning correctly? Willow (talk) 21:10, 23 September 2008 (UTC)

I see too possibilities: either you stick to the case where solutions are circles (proper, i.e. no points or lines), in which case you don't need to discuss when two lines are tangent, because at least one of the entities in question is a circle. Other case: you do allow lines as solutions (as you do in the "Resizing two given circles to tangency" section. If you do this, the lead has to reflect this somehow.

As an aside: I have never heard people calling parallel lines tangent to each other. The projective plane image you may have in mind, i.e. considering the points at infinity, does IMO not really allow to call them tangent: parallel (distinct) lines intersect in infinity, but they are not tangent there (think of two great circles on a basketball).

I understand your point, but please consider the case of two tangent circles in the plane? They will become two parallel lines if we carry out inversion in a secant circle centered on their tangent point. If we say that inversion doesn't change the tangencies of inverted objects, by adding a point at infinity (rather than a line at infinity), then the two parallel lines are tangent at that point, aren't they? I'm imagining it's like putting two tangent small circles (not great circles) on the sphere so that their tangent point is the North Pole and then projecting down stereographically? Does that seem reasonable? Willow (talk) 00:44, 25 September 2008 (UTC)

I did not check the details, but it sounds somewhat reasonable. I suggest putting some of these thoughts into that section and finding a ref for this so as to escape OR.

Update: I'm doubtful that such a projection transforms two tangent circles into parallel lines, because the intersection multiplicity of the circles is 2, whereas the intersection of two lines (intersecting at infinity) have multiplicity 1. Jakob.scholbach (talk) 11:20, 1 October 2008 (UTC)

• Figure 3: Showing the hyperbola you are talking about would be nice.

That image is much harder for me to make, but perhaps I can write a computer program to do that. I wanted to do that anyway for another article, so thank you for encouraging me! :) Willow (talk) 21:10, 23 September 2008 (UTC)

• A general markup concern: you use bold face pretty freely. I would not do this: it drags the readers attention to single symbols, it is also not standard in math books to use bold face for points, say, and also, I guess, the MOS has something in this direction. Also, the labels à la CCP don't deserve bold face, in my view.

Let me brood over that? I like the boldfacing to indicate that two things are different in character, e.g., points and lines. But I've been known to change my mind — sometimes many times before breakfast. ;) Willow (talk) 21:10, 23 September 2008 (UTC)

• Another markup thing: I am a bit irritated by switching so often between TeX markup and standard text. I personally prefer standard, because the flow is better. Most of the formulae (!hey ;) don't need TeX, for example the ones in the "Lie sphere geometry" section don't at all. I would suggest replacing every occurence of TeX, where it is possible, by standard markup.

I'm sorry, but I don't see any TeX in the text of the article? If you meant the TeX equations in math mode, I'm really sorry, but I think I'd like to keep them that way. I find the TeX formatting much better than my kludgy formatting with italics and subscripts. Willow (talk) 21:10, 23 September 2008 (UTC)

Yes, it's about

$(X_1|X_2):=v_1{w}_2+v_2{w}_1+𝐜\cdot 𝐜-s_1{s}_2{r}_1{r}_2.$

I mean, I won't insist, but why is the above nicer than

(X₁ | X₂) = v₁w₂ + v₂w₁ + c·c − s₁s₂r₁r₂

Admittedly, the difference in quality isn't glaring, but I do prefer the LaTeX version on my screen, particularly for its spacing and positions of subscripts. De gustibus nihil disputandum est, perhaps? :) Willow (talk) 00:47, 25 September 2008 (UTC)

• Why is it "Viète's reconstruction"?

Because he was explicitly trying (perhaps unsuccessfully) to reconstruct the original straightedge-and-compass solution of Apollonius? Willow (talk) 21:10, 23 September 2008 (UTC)

OK. I didn't understand that from the said section.

• "Multiplying out the three equations and canceling the common terms yields formulae for the coordinates xs and ys" - I don't see ad hoc why this gives linear equations. Could you specify what you mean by "canceling"?

When multiplied out, all three equations will have x_s² + y_s² on the left-hand side, and r_s² on the right-hand side. Subtracting one equation from another eliminates these quadratic terms, leaving only the linear ones. Should I explain it more fully in the text? Willow (talk) 21:10, 23 September 2008 (UTC)

I wrote out the fuller explanation; undoubtedly, many people would've gotten that. Thanks for catching that and improving the article; that's a very important part of the article! :) Willow (talk) 22:41, 23 September 2008 (UTC)

OK.

• "s_1–3" is usually written "s₁ ... s₃".

Yes, you're right, I was just trying to save space. Willow (talk) 21:10, 23 September 2008 (UTC)

OK, that's been fixed, I think. Does it seem OK to you? Willow (talk) 22:41, 23 September 2008 (UTC)

• In the first equation in "Lie sphere geom", why do you use the congruence symbol?

I was trying to say that this is the definition of the five-dimensional dot product. Willow (talk) 21:10, 23 September 2008 (UTC)

Yes. AFAIK, the standard symbol for this would be ":=" or $\stackrel{def}{=}$, or simply "=" and say a magic d-word in the text.

• You should introduce or at least wikilink the |...| notation of the Euclidean norm.

