Talk:Problem of Apollonius/Archive 1

Proposed merger?
I definitely think this article should not be merged with Circles of Apollonius. It is an accepted technical name for a well-defined problem in geometry that has higher-dimensional analogs (see Soddy's hexlet). By contrast, Circles of Apollonius is a kind of disambiguation article, explaining several definitions that have basically nothing to do with one another. I tried to improve it as much as I could when I was first starting here. Maybe the best solution would be to make it a formal disambiguation page, with separate articles for the subtopics, such as this one? Open to ideas, Willow 01:00, 3 April 2007 (UTC)

The two articles should be merged
According to my opinion, the two pages should certainly be merged. Quoting the rules of merging from Wikipedia itself:

"Overlap - There are two or more pages on related subjects that have a large overlap. Wikipedia is not a dictionary; there does not need to be a separate entry for every concept in the universe. For example, "Flammable" and "Non-flammable" can both be explained in an article on Flammability."

I think it's exactly the case - Circles of Apollonius defines the terms and Problem of Apollonius tries to describe a way of solution. And it's wrong, by the way - circle inversion can never convert two circles which don't have any intersection into two tangential ones. The method described is valid only in case the circles are touching or intersecting each other. Mh liv01 (talk) 11:52, 28 January 2008 (UTC)


 * Unfortunately, I don't agree, at least not yet. if you accept what I wrote there, the term "circles of Apollonius" is ambiguous, having at least four different meanings.  One of those four is this "problem of Apollonius", but the other three usages are not related.  We might agreed that all four are so short that they could be subsumed into one article, despite being unrelated, but I'm not ready to do that just yet.  There's a lot I could add to this article yet.  Willow (talk) 19:18, 29 January 2008 (UTC)


 * PS. The solution described is not wrong, as best as I can see. If you re-read the article, you'll see that we specify that the radii have to be increased or decreased by a common amount so that two of the three circles touch.  I've tried to make that clear with the new animation.  Hoping that you find it enjoyable and enlightening, Willow (talk) 19:18, 29 January 2008 (UTC)

Correct solution?
I much appreciate the effort made to extend the page and the references added, I learned really interesting new things about the backgroung of the problem. However, I am afraid the discussion about merging the two articles remains irrelevant until the content isn't corrected. I am talking about the geometric solution of the Apollonius problem - indeed in the original article you wrote about decreasing the radii of all the circles for a constant. But this is exactly the problem - doing this, you are getting completely different assignment. Unlike circle inversion which conserves intersection and angles, your transformation of the circles brings different set of solutions.

Let's say you do you transform by increasing the radii and inverting the circles by the circle inversion. If you then find your solutions and invert everything back by the circle inversion, you will still have three circles and other which will be tangential to all of them. But what next? Do you decrease the diameters of the original circles back? Then your solution circles can't be tangential to all of the original circles, any more. Also, in case of three circles you can get only four solutions (those which appear in your animation), but generally for three circles there can be up to eight solutions. Have you tried to take a pencil, ruler and compass and draw the solution? Without any intention to offend you, I must say it is not possible by this way.


 * The trick is to expand the solution circle while shrinking the original circles back to their original diameters. Willow (talk) 18:54, 4 February 2008 (UTC)

I can offer a solution of general Apollonius problem using circle inversion: The aim of this solution is to transform two of the three circles to concentric ones and the third one to their annulus - the solution is then trivial.

It uses an important feature of the circle inversion: angle conservation - any two curves intersecting each other under given angle have the same intersecting angle after they are inverted. Further it is needed to know some features of the radical axis (see e.g. http://whistleralley.com/inversion/inversion.htm).

The radical axis AC 1C2 of two given circles C1, C2 consists of centers of all circles which intersect C1 and C2 under the same angle. It means that on AC 1C2 there also lie the centers of circles which intersect both original circles orthogonally. They all are intersecting each other in two points P1, P2 lying on the common axis of C1, C2 - they are Circles of Apollonius (and here could be included the paragraph or at least reference to the article about the Circles of Apollonius, by the way - this is exactly the link which maybe wasn't clear at the beginning). If we set the circle inversion defined by a circle I having center in either P1 or P2 (let's say it will be P1. And to make it simpler the circle may pass through P2), the set of the Circles of Apollonius will be transformed to set of lines which are all passing through one point (in our case they all are intersecting each other in P1 - the center of the circle inversion). As a consequence of angular conservation of circle inversion the circles C1, C2 will be transformed into concentric circles C1', C2'. (They were orthogonal to the Circles of Apollonius, which are now transformed to the set of lines and the orthogonality must be conserved). The third circle will lie in their annulus.

In the past I made some pictures I could attach. When I get a chance to reach them, I will do it. Regards, Mh liv01 (talk) 16:50, 4 February 2008 (UTC)


 * Hi Mh liv01,


 * Thanks for your fun and really interesting letter! I wasn't offended at all, so no worries; I'm pretty cheery by nature.  :)  I like your solution very much; I'd already seen it on the web and wanted to make an animation for it, just as I did for the other one.


 * However, the best solutions I've found are those in which one circle is shrunk to a point. Then the problem becomes finding a circle that is tangent to two others and passes through a point P.  Inversion in a circle centered on P converts the solution circles to straight lines tangent to the images of the other two circles, which are easily constructed.  Then re-inversion yields the solution circles.


 * My solution does work, as follows. After finding the tangent yellow circle as in the animation, you re-invert, as you said.  Then you shrink the three original circles back to their original sizes, and expand the solution circle by the same amount.  Tangency is preserved throughout.


 * The problem with all of these shrinking and expanding solutions is that they're special for each of the eight solution types. For example, you can shrink all of them until it reaches a point; or shrink two of them and expand the third.  That gives four special cases, which may be solved as described two paragraphs above.  I believe that that would cover the solution space, but I only intuit that; I haven't proven it, exactly.  I arrived at that by thinking about the algebraic solutions, can I show you?  Let the three algebraic equations be



\left( x - x_{k} \right)^{2} + \left( y - y_{k} \right)^{2} = \left( r - s_{k}^{\alpha} r_{k} \right)^{2} $$


 * where k=1,2,3 is an index over the circles, α=1,...,8 is an index for the solution circles, and sk=±1 represents the three signs for the αth solution. It's not that hard to show that the solution centers (x, y) are linear functions of the solution radius r



x = M + Nr $$



y = P + Qr $$


 * Then we can get apparently get two solutions for r by substituting these formulae into the quadratic equation



\left( M + N^{\alpha}r - x_{k} \right)^{2} + \left( P + Q^{\alpha}r - y_{k} \right)^{2} = \left( r - s_{k}^{\alpha} r_{k} \right)^{2} $$


 * For convenience, here are the constants I got:



M = \frac{V_{3}T_{2} - T_{3}V_{2}}{U_{2}V_{3} - U_{3}V_{2}} $$



N^{\alpha} = \frac{V_{2}W_{3}^{\alpha} - W_{2}^{\alpha}V_{3}}{U_{2}V_{3} - U_{3}V_{2}} $$



P = \frac{U_{2}T_{3} - T_{2}U_{3}}{U_{2}V_{3} - U_{3}V_{2}} $$



Q^{\alpha} = \frac{U_{3}W_{2}^{\alpha} - W_{3}^{\alpha}U_{2}}{U_{2}V_{3} - U_{3}V_{2}} $$


 * where for j=2,3



U_{j} = 2 \left( x_{1} - x_{j} \right) $$



V_{j} = 2 \left( y_{1} - y_{j} \right) $$



W_{j}^{\alpha} = 2 \left( s_{j}^{\alpha} r_{j} - s_{1}^{\alpha} r_{1} \right) $$



T_{j} = \left( x_{1}^{2} - x_{j}^{2} \right) + \left( y_{1}^{2} - y_{j}^{2} \right) - \left( r_{1}^{2} - r_{j}^{2} \right) $$


 * This might suggest that there are 16 independent solutions, two solutions for every value of α. But I think there might be a sign symmetry that we can take advantage of to reduce it to 8.  If the signs sk are all flipped, then W, Q and N all change sign as well, right?  To restore equality, we need only flip the sign of r as well, and all is well.  :)  That's why we need consider only four of the eight possible special cases, I think.  Willow (talk) 18:49, 4 February 2008 (UTC)


 * Hi, it occurred to me that you might want to try out the solution yourself, so here are the coefficients of the final quadratic equation A r2 + Br + C = 0.



A = \left( N^{\alpha} \right)^{2} + \left( Q^{\alpha} \right)^{2} - 1 $$



B = 2 N^{\alpha} \Delta M + 2 Q^{\alpha} \Delta P + 2 s_{1} r_{1} $$



C = \left( \Delta M \right)^{2} + \left( \Delta P \right)^{2} - r_{1}^{2} $$


 * where the two variables are defined for brevity, ΔM = M - x1 and ΔP = P - y1. Hoping that you find this helpful, Willow (talk) 17:59, 8 February 2008 (UTC)


 * Hi Willow! It's amazing how much effort did you put into the article since I last looked at it. Nice job! You are right, the solutions using the "increase/decrease" the radii are correct - the centres of the solution circles don't change when you are changing the radii of the original circles (this was my main but odd concern). Also, it's clear that you can get entire set of the solutions - you only have to repeat the procedure with the circle inversion three times (and this maybe a bit degrades the simplicity of the solution). Good luck! Mh liv01 (talk) 15:46, 19 February 2008 (UTC)

Splendid job
Well done Willow. You are doing a splendid job. And the page is still delightfully free from vandals. I fear that won't last. However, if you did less saves between edits the history would be less cluttered. Xxanthippe (talk) 00:10, 10 February 2008 (UTC).


 * Hi Xanthippe, thank you for your very kind message! :) I'm sorry that I didn't reply earlier, I only just noticed your letter.  I'm about to add some more figures to the Gergonne solution, which I think you'll like, and then I think I'll take a rest.  Inshallah, the vandals will not think it worth their while to attack such a low-traffic article.  Or perhaps they'll be miraculously converted by its beauty and goodness, like the lioness in your namesake's story. ;)


 * I understand about saving less frequently, and usually I do. Unfortunately, I'm a little scatter-brained on this article?  It's hard for me to keep everything in my head at once, so I know that if I don't fix a problem the moment I notice it, I'm all-too-likely to forget about it.  Part of the many saves is also my obdurate optimism in using the "Preview" button too rarely, or in not noticing any errors when I do.  You'll see a lot of "typo-this" and "typo-that" in my edits. :P  Willow (talk) 13:43, 10 February 2008 (UTC)
 * One strategy is to create a draft page on your user page and do all the editing on that. When completed it can be copied onto the article page and the draft page deleted. What a nice person you are to interact with Willow! Nicer than some I have been dealing with recently. Xxanthippe (talk) 21:56, 10 February 2008 (UTC).

Numerical tables issue
I've mentioned this to Willow before, but I am unconvinced that providing numerical tables for an example adds any insight to this article. What are readers supposed to do with such tables? It would take quite a savant too look at them and see the solution, so everyone else has to plot the points painstakingly by hand, or copy the data and figure out how to paste it into some software package or computer program to display it. This is not encyclopaedic content. Geometry guy 00:02, 27 April 2008 (UTC)


 * Consider your point as taken; the unencyclopedic tables will be removed.


 * I was mainly inspired by empathy for young students (say, high-school students) coming to the article. When I first came to this problem, I didn't have an example to experiment with, which I found frustrating.  What are the readers supposed to do with such tables? Umm, actually draw out the solutions?  It's really not that much trouble to draw five circles on graph paper, especially when three of them have integer radii and center positions, no?  Personally, I find that constructing something physically with pen and paper, not merely staring at a picture, helps me to get an intuitive understanding. Also, it's sometimes easier to understand a complicated method (such as Gergonne's solution) by working backwards from the solutions, rather than forwards from the premises, no?  Finally, if someone else had provided me with a solution, I could've made the Figures more easily, without having to spend days writing and debugging a new computer program; someone else may feel a similar need someday, so I would spare them that work.  Willow (talk) 06:29, 2 May 2008 (UTC)


 * You make a good case for including the tables! Well, at least they are preserved on the talk page for now. I'm sorry I didn't dive in today. I meant to, but there were distractions. The Lie geometry section looks fun! Geometry guy 22:28, 2 May 2008 (UTC)


 * Please don't worry about it, and I hope you do have fun with it! :) I was surprised that the Lie geometry was so simple, when it looked so scary in advance. But of course, I may have made a gross misunderstanding.  I did leave out the orthogonality conditions ensuring that the solution was a circle and not a point or a line, namely, (1,0,0,0,0) and (0,0,0,0,1), respectively; that seemed like too much trivial detail for this article?


 * I think my own contributions here may be winding down, except for a picture or two and maybe something about Viete's solution, so you should feel free to work on it however seems good to you.  I did indeed preserve the tables here for posterity — well, at least for the next befuddled Willow — just as I did the solution to the algebraic equations above.  Articles come and go, but Talk pages endure. ;)  Dashing off to work once I do two more little things, Willow (talk) 22:42, 2 May 2008 (UTC)

Table 1
It may be helpful to illustrate a complete solution to one example of Apollonius' problem. In this example, the three given circles are denoted as C1, C2, and C3, as shown above in red, green and blue, respectively. The radii of these circles are 1, 2, and 3, respectively, and their Cartesian coordinates are (-2, -1), (4, 0), and (-1, 4), respectively. This example problem has eight solutions, which are given in the table below and shown in pairs, as the pink and black circles in the gallery above. The solutions in each pair are conjugate to one another, in that the pink solution encloses the given circles excluded by the black solution, and vice versa.

Here, rs equals the radius of a solution circle and (xs, ys) give the Cartesian coordinates of its center. The three signs s1–3 correspond to whether the solution circle encloses or excludes the three given circles, as may be seen in the gallery above. For example, the solution circle 3A has the signs "+ – +", which indicates that it encloses the first and third given circles, red C1 and blue C3, but excludes the second, green C2. Expressed in more technical language, the solution circle 3A is internally tangent to C1 and C3, but externally tangent to C2. In the next section, these signs determine whether the radius of a given circle should be added or subtracted in a set of three algebraic equations that describe Apollonius' problem. The points (xt1, yt1), (xt2, yt2) and (xt3, yt3) are the points of tangency between the solution circle and the three given circles C1, C2, and C3, respectively.

Table 2
This table gives the numerical parameters of the Gergonne solution for the example problem given above. The first three columns give a point (xR, yR) on the radical axis R for that pair of solution circles, and the angle θR that R makes with the x-axis. The next three columns give the angles (also with respect to the x-axis) of the three Gergonne lines L1-3 emanating from the radical center G. The next six columns give the angles (also with respect to the x-axis) of the six tangent lines at the points of tangency A1-3 and B1-3; these are the tangent lines that intersect on the radical axis R. The final six columns give the positions (xP1, yP1), (xP2, yP2), and (xP3, yP3) of the three poles P1-3 of the radical axis R in the respective given circles C1-3.