OK, I'll do that right away, that's a very good suggestion. :) Willow (talk) 21:10, 23 September 2008 (UTC)

OK, that's been fixed, I think. Does it seem OK to you? Willow (talk) 22:41, 23 September 2008 (UTC)

• I don't understand the rôle of the orientation of the circles. On the one hand you call it "for visualization", but then it is also the sign in terms of these tangencies. The only widespread meaning of orientation is the clockwise-counterclockwise thing, AFAIK.

We should consult G-guy to explain this cryptic nomenclature, but I've definitely read it in my sources. Willow (talk) 21:10, 23 September 2008 (UTC)

• The first question that comes to (my) mind in the inversion section: if the inversion is undone by applying it again, how come that a problem becomes "simpler" when applying the inversion? Jakob.scholbach (talk) 20:10, 23 September 2008 (UTC)

Because the operation of inversion can render the problem more symmetrical, e.g., by making given circles concentric, or transforming them into parallel straight lines. Another good example of that is the Steiner chain, which I've been fitfully working on these past few weeks. I'm sure that a real mathematician could give you a better explanation and better examples, though. Willow (talk) 21:10, 23 September 2008 (UTC)

Precisely; this is the usual reason to restate a problem, from the application of Cartesian coordinates to the Langlands conjecture: to get a new form which contains the same information, more conveniently arranged. Septentrionalis PMAnderson 04:17, 24 September 2008 (UTC)

OK, I have read the rest of the article:

• Pairs of solutions: the text is partly almost the same as the image caption. Consider trimming down the caption of the image? In general, your image captions are really long.

Yes, I'm kind of talkative. I also know my own weaknesses and project them onto others: I sometimes skip the main text if I can get the idea from the figures and the captions. Others might like to do that as well. ;) Willow (talk) 00:29, 25 September 2008 (UTC)

• Inversion to an annulus: Wikilinking "concentric" might be good.

That's a good suggestion, although the article concentric definitely needs some TLC. :) Willow (talk) 00:29, 25 September 2008 (UTC)

• Is "in tandem" English or Latin? I mean, is it common English use? Never heard that one.

Yes, I've heard it before; I hope that I've used it correctly! :) Willow (talk) 00:29, 25 September 2008 (UTC)

• Figure 7 and 8 have a number of captions (in the image) that don't seem to be used. e.g. r_inner, r_outer. I would remove these things, they just clutter up the image.

Actually I need to keep them becuase they're needed to define the solution radius and the distance of the solution circle from the concentric center. I think I've fixed up the captions to Figures 7 and 8 to agree with the text on that score. Willow (talk) 00:29, 25 September 2008 (UTC)

• is there a reason for not using standard header markup at "Shrinking one given circle to a point" and the next one? The equal spacing before and after the heading looks a bit odd, I find.

I didn't realize that I could go down one more level in subsections with five ='s in a row! Live and learn something new every day at Wikipedia :) Willow (talk) 00:29, 25 September 2008 (UTC)

• You mention that Euclid solved the PPP and LLL case. It might be good to say that every middle school kid (at least in Germany :) learns this, and this is apparently much easier.

To benefit students with less enlightened curricula, I explained those two cases in more detail in the article. You probably noticed that there's a daughter article that goes into all the special cases. Willow (talk) 00:29, 25 September 2008 (UTC)

• Here is, at last, a critical (in the original sense) point: the lead is, I have to say, fairly unbalanced. The generalizations, strangely, occupy a third of the lead, but much less of the article. Another third is occupied by the historical stuff. The current lead section gives the impression: it is an old problem, somehow there are 8 solutions, this can be done more generally.

I read above that your intention is to make the lead interesting or readable to the general public, but this should not be at the expense of matching the content. Please consider adding some hints how the solutions were achieved. You don't have to go into the details, obviously, but somehow the flavor of the many pretty complicated thoughts of the "ancient" geometric guys should be in the lead. This wish corresponds to some guideline "do not dumb it down", right?

Dang, the more I think about it, especially in light of the question which "glasses" (acc. to Gg below) one wears, the more I come to the conclusion that outlining the solution methods, and somehow showing that the problem is a very nice example of how (some techniques in) mathematics evolved over time, is a must. Jakob.scholbach (talk) 20:15, 24 September 2008 (UTC)

• Another statement I miss a bit in the article is a qualitative comparison of the methods. As far as I personally see it, the geometric methods are terribly complicated, the algebraic is pretty nice (and by way of introducing these coordinates, the problem is almost trivial, which could be a nice info for the lead, too), finally the Lie sphere idea puts the other ones to shame with respect to elegance and simplicity.

• Of course I share your view that the Lie sphere approach puts others to shame by its elegance and simplicity, but it is important to realise that the beautiful algebra involves solving quadratic equations, and classical geometers were interested in "geometrical" (for them, straight edge and compass) constructions. Solving quadratic equations is ugly from this point of view, and so the other geometrical solutions (which are indeed terribly complicated) are actually simplifications of the direct translation of the algebraic solutions into classical geometry. What is simple and what is beautiful depends upon the glasses we use. Geometry guy 20:00, 24 September 2008 (UTC)

• Yes, I'm aware of this. Anyway, some evaluation should be in the article. Even saying that the problem amounts to solving quadratic equations is a good hint, and should be understandable to many, what the problem's nature is like. Jakob.scholbach (talk) 20:15, 24 September 2008 (UTC)

  I agree. However, our ability to include evaluation is limited to what reliable sources have to say. Here I leave the discussion in Willow's capable hands. Geometry guy 20:24, 24 September 2008 (UTC)

We're getting a little subjective here, no? Perhaps there are other ways of viewing the the complexity/elegance of the solutions? I need to 'fess to my own pair of glasses here. I've a childhood love for geometry (at least those that I can visualize easily) and true love lasts a lifetime; I find the geometrical solutions cool and fun, even a little bit racy, like a good paranormal romance novel. On the other hand, I find the algebraic solutions, well, a little anticlimactic, rather like a textbook on accounting. Sure, they're true — but are they Art? ;) I'm not this way, but a jaundiced eye might see Lie spheres as just a convenient tool for 5-column bookkeeping. (gasp — forgive me, G-guy! ;)

That said, I'm open to discussing the methods more in the lead, provided that it engages and doesn't scare off potentially interested readers. I think that might be possible; where there's a will — and people of good will and good sense — there's surely a way, don't you agree? :) Willow (talk) 00:29, 25 September 2008 (UTC)

Subjective, moi?! ;) Alas our poor eyes, jaundiced or otherwise, cannot fully appreciate the geometrical beauty of the Lie quadric, constrained as they are by having two dimensional retinas. If these were three dimensional instead, then those 5d column vectors would come to life as points in projective four space. We would then see the Lie quadric, a three dimensional hypersurface, criss-crossed with lines, stretching majestically out to infinity, just as easily as we can see its two dimensional analogue, the hyperboloid of one sheet. Now that's geometry! And perspective it is best painter's art. Geometry guy 17:58, 25 September 2008 (UTC)

• The wording "The eight solution circles differ from one another in how they enclose (or exclude) the three given circles; the eight circles correspond to the eight ways of enclosing or excluding the given circles." is a bit repetitive. Jakob.scholbach (talk) 19:16, 24 September 2008 (UTC)

• Comment on orientation. In plane geometry there are two ways that two circles can be tangent: from the inside (one circle inside the other) or from the outside. In Moebius (or inversive) geometry, these two ways are equivalent, because an inversion, centered on a point which is outside one of the circles but inside the other, exchanges the two pictures. Lie sphere geometry resolves the ambiguity in a different way: by orienting the circles. For visualization, such an orientation can be viewed as little arrows or chevrons pointing either clockwise or anticlockwise around the circle. Now we can redefine "tangent" to mean "oriented contact" rather than just "contact": two circles make oriented contact iff they touch and the arrows on the two circles are pointing the same way at the contact point. Lie sphere geometry keeps track of the orientations by letting the radius of a circle have a sign. Willow has denoted these signs by the s's. Geometry guy 08:44, 24 September 2008 (UTC)

Would epsilon or sigma be clearer than s? Some people who do not understand Lie groups nevertheless have met the usual conventions of mathematical notation. Septentrionalis PMAnderson 00:34, 25 September 2008 (UTC)

Personally, I would absorb the signs into the radii, but this may be a too advanced viewpoint. An ε would be preferable to an s at the college level, at least for math majors who've done permutations or tensor analysis. I'm not sure it would help the rest of the readership. Geometry guy 20:21, 25 September 2008 (UTC)

Not particularly arbitrary section break

• Qualified support. I've given Willow way too much grief and not much help over this article! She has made it into an impressive piece of work. I'm happy to support it now modulo the scientific citation guidelines. A general citation would be helpful to the reader at the beginning of the sections on : "Statement of the problem", "Intersecting hyperbolas", "Inversive methods", "Pairs of solutions by inversion", "Gergonne's solution", "Ten combinations of points, circles and lines" and "Number of solutions". I also added two tags. I think these are easily fixed, and hope to cross out "qualified" very soon! Geometry guy 21:59, 28 September 2008 (UTC)

  I've fixed all my concerns and a lot more besides. Change to "enthusiastic support". Geometry guy 13:30, 11 October 2008 (UTC)

• Comment - these could easily be opposed issues, but I believe they will be easily remedied: 1. "Statement of the problem" has no citations, and a citation needed tag. Please provide at least a primary source to look at. Two or three references, even if broad, would make the section better. 2. "History" End of the second paragraph in this section needs a citation. 3. "Intersecting hyperbolas" there is text sandwiched between two images. I don't know what can be done to fix this, but it may cause problems. 4. The captions of the "figures" are lengthy, and might be best to be mentioned in the article and point to (figure 1, 2, etc) instead of being in the caption. 5. "Algebraic solutions" should have some citations to where these equations come from so people can look at them later. If they, and the next sections, are using the references at the very end, could you please make that clearer at the beginning (something to say, according to ___ the equations are: ). 6. Figure 7 and 8 are left formatted and cause some formatting problems. Perhaps move to the right, or cut the captons down as I suggested above? 7. Sandwiching problems with figures 9 and 10, which could be fixed if you cut the captions down in the way suggested above. 8. Figure 11 causes the subheadings to format improperly. It is because it was an image put before a subheading. Could you fix this? Thats it. All formatting issues and a few primary sources needed to be refered to more often for people like myself to go back and check more easily. Ottava Rima (talk) 16:14, 30 September 2008 (UTC)