The lead
I think the lead may need the Geometry guy treatment. My feeling is that the article needs a first section which explains the problem, based on the current lead, and then the new lead would be able both to introduce and to summarize the article more effectively and concisely. Do other (highly esteemed :) editors agree? Geometry guy 23:37, 6 May 2008 (UTC)


 * Bien sur, avec grand plaisir, Votre Eminence! :) Or maybe I should say: with goodly gree, your Grace!


 * Concision is not my strength, it's really plain to see. :P My prose is more like a spider plant, or maybe more nicely like a typical strawberry: sprawling and sending off new shoots in every direction. ;)  I fear there's no hope for a cure, I tell stories the same way: "Oh, yes, and then there was the time..."  Ordinarily, I'd worry that an expert would make it too concise or too elevated for our friends to read; but I know that the article will be in good hands with you. :) Willow (talk) 11:08, 7 May 2008 (UTC)


 * Yes, I think there's some more room to summarize. Paragraph 2 could lose most of its details and merge with 3. Also, the "Applications" section should get at a sentence of two in the last paragraph. Melchoir (talk) 06:47, 14 May 2008 (UTC)


 * I took your advice, Melchoir, thank you! It made sense to remove the parts of paragraph two that were redundant with the caption of Figure 1.  Does it seem OK now?  I'll add something about the Applications. Willow (talk) 20:07, 14 May 2008 (UTC)


 * Yes, much better, but I actually don't think the caption needs most of that info either. (See below). In other news, are the citations in the lead really necessary? The same citations are repeated in the body, apparently for the same facts. Melchoir (talk) 08:03, 15 May 2008 (UTC)


 * Probably lead issues are best addressed after restructuring issues are considered below. However, one restructuring issue that I would emphasise (and would have tried to tackle days ago if I had more energy and time) is the need for a preliminary section on the problem in general terms. Starting out with "limiting cases" is just unfair to the reader, and contrary to WP:LEAD. The lead section may set the scene and put the reader in the mood, but the rest of the article needs to reflect the lead (aka, the lead summarizes the article). I suggest imagining that you had been given a general idea what the Apollonius problem was about, and had decided to check up on Wikipedia, but for some reason the lead was invisible. The rest of the article needs to fill in the details of the ideas that stimulated your curiosity. A preliminary section on the problem and its generic solution is essential for that. Such a section will, in turn, help to make the lead more concise so that it attracts more readers and stimulates them to read on. Geometry guy 22:35, 15 May 2008 (UTC)

Random issues
Overall it looks like a great article! Sorry about the bullet format, but it's easier on me...


 * Hey Melchoir, I can't tell you how grateful I am that you came and gave such an excellent review! I'll try to address your concerns ASAP. :) Willow (talk) 20:04, 14 May 2008 (UTC)






 * In the lead image Image:Apollonius all solutions.png, it wasn't immediately obvious to me that the given circles are the same in all four corners. (To be honest, my first impression was that the image suggests that any given configuration has only two solutions, which obviously conflicts with the text.) There's a lot of conflicting visual information that gets in the way, relative to something like Image:Apolloniuscircle.gif. Have you tried dropping the labels and putting all eight solutions on a single frame?


 * I originally wanted to do this, but it gets hard to see the individual solutions. I'll tinker with it a little; maybe if I made the three given circles all black and re-used red, green and blue for the solution circles?  Your solution below is pretty good; if you wanted to improve that, you could use the exact solutions in Table 1 above.  I also made an animation, but it's over 6 MB and I dreaded uploading it here, where there are already so many figures.  I'm not really fond of  Image:Apolloniuscircle.gif, because (1) I have doubts about its copyright status, (2) it's not in color, so it's harder to describe in the caption (e.g., "Which black circle?") and (3) because of its landscape aspect ratio and thin lines, the picture either squeezes the text too much (if big) or is illegible (if small).  (talk) 20:04, 14 May 2008 (UTC)


 * Sure, I'll give it another shot. The only point I would add is that it's not really necessary for the caption to describe in detail which elements of the diagram play which roles. That seems like a job for the article body, particularly "Pairs of solutions". In the lead, I'd rather see a simple illustration that doesn't compete with the text for attention. Melchoir (talk) 07:24, 15 May 2008 (UTC)
 * (See V2 on right) Melchoir (talk) 03:08, 16 May 2008 (UTC)


 * In Image:Apollonius solution 3B breathing.gif, it's a little strange that 1 and 3 come so close to intersecting, but don't. Actually none of the images in the article have intersecting given circles; I think at least one should, and this is one is a good opportunity.


 * Sure, I can do that. Thanks for the idea! :) Willow (talk) 20:04, 14 May 2008 (UTC)
 * OK, that's done! :) Willow (talk) 21:15, 14 May 2008 (UTC)


 * Great! Melchoir (talk) 07:24, 15 May 2008 (UTC)


 * The lead image in "Pairs of solutions" is awfully wide.


 * Do you think it would be better to use the present lead image instead? Or arrange them vertically instead of horizontally?  The latter seems impractical.  Willow (talk)


 * Yes, a vertical arrangement would probably be too tall for the small amount of accompanying text. The present lead image might work, or a table format similar to it... this is a tough one. Melchoir (talk) 07:24, 15 May 2008 (UTC)


 * Is there any particular reason why some of the images are SVGs and some are PNGs?


 * I made the SVG's using Xfig, and the PNG's with Blender. I can re-create my PNG's as SVG's, but it will take time.  Can you position circles arbitrarily in Inkscape?  I have to learn how to use that for more than just checking my SVG's!  Willow (talk) 20:04, 14 May 2008 (UTC)


 * Sorry, I'm not familiar with Inkscape. Melchoir (talk) 07:24, 15 May 2008 (UTC)


 * No Möbius transformations?


 * Where should I introduce them? I do mention the mapping from the planar problem onto the sphere (under "Extensions").  I could talk more about how a single spherical problem corresponds to a host of planar problems by rotating the sphere problem and re-doing the stereographic projection.  Willow (talk) 20:04, 14 May 2008 (UTC)
 * They could probably be blended into scaling and inversion. JackSchmidt (talk) 21:47, 14 May 2008 (UTC)
 * Yeah, probably the only worthwhile part of a Möbius transformation is circle inversion. Explicitly using Möbius transformations may or may not add any real substance to the article, but I just get the feeling that they might provide a useful formalism. There's a journal article titled "Apollonius tenth problem via radius adjustment and Möbius transformations", but I haven't read it. Melchoir (talk) 07:24, 15 May 2008 (UTC)
 * I've added a parenthetical mention. Melchoir (talk) 02:11, 19 May 2008 (UTC)


 * "Scaling and inversion" could use a citation.


 * Sorry, where do you mean? Willow (talk) 20:04, 14 May 2008 (UTC)


 * Actually I'm sorry, I just noticed that "Scaling and inversion" has two subsections, which are both cited. Melchoir (talk) 07:48, 15 May 2008 (UTC)


 * The first sentence of "Hyperbolic solutions" is a little confusing. It hasn't yet been established that we're talking about a pair of given circles, so the phrase "the centers of the given circles" evokes an image of three points.


 * I'll try to improve that wording; I was conscious of the possibility for confusion, but apparently I wasn't careful enough. :( Willow (talk) 20:04, 14 May 2008 (UTC)
 * That's done, too; is it better now? Willow (talk) 21:17, 14 May 2008 (UTC)
 * Yes! Melchoir (talk) 07:48, 15 May 2008 (UTC)


 * The overview mentions that Viète's solution was the first to be constructable by straightedge and compass, but it isn't clear which solution in the article is the closest to Viète's. I'd like to see a reiteration of the straightedge and compass point at the appropriate place.


 * The Gergonne solution and the annular solution are both pretty easy to see as compass-and-straightedge constructions — perhaps the latter more so? As for Viète's solution, I haven't been able to find it stated explicitly.  Apparently, he built up to the general solution from the limiting cases.  Willow (talk) 20:04, 14 May 2008 (UTC)


 * Okay, I don't think this is obvious from the text. For the section "Inversion to an annulus", the last paragraph starts out by constructing the "radical axis" of two circles, a concept with which I'm not familiar. Now, if I click through to Radical axis and read the first paragraph of the body, I learn a compass-and-straightedge construction. That's fine for that one step, but this article could have been more helpful in telling me where the constructable steps are. Melchoir (talk) 07:48, 15 May 2008 (UTC)


 * For that matter, it would be awesome to see the explicit construction, as in commons:Category:Ruler-and-compass_construction.


 * Sure, I could do the animation; it'll take a while, though. Willow (talk) 20:04, 14 May 2008 (UTC)
 * Where would we put such an animation? Willow (talk) 21:17, 14 May 2008 (UTC)


 * Is there room at the end of "Gergonne solution"? Melchoir (talk) 07:48, 15 May 2008 (UTC)


 * "Inversive methods" says "Inversion and scaling are both conformal transformations, meaning that they always transform a circle into another circle or a straight line.", with a link to Conformal map. Is that really what you meant?


 * Sorry, is there anything wrong with that? Conformal transformation redirects there, so I was avoiding the re-direct.  The mathematics seems correct to me, although it doesn't say "circle-preserving" in the lead.  Willow (talk) 20:04, 14 May 2008 (UTC)


 * The math is basically fine in one direction: scaling and inversion are conformal (or at least anti-conformal), but conformals need not take circles to circles or lines. For instance f(z)=exp(z) is conformal, but does not take circles to circles/lines.
 * As far as "circles *or* lines" being needed, I agree with WillowW. Scaling always takes circles to circles, but inversion can take a circle to a line (if the inversion is in the circle with center P, then any circle passing through P is taken to a straight line).  This could be referenced by quoting Möbius transformation, or circle inversion, which might address 3 points at once. JackSchmidt (talk) 21:47, 14 May 2008 (UTC)


 * That clarifies it for me, thank you! :) I'd thought that every conformal mapping was equivalent to a rotation (or scaling?) of the Riemann sphere, but I guess I'd misunderstood/misremembered that from my friend who'd tried to explain it to me. :P I'll fix that tomorrow; I have to run off to work now! Willow (talk) 23:40, 14 May 2008 (UTC)
 * No problem. In fact, you are quite correct, but people (including wikipedia) use the word "conformal" a little more generally than that.  The link to conformal transformation is certainly not wrong, but it could be made into a proof in disguise just by changing "conformal transformation" to "moebius transformation" (because the latter is more specific).
 * It's been a while since I studied these things, but I took the opportunity to look at Tristham Needham's excellent book, Visual Complex Analysis. If only to cement things in my own mind, let me emphasize how correct your understanding was:
 * Every conformal mapping of the *sphere* to the sphere is a moebius transformation. The trouble with exp(z) is it is only conformal on the plane (so the sphere minus one point), and that allows it to do awful things like take circles through the origin and pinch them near 1+0i, as in one of the first poorly labelled pictures on that wolfram link.  It is very common to talk about conformal maps of the plane (or equivalently, of the unit disk), and I think not so common to use the word "conformal transformation" when talking about the whole sphere -- probably only because the name "moebius transformation" sounds cooler.
 * One definition of conformal is "takes infinitesimal circles to infinitesimal circles". Not only that, but every conformal mapping is *locally* equivalent to a rotation+scaling, much as you said for the sphere.  Even when the conformal transformation is not defined on the whole sphere, it must locally look like it is.  However, overall it may distort distances unevenly, producing those "pinched circles."
 * In both ways then, the only problem with using the word "conformal" is that it is easy to confuse the local with the global. By the way, if you have not seen Needham's book, I highly recommend it. It is heavily influenced by geometry, has extremely motivating, clear, and educational illustrations, and even has sections on Ptolemy's theorem, spherical, and hyperbolic geometry. JackSchmidt (talk) 00:30, 15 May 2008 (UTC)


 * Thank you again, that clarifies it even more! :D I'd been worried that I'd mis-remembered, or that my friend had over-simplified it so that I could visualize it easily, knowing how much I liked map projections.  Actually, I guess he did do that, since we only ever talked about Möbius transformations, and he never called them by name; I picked that up later.  Now I have something new to learn about!  I haven't seen that Needham book—it would be an amazing coincidence if I had—but maybe my local librarian will be kind enough to order it for me; it sounds great!


 * Oh, I'm pretty sure you didn't mean this, but I think the "scaling" here can't be done by a Möbius transformation, since some circles swell, while others shrink; previously disjoint circles can intersect. Did I understand you rightly?  Willow (talk) 06:27, 15 May 2008 (UTC)


 * I'll just jump back in here... this edit should fix the problems, although the price is that the word "conformal" no longer appears. This is because, strictly speaking, inversions are anti-conformal, and it takes two inversions to produce a Möbius transformation. It may be necessary to leave the connection vague in this article. Melchoir (talk) 02:11, 19 May 2008 (UTC)


 * I'd love to see more in "Applications", if possible. (Which problems of celestial mechanics?)


 * Umm, I would need to track that down, perhaps in Chandrasekhar's commentary on Newton. As for other applications, I'll keep looking; I was surprised to find so few after a diligent search.  Willow (talk) 20:04, 14 May 2008 (UTC)


 * I certainly know that feeling, and maybe there aren't any other applications. All the more important, then, to milk the ones we've got! Of course this article doesn't have to restate the entire problems, but if they have proper names or Wikipedia articles, those would already be quite valuable to reference. Melchoir (talk) 07:57, 15 May 2008 (UTC)


 * Some more time has to be spent on copyediting, as with any article. At a glance, there's some missing punctuation before and after math displays.


 * That's my prejudice showing through. :( To me, English punctuation in math equations seems superfluous and out-of-place. However, I see the value of a consensus style. Willow (talk) 20:04, 14 May 2008 (UTC)


 * If it's any consolation, I think punctuation certainly looks less out-of-place in handwriting or in proper typesetting. It's a little awkward on Wikipedia, but it should still help the reader follow. Melchoir (talk) 07:57, 15 May 2008 (UTC)

Thanks for all the work so far; I'm looking forward to seeing this hit TFA! Melchoir (talk) 06:42, 14 May 2008 (UTC)

Resection?
I've tried a bit of rearranging at Talk:Problem of Apollonius/Resection. This is a proof-of-concept demonstration that eliminates single-paragraph sections and provides a hierarchical TOC, per 2(b) of the FA criteria. Thoughts? Melchoir (talk) 08:34, 15 May 2008 (UTC)


 * I like your re-sectioning very much! :) I'd been feeling that Descartes theorem was stuck in the hinterlands, and your solution brings it up front again.  I'm not sure if I understand the heading, though: "Solution forms"?