  I've fixed 1, 2, improved 3, and fixed 8. Regarding 4, other reviewers have pointed to the benefit of long captions, so I'm going to respect Willow's editorial choice here. Regarding 5, the general reference is at the beginning of the section per scientific citation guidelines. With no disrespect intended, the equations in this section are trivial. Regarding 6, what counts as a formatting problem is somewhat subjective. Regarding 7, I only get sandwiching problems on a widescreen browser, where it is not a problem: as I narrow the browser the sandwiching goes away. I've tested my improvement of issue 3 in a similar way. Geometry guy 13:30, 11 October 2008 (UTC)

  Further to this and Randomblue's comments, I've copyedited the captions (over 700 characters removed). Geometry guy 19:23, 11 October 2008 (UTC)

• Update: Thank you for everyone's thoughtful reviews and suggestions! I'll return to improve the article tomorrow. Per Sandy's advice, I've been working with Graham87 to address the accessibility of the images; I made Template:Tl as a workaround, but the ever-resourceful Simetrical heard our plea and updated the MediaWiki software itself in Template:Rev, which should go live soon (keep checking Special:Version!) I'm brooding over the lead and the formatting issues, which I'll hopefully amend to all your satisfactions tomorrow. Thanks for your patience! :) Willow (talk) 23:04, 30 September 2008 (UTC)

I need an accessibility expert reviewing FACs :-)) I just sent the Germanium editors to see Graham87; maybe you can help out if you become knowledgeable in this area? So far, I suspect I'm the only one reviewing for WP:ACCESS. SandyGeorgia (Talk) 06:07, 2 October 2008 (UTC)

• Support as I trust the concerns listed above (see G-guy, , Ottava etc.) will be addressed. Some potential wikilinks include:

• shrunk to zero [[radius]]
• generalizations to even [[higher dimension]]s
• tangent, although two [[Parallel (geometry)|parallel lines]]
• of distances to two [[fixed point]]s
• on the [[Sides of an equation|right-hand side]]
• [[Linear independence|linearly independent]]
• must lie on the [[Bisection|perpendicular bisector]] line
• pertain if the [[speed of sound]]
• is not [[Isotropy|isotropic]]
• boundaries of an [[infinite set]]

• are the [[curvature] and radius
• Ling.Nut ^{(talk—WP:3IAR)} 03:00, 1 October 2008 (UTC)

  Done except fixed point (mathematics), as this isn't the meaning here. Geometry guy 13:30, 11 October 2008 (UTC)

• Support in terms of prose. I do hate whatever template produces those lines of clunky equations. Why do they have to be so big? Why does the font have to be so unattractive? I see at least one example where the template hasn't been used on stand-alone equations, and it's much nicer (although almost too small now). Tony (talk) 05:06, 2 October 2008 (UTC)

Support per Jakob. Randomblue (talk) 16:41, 13 October 2008 (UTC)

The points I have raised are just examples, they certainly don't form an exhaustive list.

• 1) ref 34 has a date problem

  Fixed. --jbmurray (talk • contribs) 08:04, 6 October 2008 (UTC)

• 2) ref 43 has an inconsistent dot

  Sorry, I don't understand this. --jbmurray (talk • contribs) 08:04, 6 October 2008 (UTC)

  Willow's convention is to not put any dot in abbreviations of names. Fair enough, but it should be consistend throughout. For example, ref 43, "McMullen, Curtis T." needs to be changed. Randomblue (talk) 10:28, 6 October 2008 (UTC)

  Fixed. Ben (talk) 13:59, 6 October 2008 (UTC)

• 3) Why are Reye T and Gosset T redlinked?

  Unlinked. Ben (talk) 13:48, 6 October 2008 (UTC)

• 4) ref 52 and 53 have inconsistent dots

  And I don't understand this, either. --jbmurray (talk • contribs) 08:04, 6 October 2008 (UTC)

  Schmidt, R. O. -> Schmidt RO, etc. Randomblue (talk) 10:28, 6 October 2008 (UTC)

  Fixed. Ben (talk)

• 5) the four instances of 'Ecole' should be 'École'

  Fixed. --jbmurray (talk • contribs) 08:04, 6 October 2008 (UTC)

• 6) '(d=1)' -> bad spacing

  I don't see this, I'm afraid. --jbmurray (talk • contribs) 08:04, 6 October 2008 (UTC)

  "which is higher than that of a regular (or rectifiable) curve (d=1) but less than that of a plane (d=2)" there is also d=2 which should be d = 2 :) Randomblue (talk) 10:28, 6 October 2008 (UTC)

  Fixed. Ben (talk) 13:59, 6 October 2008 (UTC)

• 7) "Apollonius of Perga (ca. 262 BC – ca. 190 BC) posed and solved this famous problem in his work Επαφαι ("Tangencies"), which has been lost." How can we know he actually solved the problem? Maybe write 'thought to have solved'.