 * I suggest that we write an introductory section, as G-guy suggests, and maybe work the "Pairs of solutions" into that as an illustrative example? Then the "Limiting cases" and "Descartes theorem" could be grouped under a heading like "Special cases", followed by "Solution methods", etc. just as you have.  Does that seem good to you?  I'm rather tired tonight, though, and have to get up early, so I can't do it tonight. :P


 * I love the new picture, and agree that it's better for explaining by far. A few solutions are still hard for me to make out, but I think that'll be better once the picture is scaled up.  Thank you so much again; it's fun to have people to bounce off of! :D Willow (talk) 03:32, 16 May 2008 (UTC)


 * Hey, my pleasure, you've been very helpful yourself!
 * Yeah, "Solution forms" isn't optimal. I just wanted to distinguish between the word "solution" meaning the circles and "solution" meaning methods and constructions, as in the following section. But that's not a big deal. Your proposal sounds good; you want to try it out on the article? Melchoir (talk) 20:15, 16 May 2008 (UTC)
 * I like this, both the original idea, and Willow's modifications. I'm sorry I can only be a cheerleader for now: I hope to get involved in the nitty-gritty soon, while recognising that you guys really know the article and how to make it work. If I were editing this first section, I would try to find a better title for the first section than "Introduction", and ensure that it gives a precise description of the problem as well as an introduction to its solutions, so that the lead can be made more accessible and concise by giving a less precise description of the problem. This concision would provide space for the lead to summarize other points in the article in appropriate detail. (This is not a criticism: the lead is good now. It is an opportunity to make it even better using the reorientation of the first section.) Geometry guy 21:59, 16 May 2008 (UTC)
 * On reflection, defining the problem in the first section would save very little from the lead. I still think it should be done, however, and any other ways in which section 1 can help the lead are worth considering. Sorry for my unfocused advice. Geometry guy 22:59, 17 May 2008 (UTC)
 * I'll try my hand at some section renaming, although I'm not doing much writing:
 * Introduction -> Statement and general solutions (more precise)
 * Pairs of solutions -> (nothing) (Presently there isn't enough material to warrant a subsection here.)
 * Number of solutions for special arrangements -> (nothing) (This subsection and the one above it are just a paragraph long, and are very nearly the same topic.)
 * Overview -> (nothing) (The first text in "Solution methods" is already implicitly an overview of the rest of the section.)
 * Hyperbolic solutions -> Intersecting conics (This covers both the hyperbolae in 1p and the line/circle in 2p. Also, the adjective "hyperbolic" has connotations that aren't necessarily relevant.)
 * Lie geometry solution -> Lie sphere geometry (More common name, whereas "Lie geometry" is somewhat ambiguous. "Solution" is unnecessary, given the parent section.)
 * Melchoir (talk) 05:00, 18 May 2008 (UTC)

Copyedit questions for Willow

 * Regarding: "\left( k_{1}+k_{2}+k_{3}+k_{s} \right)^{2} = 2\, \left( k_{1}^{2} + k_{2}^{2} + k_{3}^{2} + k_{s}^{2} \right)

where rs is the radius of the solution circle and r1-3 those of the three given circles, and where the curvatures k"

didn't you mean r_{1} etc?


 * There are several places where I used the notation r1–3 to mean "r1, r2 and r3", and similarly for s1–3, C1–3, etc. It seemed like an OK shorthand, but maybe it should be changed. Willow (talk) 12:11, 18 May 2008 (UTC)


 * I'm not sure if we're talking past each other here; you gave an equation in which no r's of any kind appear, and then you said "where rs..." Shouldn't there be an "r" somewhere? - Dan Dank55 (talk)(mistakes) 14:31, 18 May 2008 (UTC)


 * I did mis-understand you! :) I see now that we're missing the transitional fossil; the explanation was left over from an earlier, fuller explanation, and didn't make sense in this modern, streamlined world.  I'm updated it, and focused more on the curvatures; perhaps it's OK now? Willow (talk) 03:21, 19 May 2008 (UTC)


 * Regarding "Avoid joining two words by a slash...as it suggests that the two are related, but does not specify how": IMO there are lots of exceptions to that suggestion from WP:MOS. I thought "internal/external" was probably better as "internal and external", but let me know if I misunderstood.


 * I usually do avoid that construction; sorry if I succumbed here! :P Willow (talk) 12:11, 18 May 2008 (UTC)
 * I found only one instance, "shrinking/swelling", which I converted into "resizing" and touched up the surrounding wording as well, just a bit. Please let me know if I missed another instance! :) Willow (talk) 04:05, 19 May 2008 (UTC)


 * "the four solutions



\theta = \pm 2 \ \mathrm{atan}\left( \sqrt{\frac{1 - C_{\pm}}{1 + C_{\pm}}} \right). $$"

Isn't that eight solutions?


 * It's four because you have to choose the same sign for C in the numerator and denominator. I was worried about that, too, which is why I specified "four", but maybe we need to be more explicit. :(  Sorry about that, Willow (talk) 12:11, 18 May 2008 (UTC)


 * WP:MOS says "Avoid sandwiching text between two images facing each other", which has often (but not always) been interpreted to mean, don't have right and left images on either side of any one line of text.


 * I know that, but the images seem all necessary, or at least highly desirable, so we shouldn't remove them. I couldn't see where else to place them except close to where they're being discussed. Willow (talk) 12:11, 18 May 2008 (UTC)


 * Does "center of similitude" mean http://mathworld.wolfram.com/ExternalSimilitudeCenter.html?


 * Yes, but also http://mathworld.wolfram.com/internalSimilitudeCenter.html. The picture they use for the External case is awful, although I suppose it would be enlightening if you thought about it long enough. Willow (talk) 12:16, 18 May 2008 (UTC)


 * Okay, I don't think scientists are going to know the term, so I would prefer either a quick description in the text, or a wikilink, or a note or ref with an external link here. - Dan Dank55 (talk)(mistakes) 15:17, 18 May 2008 (UTC)


 * "solved in 1937 by several people": independently? If not, I would probably omit "several people".


 * Yes, independently; at least, that's the impression I drew from the Nature sequel article. Only Dr. Gossett's poem was published, but several people wrote in with the solution.   Willow (talk) 12:16, 18 May 2008 (UTC)

- Dan Dank55 (talk)(mistakes) 19:59, 17 May 2008 (UTC)

Copyedit questions for the FA Team
I haven't reviewed math articles on Wikipedia before, so there's a lot of stuff I don't know, and I'd appreciate feedback to get me up to speed. I have read Manual of Style (mathematics) and WP:Mathematics, and didn't find answer to my questions. There were some things in this article that didn't seem right to me, but this is mostly based on what I saw in a past life as a math grad student; maybe we do things differently on Wikipedia. Everything I put in quotes appears in this article, and the quotes are in the order they appear in the text, so you can get from one to the next easily by searching.


 * Is it better to write "the curvatures k" when referring to k1, k2, etc in a math or science article, or should we write "the curvatures ki"?
 * I personally prefer "the curvatures ki", but I can't speak for other readers. Melchoir (talk) 04:43, 18 May 2008 (UTC)


 * Just a suggestion, but "limiting cases" has a very specific and logical meaning in math that's going to fly right past the average science-literate reader (a case you arrive at by either actually or theoretically taking a limit), and if this were my article, I would create an entry on either Wiktionary or (if there's sufficient interest and material) Wikipedia and link the first occurrence of the phrase. Neither "limit case" nor "limiting case" appears currently in either WP or Wiktionary, and I can't think of a suitable synonym.  Although "limit" doesn't quite capture the full meaning of "limiting case", as I vaguely remember, linking to Limit (mathematics) might be a satisfactory solution.


 * How about "edge cases" or "degenerate cases"? Melchoir (talk) 04:44, 18 May 2008 (UTC)


 * Personally, I'm more interested in reaching readers not quite so thoroughly trained in math, since experts such as G-guy presumably won't need our help in solving or understanding Apollonius' problem. ;) I'm worried that "edge cases" or "degenerate cases" won't be understood by those readers.  The "limiting case" terminology is an  admittedly imperfect compromise, but what I had in mind was that the limit was being taken as the radius of at least one of the given circles went to zero or to infinity, while holding one point fixed.  That's mathematically OK (I think?), but also reasonably intelligible to the lay-reader (I hope?). :) Willow (talk) 04:38, 19 May 2008 (UTC)


 * Concur with Willow. I linked the first occurrence of "limiting case", on the third line, to Limit (mathematics), so that the reader will get that we mean something special, and so that they can look it up. - Dan Dank55 (talk)(mistakes) 23:49, 19 May 2008 (UTC)


 * This may just be a personal preference, but something about "solved by Apollonius of Perga in his now-lost work Επαφαι ("Tangencies"), also known by its Latin title, De Tactionibus" doesn't sound right to me. (And btw, what's the reason for not italicizing the title?)  This is more or less the information in the lead, and it seems to me that either the lead or a historical discussion is the right place for this; other sections should then just make a quick reference to that information.  This seems to be the way that other articles do it, but I don't have a strong feeling about this.


 * I must've made a mistake when I moved it, if I didn't italicize the title; it's italicized now. Perhaps it was TMI to put both the Greek and Latin title into one sentence; I like Septentrionalis' solution.  Willow (talk) 04:38, 19 May 2008 (UTC)


 * Since non-mathematicians will never guess that "Lie geometry" is pronounced "Lee", my preference would would be to insert "(pronounced "Lee")" at the first occurrence of "Lie geometry". WP:Pronunciation insists that IPA is mandatory and "Lee" is then optional, but for a pronunciation this simple, I think that IPA is overkill; is anyone with me on this?  If people would prefer a more subtle approach, perhaps we could simply link the word "Lie" to Sophus Lie, which gives the pronunciation.
 * I definitely prefer a more subtle approach. The pronunciation of Lie's name is tangential (har!) to the topic. Melchoir (talk) 04:46, 18 May 2008 (UTC)


 * Okay, linked to Sophus Lie. - Dan Dank55 (talk)(mistakes) 23:50, 19 May 2008 (UTC)

*"Adrianus Romanus solved for the centers of the solution circles". My instinct, which might be completely out of line with what you guys want, is to try to make any math article accessible to any reader who is literate in any field connected to math, science or tech, or at least to make the attempt to use language they're familiar with when possible. It seems to me that people interested in science or tech may not get the way mathematicians use "solved" as an intransitive verb, "solved for the centers"; they might need to see "solved the problem for the centers" or "obtained the solution for the centers". But "solved for the centers" is a faster and more elegant way to say this, if you're speaking to mathematicians, and many scientists. Thoughts?
 * I think I agree with the principle, but this example isn't so bad that it needs an alternative. Just about everyone carries memories of being asked to "solve for X", right? Melchoir (talk) 04:48, 18 May 2008 (UTC)
 * I think I agree. As they sometimes do at FAC, I'll strikeout to show I'm removing the objection. - Dan Dank55 (talk)(mistakes) 14:53, 18 May 2008 (UTC)


 * "either rs+r1 or simply": Spacing is sometimes a judgment call. In the last long discussion on scientific notation at WT:MOS, we decided that we would usually put spaces on either side of a multiplication symbol, but we left open the possibility of dropping the spaces in dense mathematical expressions (although no one actually pointed to an example in a current Wikipedia article).  The examples in WP:MSM suggest that the given expression should be spaced, and I did so, but I don't think it's terribly important.  (Or rather, the spacing used in handwritten math is important, and the spacing used on web pages will always be an inelegant approximation.)  Btw, I don't mind the lack of spacing in "A1/A2" or "(d=1)", both in this article.  I can't say why in the first case, other than the fact that on paper and on whiteboards that would be written with very little spacing, and so it's not a problem for me if Wikipedia reflects that.  In the second case, the parentheses make the expression a kind of label; it's not a subscript, but it serves a similar purpose, and I am also used to seeing that with very little spacing when handwritten.


 * Oops, I didn't read this until just now ( blush ). I put a non-breaking space into the rs + r1 expression this morning, when I fixed the mistake below; that should be OK, right? Willow (talk) 04:38, 19 May 2008 (UTC)


 * The non-breaking space is great. I think the principle on a non-breaking space in this case is that you don't want to have something at the start of a line which forces the person who's skimming down the page to stop and go "Huh?"  I haven't given up on the software solution at bugzilla which would obviate the need for nbsp in this case, but we still need nbsp's for now. - Dan Dank55 (talk)(mistakes) 23:53, 19 May 2008 (UTC)

- Dan Dank55 (talk)(mistakes) 20:00, 17 May 2008 (UTC)
 * "their norm, in technical language": I removed "in technical language" (here and two other places), which would have been perfectly fine in an article not about mathematics. It seemed to me that it was clear both from the link and from the context that some kind of mathematical definition for "norm" was intended.  "In technical language" seemed like the kind of phrase that you sometimes hear people ask for at WP:FAC, and my feeling is that it is a way of asking people to apologize for the fact that they're using mathematical language in a math article; no apology is needed.
 * I'm not on the FA team, but I'll reply inline to a few points. Melchoir (talk) 04:43, 18 May 2008 (UTC)


 * My own feeling is that it's not apologizing; it's merely a flag, or parenthetical, to alert some readers that what follows is a technical definition, and they needn't worry themselves about tying to figure out why it's true. I think I won't abandon my custom of doing that, but if the consensus here is that it's better without, then I've no objection ot deleting it.  Willow (talk) 04:38, 19 May 2008 (UTC)


 * I'll defer, Willow; you can search for "technical" in the edit history to see what I removed. I just didn't think that "externally tangent" in a math article was "technical".  I also wanted to make sure you weren't saying that to make the reviewers happy; I'll support your position if they disagree. - Dan Dank55 (talk)(mistakes) 23:55, 19 May 2008 (UTC)

Okay, it looks like the rest of the FA-Team is a no-show; I was hoping they wanted this feather in their cap. O tempora o mores. But Melchoir and Sept are here; nice work all around, and congratulations to the eventual victors. Veni, vidi, wiki. (Surely this is trite, but I haven't seen it before.) I linked the first occurrence of "Lie", both for the bio and the unexpected pronunciation. I deleted that overlapping information I mentioned above in the section on "Limiting cases", and I moved the little bit that wasn't in the lead into the lead; feel free to revert if you had something else in mind. I confess that I don't understand the purpose of the Latin that I moved, but I trust you on this. Now, when you decide if you want to give a reference to "similitude", Willow, and you will explain what "where rs" means in reference to an equation that contains no r's, then we r r-right. - Dan Dank55 (talk)(mistakes) 03:10, 19 May 2008 (UTC)


 * That's alright, Dan; perhaps they saw that you and User:Melchoir and the North Star were here, and thought that we could want no better team? :) Anyway, I don't feel exactly victorious when articles like this one come to fruition. Rather, we seem more like midwives to something beautiful and noble, and I hope that other readers stumbling across our article will see our little princess in the same radiant light. :)  Thank you again for your blessed help, Willow (talk) 04:38, 19 May 2008 (UTC)


 * Maybe they were afraid we would put them to shame. Weenies.  Speaking of weenies, G-Guy hasn't answered my question yet about kappa; maybe you guys know.  Back in the day, we always used a kappa to express scalar curvature; k was reserved for signed and vector quantities.  Is this still true?  I'd like to think that we can get by with k, as Willow has done here. - Dan Dank55 (talk)(mistakes) 23:45, 19 May 2008 (UTC)


 * Back in the day? I don't know, either seems fine. Melchoir (talk) 07:14, 20 May 2008 (UTC)

Scaling
I'm beginning to get worried about the use of the word "scaling" to describe resizing the given circles. I think "scaling" is better reserved for a proportional dilation of the entire plane. Here, the word is being applied to a transformation that acts only on the given circles, isn't proportional, and sometimes means both shrinking and expanding at the same time.