  His solution was described by others in works that haven't been lost. This is noted in the history section. Ben (talk) 13:48, 6 October 2008 (UTC)

  • Ok, but to what extent can we affirm he solved the problem using "descriptions" of his proof? Randomblue (talk) 14:07, 7 October 2008 (UTC)

    To the extent that this what the reliable sources say. :-) Geometry guy 13:30, 11 October 2008 (UTC)

• 8) "Many eminent mathematicians have developed various geometrical and algebraic methods for solving this problem". Why 'eminent'? Wikipedia:Avoid peacock terms

  I don't think many people would question it, but I've removed it anyway. Ben (talk) 13:48, 6 October 2008 (UTC)

• 9) "which has practical applications in trilateration". Why 'practical'? This seems redundant.

  Theoretical applications exist for many pieces of mathematics. Ben (talk) 13:48, 6 October 2008 (UTC)

• 10) "These eight circles represent the solutions to Apollonius' problem and are sometimes called Apollonius circles" I find this is misleading. When there are less solutions we still call them 'Apollonius circles', right?

• Reworded this the reduce the number of occurrences of "eight". Ben (talk) 13:48, 6 October 2008 (UTC)

• 11) In the second paragraph of the lead I count 5 occurrences of "eight"

• See above. Ben (talk) 13:48, 6 October 2008 (UTC)

• 12) "difference d1−d2" wrong spacing

• I think I fixed this. Ben (talk) 13:48, 6 October 2008 (UTC)

• 13) "Viète solved all ten of these cases using only compass and straightedge" needs rephrasing. Viète certainly used other things to solve these cases (such as hard work, thought, paper, ...). What you mean is "Viète found all solutions to these ten cases, and the constructions can be made using only compass and straightedge."

• Ahh, I think the current wording is better. To solve a problem using only a compass and straightedge has a precise meaning, and a wikilink is provided earlier in the text to explain this if someone isn't sure. Ben (talk) 13:48, 6 October 2008 (UTC)

• I didn't say my wording was better, just that the current wording needs to be changed (to be specific, the verb "solved" needs to be changed). Randomblue (talk) 14:07, 7 October 2008 (UTC)

  Fixed by adding "constructions". Geometry guy 14:05, 11 October 2008 (UTC)

• 14) in the fourth paragraph of the "Viète's reconstruction" section, I don't understand the convention for explicitly describing the cases. For example, "CPP case (a circle and two points)" but "CLP case".

• A circle, a line and a point. The convention is noted in the first paragraph. Ben (talk) 13:48, 6 October 2008 (UTC)

• Oops, I read over this again this morning and I understand what your point was now. It looks like the convention is to explicitly describe the cases the first time they're mentioned, and then just use the abbreviation after that. Ben (talk) 05:25, 7 October 2008 (UTC)

• 15) "and again in 1936 by Nobel laureate Frederick Soddy" why is 'Nobel laureate' relevant?

• I guess it's not, but it is interesting. Removed it. Ben (talk) 13:48, 6 October 2008 (UTC)

• 16 "to construct H. Rademacher's contour for complex integration" is the text going to be consistent with the references with regard to dots in names?

• I can't find this sentence. Ben (talk) 13:48, 6 October 2008 (UTC)

• Very last section. Randomblue (talk) 14:07, 7 October 2008 (UTC)

No it isn't. The text uses full names or surnames only. I've fixed H. -> Hans. Geometry guy 14:15, 11 October 2008 (UTC)

• 17) "Apollonius' problem can be framed as a system of three coupled quadratic equations." what is a 'coupled quadratic equation'?

• A quote from a book I came across while looking for a way to be precise: "problems describing different types of mechanics may be coupled through a variety of mechanisms with varying degrees of interaction. Both of these characteristics are difficult to generalise and quantify and lead to a certain vagueness when discussing coupled problems in general terms." Nevertheless, I had a go at clearing this up. Ben (talk) 13:48, 6 October 2008 (UTC)

• Sorry, I feel a bit stupid, but I still don't understand what this means. Randomblue (talk) 14:07, 7 October 2008 (UTC)

  Nor do I, but I do understand what a quadratic equation is and have wikilinked it. Here "coupled" simply means that the three equations involve all three unknowns x_s, y_s and r_s, i.e., they are not separate quadratic equations for each unknown. I've wikilinked it. Geometry guy 14:05, 11 October 2008 (UTC)

• 18) "The three signs s1, s2 and s3" -> s1, s2 and s3 are not signs

• It's useful to be able to refer to the "signs" as a group, so I've reworded it a bit to make the definition explicit. I hope that is ok. Ben (talk) 13:48, 6 October 2008 (UTC)

• 19) "(2 × 2 × 2 = 8)" Maybe 2^3 would be a more appropriate.

• I think the point here is to visually highlight that each sign gives rise to two possible solutions. There is nothing wrong with 2^3 if you really think it would be better though. Ben (talk) 13:48, 6 October 2008 (UTC)

• 20) "The general system of three equations may be solved by the method of resultants." Good to know! So what is a resultant? I would like to take this as an example to show that the level varies extremely within the article. Some trivial points sometimes take 3 lines to explain ("Apollonius' problem can be framed as a system of three coupled quadratic equations.[23] Since the three given circles and any solution circle must lie in the same plane, their positions can be specified in terms of the (x, y) coordinates of their centers. For example, the center positions of the three given circles may be written as (x1, y1), (x2, y2) and (x3, y3), whereas that of a solution circle can be written as (xs, ys). Similarly, the radii of the given circles and a solution circle can be written as r1, r2, r3 and rs, respectively."), and then bluntly, in half a line, the reader is assumed to know resultants.