This issue may seem like a nit, but there's a possibility for confusion. For example, in the sentence "Inversion and scaling are both conformal transformations...", which sense of the word should be read? The sentence only makes sense for dilation, but the context is talking about resizing.

Is it acceptable to replace "scaling" with "resizing"? I don't know what language the sources use. Melchoir (talk) 05:38, 18 May 2008 (UTC)


 * "Resizing" is fine with me! :) The sources unfortunately don't use a general verb; they discuss the two types very specifically and separately, saying either "expand" or "shrink" depending on context. "Scaling" was the best word I could think up to cover both cases, but I see the potential for confusion, especially since the transformation is more additive (r→r+Δr) than multiplcative ( r→αr ), and since some circles shrink while others grow.  Thank you for another thoughtful and helpful contribution! :) Willow (talk) 12:03, 18 May 2008 (UTC)


 * Okay, done! --Except for the intro in "Inversive methods", which needs some more work; I'll try it out... Melchoir (talk) 01:18, 19 May 2008 (UTC)
 * There. Melchoir (talk) 02:05, 19 May 2008 (UTC)

Hyperbola formulas
The current text has
 * "The distance d1 between the centers of the solution circle and C1 is either rs + r1 or simply rs, depending on whether these circles are chosen to be externally or internally tangent, respectively."

I really think it should be this:
 * The distance d1 between the centers of the solution circle and C1 is either rs + r1 or rs &minus; r1, depending on whether these circles are chosen to be externally or internally tangent, respectively.

Melchoir (talk) 05:40, 18 May 2008 (UTC)


 * Is it OK if I join the Peace Corps for a few years to live down the embarrassment? :P I don't know how I made such a simple mistake, especially when I'd gotten it right in the algebraic equations. :(  Thank you, thank you, thank you for catching that, Melchoir! :) Willow (talk) 11:44, 18 May 2008 (UTC)

Of course, no worries! Melchoir (talk) 21:06, 18 May 2008 (UTC)


 * PS. Oh, but why is it "conics" — shouldn't the heading be "Intersecting hyperbolae"? The method never uses other conic sections, or have I misunderstood that? Willow (talk) 11:54, 18 May 2008 (UTC)


 * There are two solutions with centers at infinity, the two lines on each side tangent to both given circles, so the locus must be a hyperbola (or if the circles are the same size, a straight line, which is a degenerate hyperbola.) Septentrionalis PMAnderson 17:39, 18 May 2008 (UTC)

True, "hyperbolae/s" would suffice for the first paragraph. The second paragraph mentions a related solution in which "...solution-circle centers were located at the intersections of a line with a circle", so "conics" covers all the bases. Perhaps you're saying that the first paragraph is more important to represent in the section title than the second, which would be reasonable. Melchoir (talk) 21:06, 18 May 2008 (UTC)
 * Can we please use hyperbolas? "If you are particularly proud of something, cut it out." Septentrionalis PMAnderson 17:33, 18 May 2008 (UTC)

Agree; "hyperbolas" appears to be more common. Melchoir (talk) 21:06, 18 May 2008 (UTC)


 * Oh Melchoir, I only just now realized that your section heading covers both the Adriaan van Roomen solution and Newton's — d'oh!  With all these little mistakes and oversights, I appreciate more and more why tradition encyclopedias are written by experts.  (eyes roll at myself)   Still, I'm very glad and grateful that all of you are here to set the article straight. :)  Nevertheless, I still think — and I think you agree? — that focusing the section heading on the van Roomen solution (Intersecting hyperbolas) might be better for readers like myself, who need to understand one thing well, before we can generalize. :) Willow (talk) 00:36, 19 May 2008 (UTC)


 * Well, I'm not an expert in inversive geometry by any means, but I'm glad to help. (Perhaps someone else is an expert?)
 * Focusing on van Roomen and using "Intersecting hyperbolas" is fine by me. Of course, I'd also be interested to know how Newton relates to it. There's certainly room for a couple more sentences at the end of the subsection... Melchoir (talk) 01:29, 19 May 2008 (UTC)


 * I agree but I'm embarrassed to say that I didn't read up on Newton's solution. I only took the word of Altshiller-Court's historical article.  I'll look it up tomorrow and try to understand it, and to write it up! :) Willow (talk) 04:45, 19 May 2008 (UTC)


 * Thanks, the section is looking much better! The historical link with Viète is especially gratifying. Of course, I'd still like Viète's solution to be more clearly identified below... Melchoir (talk) 01:54, 21 May 2008 (UTC)

Adrianus Romanus
I would prefer to use Adriaan van Roomen, as we and MacTutor do. MacTutor uses Regiomontanus, so they are not inventing something novel here, and we don't want readers slipping to the conclusion he was Roman, which would help nobody. Septentrionalis PMAnderson 17:33, 18 May 2008 (UTC)


 * οι μοι, ο ποποι! Alas, alas that the mother languages of scholars, Latin and Greek, have become so disregarded in favor of these inelegant modern languages. ;)  But in deference to the younger crowd, I've replaced Adrianus Romanus throughout by Adriaan van Roomen.  Kol b'seder? ;) Willow (talk) 00:46, 19 May 2008 (UTC)

But we should mention Adrianus Romanus. Some book will doubtless use only the Latin name, and it will avoid potential confusion — especially if an enterprising editor then adds him to the article under the impression these are two people. Septentrionalis PMAnderson 16:27, 21 May 2008 (UTC)

Can the given circles solve the problem?
It doesn't matter, really, which way we put this; but the section immediately after the lead says that the given circles can be solutions to the problem, and the next section (about three mutually tangent circles as data) says they can't. One or the other. Septentrionalis PMAnderson 17:46, 18 May 2008 (UTC)


 * You're referring to "Unless one of the given circles is already tangent to the other two (which would make it a solution)" versus "Figure 3: If the given three circles (shown in black) are mutually tangent, Apollonius' problem has only two solutions (shown in red).", right? Melchoir (talk) 21:11, 18 May 2008 (UTC)


 * It's so nice to see you again, North Star! :) I never took the chance to thank you for your support at Action potential; thank you, thank you with every fiber. :)  I think you can appreciate how hard that all was.  I half-expected to see you first at Featured list candidates/List of scientific publications by Albert Einstein, but I'm doubly happy to see you here, unexpectedly.  Per the Bruen et al. reference (1983), we may choose the convention that a circle is tangent to itself; therefore, you're quite right that the Soddy-circle case has five solutions, not two.  I tried to spruce up the wording to match that convention; I hope it's OK? :) Willow (talk) 00:26, 19 May 2008 (UTC)

St. Petersburg
Did Willow actually consult the Memoirs of the Saint Petersburgh Academy of Sciences? (I wouldn't put it past her.) If so, she should specify volume number and title of paper (or "untitled"); if not, we should indicate that we are relying on some other source for the location of the papers (which can still be a footnote; it would clutter the text).