• The method is described in detail following that sentence. Ben (talk) 13:48, 6 October 2008 (UTC)

• Ok, but I still don't know what a resultant it. Maybe add a wikilink. Randomblue (talk) 14:07, 7 October 2008 (UTC)

  Done (not by me). Geometry guy 14:05, 11 October 2008 (UTC)

• 21) images: 'succinct captions' is part of criterion 3. Clearly not satisfied for nearly all images.

• Agreed, but I think some leniency should be given for mathematical images. According to WP:CAP, make sure the reader does not miss the essentials in the picture. The images in this article contain a lot of information that, with a good explanation, go a long way to help the reader see what is going on - even if the article is over their head. Ben (talk) 13:48, 6 October 2008 (UTC)

• Agreed, but a lot of useless (by which I means already explained in the article) appears in the captions. For example, in Figure 2, the following is useless: "In each pair, one solution circle encloses the given circles that are excluded by the other solution, and vice versa. For example, the larger blue solution encloses the two larger given circles, but excludes the smallest; the smaller blue solution does the reverse." Randomblue (talk) 14:07, 7 October 2008 (UTC)

  I've copyedited the captions (a reduction of over 700 characters), but I don't want to remove information that is useful to the reader in interpreting the symbols, colours, etc. appearing in the image. Geometry guy 19:23, 11 October 2008 (UTC)

• 22) "one example is the annular solution method of H. S. M. Coxeter" dots?

• Not sure about this one?

• Fixed. Ben (talk) 13:59, 6 October 2008 (UTC)

• 23) "using only a compass and straightedge" delete a before compass or add a before straightedge

  Done. Ben (talk) 13:48, 6 October 2008 (UTC)

• 24) "Many simple constructions are impossible using only these tools" 'simple'? POV. Actually, because they can't be perfomed using compass and straightedge, they could be described as difficult!

  Removed simple. Ben (talk) 13:48, 6 October 2008 (UTC)

• 25) "Therefore, van Roomen's solution—which uses the intersection of two hyperbolas—might violate the straightedge-and-compass requirement." 'might'. So we're not actually sure?

• Reworded. Ben (talk) 13:48, 6 October 2008 (UTC)

• 26) "Prior to Viète's solution, the mathematician Regiomontanus doubted whether Apollonius' problem could be solved by straightedge and compass." Is it necessary to point out that Regiomontanus was a mathematician?

• I guess not. Ben (talk) 13:48, 6 October 2008 (UTC)

• 27) "Viète first solved the simpler special cases of Apollonius' problem" What do you mean by 'the simpler special cases'? Do you 'some of the special cases'? Also, I'd avoid using "simpler".

• Reworded and added to the example simple case to show why it's simple. Ben (talk) 13:48, 6 October 2008 (UTC)

• 28) "Apollonius' own book on this problem—entitled Επαφαι ("Tangencies", Latin: De tactionibus, De contactibus)" why is the Latin title relevant?

• Some older books will refer to it by its Latin title. Ben (talk) 13:48, 6 October 2008 (UTC)

• 29) "Whereas Poncelet's proof relies on homothetic centers of circles and the power of a point theorem, Gergonne's method exploits the conjugate relation between lines and their poles in a circle. Methods using circle inversion were pioneered by Julius Petersen in 1879;[15] one example is the annular solution method of H. S. M. Coxeter.[16] Another approach uses Lie sphere geometry" In two lines you manage to fit in many 'sophiticated' tools with no explanation whatsoever. This could frighten people, who might expect a 'soft' introduction in the 'History' section.

• I think up until this point, the history section keeps things fairly straight forward for the interested reader. The point of these last two paragraphs is to note that other methods have been developed (with a hint as to how), with supporting wikilinks and references for the more experienced reader, and discussion of the methods later in the article. The less experienced reader can still take something away from the paragraphs, by noting that people have worked on the problem even in (relatively!) recent times. On the other hand, not mentioning these would leave a gap in the history in my opinion. Ben (talk) 13:48, 6 October 2008 (UTC)

• 30) "(In complex analysis, "infinity" is defined precisely in terms of the Riemann sphere.)" 'defined precisely', sounds redundant.

• Removed precisely. Ben (talk) 13:48, 6 October 2008 (UTC)

• 31) "In general, any three distinct circles have a unique circle—the radical circle—that intersects all of them perpendicularly" citation needed. Also, 'radical circle' could do with a wikilink.

• wikilinked Ben (talk) 13:48, 6 October 2008 (UTC)

• 32) First sentence of 'Pairs of solutions by inversion' : "solutions to Apollonius' problem generally occur in pairs"; later in the section, "solutions to Apollonius' problem generally occur in pairs"

• Fixed Ben (talk) 13:48, 6 October 2008 (UTC)

Ok, I tried to deal with as many of these as I could. Ben (talk) 13:48, 6 October 2008 (UTC)

• 33) "Inversion has the useful property that lines and circles are always transformed into lines and circles, and points are always transformed into points." -> reference needed

  This is standard, and covered by the general ref that I added to the beginning of the section. Geometry guy 14:05, 11 October 2008 (UTC)

• 34) the serial comma is not respected throughout, see for example "three given circles of radii r1, r2 and r3"

  I prefer to neither respect nor disrespect the serial comma :-) It is useful where it adds clarity. Geometry guy 14:05, 11 October 2008 (UTC)

• 35) still remains point problems in references 34, 54 and abbreviation problems in references 52, 53

  I've fixed what you call the "point problems." I don't know what you mean by "abbreviation problems." --jbmurray (talk • contribs) 20:58, 9 October 2008 (UTC)

  First names of authors in full. I've fixed them I think. Geometry guy 14:05, 11 October 2008 (UTC)

• 36) the citations needed need to be dealt with

  I added the citations needed. And I've fixed them :-) Geometry guy 14:05, 11 October 2008 (UTC)

• 37) Mathematical facts should be referenced, as for the article Group (mathematics). For example, the whole "method of resultants" goes unreferenced. Randomblue (talk) 14:07, 7 October 2008 (UTC)

  I've added some general citations to cover standard facts. Geometry guy 14:05, 11 October 2008 (UTC)

• 38) "Célèbres Problêmes mathématiques." The current spelling for 'problem' in French is 'problème', not 'problême'. However, the old spelling might have been 'problême', although I doubt it. Also, French titles usually only have a capital on the first word.

  Fixed. --jbmurray (talk • contribs) 20:47, 9 October 2008 (UTC)

• 39) "Pappus d'Alexandrie: La Collection Mathématique" check capitalization.

  Fixed. --jbmurray (talk • contribs) 20:47, 9 October 2008 (UTC)

• 40) "Traité de Géométrie" capitalization

  Fixed. --jbmurray (talk • contribs) 20:47, 9 October 2008 (UTC)

• 41) "Inversion in a circle with center O and radius R consists of the following operation (Figure 5): every point P is mapped into a new point P' such that O, P, and P' are collinear, and the product of the distances of P and P' to the center O equal the radius R squared" -> 'inversion' and 'collinear' could do with wikilinks; why 'new'?, the point need not be new; why 'every'?, your definition does not apply to O
• 42) 'Lie quadric' could go with a wikilink

  Added a wikilink, but there is no page on this yet. I don't know enough about the space to create anything more meaningful than what is given in this article, but the description given in this article suffices for what follows. Ben (talk) 11:01, 10 October 2008 (UTC)

  I've made this into a redirect to the beautiful article on Lie sphere geometry :-) Geometry guy 14:05, 11 October 2008 (UTC)

• 43) "This formula shows that if two quadric vectors X1 and X2 are orthogonal (perpendicular) to one another—mathematically, if (X1|X2) = 0—" why is 'orthogonal' not 'mathematical'?

  Reworded. Ben (talk) 11:01, 10 October 2008 (UTC)

• 44) Geometry: Euclid and beyond -> capitalization of English titles

  Fixed. --jbmurray (talk • contribs) 20:49, 9 October 2008 (UTC)

• 45) "Apollonius problem is to construct one or more circles tangent to three given objects in a plane, which may be circles, points or lines. Thus, there are ten types of Apollonius' problem" 'Thus' seems to be the wrong connecting word. The result ("there are ten types of Apollonius' problem") doesn't follow immediately from the premise. Also, the result needs a reference.

  Reworded. Still to look for references for this and the citation needed tags mentioned above and below. Ben (talk) 11:15, 10 October 2008 (UTC)

Comments on the sources:

• "Statement of a problem" section could use a citation or two. Especially where the article cites GPS as an application of the problem, a citation could go there.
• There also is a citation needed tag in the applications section. I think a reference belongs in the beginning section of that paragraph.
• Otherwise, I think this article meets the FAC criteria 1c, which concerns sourcing. I did some looking around on Google Books and Google Scholar to see if key papers and books are included as citations in this article, and that appears to be the case. --Aude (talk) 04:51, 4 October 2008 (UTC)

  Done. (Fixed my own tags and more besides.) Geometry guy 14:05, 11 October 2008 (UTC)

• Status? Does anyone know the status? WillowW hasn't edited since the 30th. SandyGeorgia (Talk) 19:49, 4 October 2008 (UTC)

The lead

• Reminder Since Willow seems to be on a wikibreak I just want (in case it's somehow lost in the long running of the candidacy) to remind those of you who kindly work on this article at the moment that there is still a central problem with the article, namely the lead section. Please see my comments above. For the moment this issue still prevents me from supporting this article being featured. Jakob.scholbach (talk) 19:48, 11 October 2008 (UTC)

  Ah, yes, good point. Luckily, I'm allegedly an expert on WP:LEAD. How does it look now? Geometry guy 20:58, 11 October 2008 (UTC)

Yes, is better. But could still be improved: first, it's still unbalanced: generalizations get lot of attention, inversive methods are not mentioned. I mean I don't wanna insist that every subsection has to be covered proportionally, but it's a point. More importantly, the lead doesn't flow that well (e.g. I'd incorporate the limiting cases in the description of the solutions, since this is factually linked. Likewise the number of solutions (8) should be connected to the (not yet extant) mentioning of the inversive strategy). Another example where the flow is interrupted is the line break after the first paragraph. It closes with history, and the next paragraph starts with history, too. Finally, I'd love to have a qualitative description of the development of the solution methods. Currently they are soberly listed, but convey little or no enthusiasm to people not acquanted to this stuff. Also "..., or additional symmetries" doesn't speak to me at all, instead it may even be distracting since people may wonder: "he? a circle is already pretty symmetric. what additional syms?". Jakob.scholbach (talk) 22:42, 11 October 2008 (UTC)

Do you want to have a go at it? Geometry guy 22:48, 11 October 2008 (UTC)

Okay, I've given in another go, incorporating some of your suggestions, while also following Willow's view. Geometry guy 19:42, 12 October 2008 (UTC)

OK, just at the same time I pasted my 2c (edit confl.). I tried to emphasize the evolution of the ideas behind the solutions, thus somewhat merging historical order with mathematical stuff. What do you think? Jakob.scholbach (talk) 20:01, 12 October 2008 (UTC)

My rewrite also merged history with solution methods, so we seem to be broadly agreed on that. I like the "solutions by algebraic means" and the "pairs of solutions" stuff, but I find the last three sentences of paragraph three somewhat empty of content and would rather tell the reader something interesting instead. What do you make of my version? Geometry guy 20:28, 12 October 2008 (UTC)

Nice. Good that we have the article history! OK, my sentence with Gergonne is not great. In principle I do like the following two ones, though. Rewording and adding a little more flesh may be in order, certainly. Your version: Giving some trendy keywords as you do with GPS and Hardy-littlewood (I don't know if the latter rings a bell with general readers, though) is certainly a good method to remedy aridity or emptiness. I would personally refrain from putting two much folklore like the poem and the letter to the princess. Are you up to merging the two drafts? Jakob.scholbach (talk) 20:42, 12 October 2008 (UTC)

Thanks. I can give it a go. One reason I added the (reliably sourced) folklore is to replace assertions like "important special case" with some facts about the special case that might indicate to the reader that it is important and interesting, but I guess I overdid it. I similarly don't like "to great effect", but there are facts that can be deployed there. Geometry guy 21:06, 12 October 2008 (UTC)

Okay, I gave it a shot. Inevitably, incorporating both our thoughts has added a few hundred bytes. Also inevitably, I've probably tended to favour my own structure. Geometry guy 22:41, 12 October 2008 (UTC)

Looks great. Another issue I just refound when striking my above comment is the question with tangency of parallel lines (see above). I personally think they shouldn't be considered tangent (since their intersection multiplicity is one, not two, at infinity) but Willow seemed to be of this opinion. This is not that big of a deal, one could simply remove this statement if it is wrong (which I believe). Jakob.scholbach (talk) 06:17, 13 October 2008 (UTC)

I had a quick look above, but the discussion I found was under a stricken comment. Are you concerned that two distinct parallel lines are described as being tangent at infinity? Ben (talk) 06:41, 13 October 2008 (UTC)

Yes, exactly. Perhaps I shouldn't have striked out the comment, but later I became aware of this being still an issue. For convenience I post it here again:

• Statement section: I don't understand why you talk about two lines being tangent or not. I did not check the details, but based on the common interpretation of tangency meaning intersection multiplicity two or higher, two parallel (distinct) lines do not qualify as tangent (use Bezout's theorem, for example, which computes their intersection multiplicity (at infinity) to be 1). I suggest finding a ref for this claim, if you consider it still right, so as to escape OR. Jakob.scholbach (talk) 06:47, 13 October 2008 (UTC)

  You're thinking in terms of projective plane geometry, where there is a line at infinity, and parallel lines do indeed meet with multiplicity one. That's not relevant here: the natural geometry is Moebius or inversive geometry, where there is just a point at infinity (as in the Riemann sphere). In that case all lines meet at infinity (obviously!) and parallel lines are tangent at infinity. I corrected this issue some time ago. Is further clarification needed? Geometry guy 08:29, 13 October 2008 (UTC)

  • All right. That makes sense. I hadn't caught your edit there. The current wording "two parallel lines can be considered as tangent at a point at infinity." is still vague. With your explanation here it becomes understandable. So, perhaps add what you just explained me. A precise reference for this idea would be good, too. So, I'm happy to support this article being featured. Jakob.scholbach (talk) 08:43, 13 October 2008 (UTC)

  I think Geometry Guy has already added a reference, but I've checked that it's discussed in Needham's Visual Complex Analysis so I can add another if you want. Ben (talk) 10:38, 13 October 2008 (UTC)

  One ref per fact is fine! Jakob.scholbach (talk) 16:07, 13 October 2008 (UTC)

  The Needham ref might be more accessible (in every sense) for more readers, so I don't mind if you want to substitute it. Anyway, thank you Ben, Jakob and Randomblue (et al.) for helping to do justice to Willow's fine work. Geometry guy 18:18, 13 October 2008 (UTC)

The above discussion is preserved as an archive. Please do not modify it. No further edits should be made to this page.