This, unfortunately, is the sort of point FAC is likely to catch; it is fundamentally unimportant (as is the matter of whether we are using the English or French spelling of Memoires), but it requires no knowledge of the subject and no research. Septentrionalis PMAnderson 14:42, 20 May 2008 (UTC)
 * Being done, even as I type. Good. Septentrionalis PMAnderson 18:53, 20 May 2008 (UTC)

Applications
Thanks for the expanded material, Willow! Just a couple comments:


 * The phrase "although the connection to Apollonius' problem is not always recognized" sounds a little... argumentative? I don't know. What do Schau and Robinson say?
 * I didn't mean for it to sound argumentative. The problem is that they use the solution to Apollonius' problem, without actually mentioning it.  Maybe I should re-word it? Willow (talk) 22:57, 27 May 2008 (UTC)
 * I'll delete "although"; that might be enough. Melchoir (talk) 00:59, 3 June 2008 (UTC)


 * Newton's Proposition XXI in book I of the Principia... it seems like a bit of a stretch to call it a problem of celestial mechanics. I suppose you could say that Apollonius gives you the empty focus of an orbit, and hence the path of the orbit, from observations of a body's position and direction of motion. I don't know if that's the purpose Newton had in mind, or if it's just a geometrical diversion for him.
 * I meant to change that, but I forgot. As far as I have found, Newton uses the solution to Apollonius' problem only in the following section of the Principia (Lemma 17? Proposition 21? I forget now) to derive the orbit given a center of attraction and a tangent line, presumably the instantaneous velocity.  Willow (talk) 22:57, 27 May 2008 (UTC)
 * Right, it's Lemma (some number) that references Appollonius, while the following Proposition 21 uses the Lemma for orbits. I'll just retouch it a bit. Melchoir (talk) 01:01, 3 June 2008 (UTC)

In other news, the introductory section still needs to be expanded. I for one will be away for a week or so. Possibly Geometry guy can be conned into doing the dirty work persuaded to help? Melchoir (talk) 01:56, 24 May 2008 (UTC)

Circle inversion
WillowW, under your organization of the article there's a problem with the subsection "Pairs of solutions". To understand why solutions are paired, the reader needs to know what circle inversion is. That's why I moved circle inversion up and merged it with the pairing section in this edit. Without this, no explanation can make sense.

Also, I see that you reverted most of my changes. That's fine, but some of them are fixes for honest errors. The displayed equation in the description of circle inversion should use O, not C; and the first straightedge and compass section should say that things are constructible if the ratios of the distances involve only square roots, not cube root, higher roots, or (and this is the fix:) transcendental numbers such as π. Ozob (talk) 19:32, 28 May 2008 (UTC)


 * Hi Ozob,


 * I'm really glad that you came back and aren't mad at me, and I'm really grateful for your help finding and fixing the errors. I'm sorry about the reversion; it was a difficult decision for me.  There were some things I disagreed with, or thought were misplaced, but I'll confess that the final decision was made more emotionally than professionally.  Ever since action potential last month, where I was working like crazy to save an article I didn't even like, people have been routinely and inexplicably making formatting changes that are invisible to the reader but make it harder for me to maintain the article.  For instance, people will systematically edit the interiors of my &lt;ref&gt; tags to make them less legible, or they'll convert all my & n d a s h ;'s to –, or they'll convert math-mode r_{3} to r_3.  Ordinarily, I wouldn't have reverted you and the DOI bot, but rather sifted the wheat from the chaff; but yesterday I was in no mood to have extra stuff to deal with. :P  I'm really sorry for that, and I'll try to make amends.  I'll go through and try to do that sifting right now, but if I miss anything, please let me know.  Appreciating your advice and looking forward to working with you in the future, Willow (talk) 19:50, 28 May 2008 (UTC)


 * PS. I thought the link to circle inversion near Figure 2 would suffice? I kind of want to hold off on the details of inversion until the reader reaches the section of the Solutino methods on inversion.  Otherwise, I'm afraid that it won't be fresh in their minds.  Willow (talk) 19:53, 28 May 2008 (UTC)


 * I looked through both Ozob's and DOI bot's edits. I think they were basically good edits.  The main complaint I would have is:
 * Make each logical change its own edit.
 * Never do a "move" and anything else in the same edit.
 * Avoid format changes mixed with "substantial" changes.
 * This is solely to help build consensus. Let them agree with each one of your logical changes individually, or at least disagree with each one on its own terms.  On the other extreme, try to group changes that are logically the same into one edit (so all the r_{x} -> r_x would be a nice simple edit, Im not suggesting one edit per math tag :).  That way each logical change can be agreed with, ignored, or undone separately from the others.  This is especially useful to separate formatting changes from substantial/content/organization changes.
 * The "move" thing is more or less a bug in mediawiki, but a bug that is going to be with us for quite a few years. A tiny change to headings or a paragraph reorder confuses the diff'er and all your changes look like complete rewrites after that.  This is scary even for an article not about to be a featured article candidate!
 * I tried to go through and select the obviously good edits, but the diffs are HUGE and hard to manage. I'll pick on DOIbot because his robotic nature is both harder to damage and easier to fix.  The &amp;ndash; to – change is of no consequence, but it obscured the other major content edit: adding electronic copies of lots of the bibliography.  Even that edit had problems: it created ugly links appended to the bib-entries rather than making the title of the articles hyperlinks to the online versions.  I wanted to fix that edit, but DOIbot made it hard to sift through 20 pages of diffs just to find the DOIs.  Even after I did that, I would still have to see why the current citation template was producing ugly results.  It just wasn't worth it.  Had it been two edits, it would be easier to decide whether the ndash "fix" was good or bad for the article on its own, and decide how to handle the DOIs separately. JackSchmidt (talk) 20:21, 28 May 2008 (UTC)

JackSchmidt: Yes, I realize that edit was much, much too big. I didn't intend that initially, but then it just grew and grew and ... Well, it was a mistake.

Willow: You sound like you're having a rough time, like you've been trying to work with people who don't appreciate your editing. But I know that you've made this article much, much better than it was, and I respect your opinions about it. That's why I came to the talk page rather than edit warring: I knew from your past edits and from this talk page that you were reasonable and well-intentioned. If you will indulge my inclination to flattery for a moment: For your work on this article, I should not merely forgive you, I should kneel and kiss your feet!

Regarding circle inversion, I did wonder if it was a good idea to put circle inversion so far apart from its use in solving Apollonius' problem. But I also felt like it was necessary to say something detailed about it before we use it; it ought to be possible to read the article straight through from start to finish and have everything make sense. I thought about trying to mix the pairing property into the section on solutions to the general problem, but it felt to me like it would be extremely awkward. The best solution I could think of was to put circle inversion and pairing together near the start. I see that you've moved in inversion back to where I had it, but if you or anyone else has an idea for how to keep all of that material in the same place, I'd be interested. Ozob (talk) 22:10, 28 May 2008 (UTC)


 * Hi all,


 * I'm sorry, I'm just getting back here after a micro-panic attack over some poetry of Catullus. Thanks for all your kind words and thoughts, although you're really over-the-top, Ozob!  After a day in the garden, my feet can't be even looked at, except askance. ;) For my part, I'm really grateful that you're here and elsewhere in Wikipedia.  Have you noticed my friend Scartol's valiant effort to ready Emmy Noether for its Featured Article candidacy?  I've been gamely trying to help, but that article and the associated list of publications by Emmy Noether could use the graceful touch of an expert! :) Willow (talk) 19:27, 30 May 2008 (UTC)

Tangency of parallel lines?
Several sources, including all those that count the number of solutions, allow two parallel lines to be tangent at infinity. That's useful, since otherwise we have to make a special case for mapping solutions of Apollonius' problem on the Riemann sphere onto the plane by stereographic projection when a tangent point happens to fall at the North Pole. Is it OK if I likewise allow that definition here in this article? Willow (talk) 19:33, 30 May 2008 (UTC)


 * This bothers me. I am used to calling two algebraic curves tangent at P when their intersection multiplicity at P is at least two, and nothing remotely like this is ever true for two algebraic (= complex) lines. But real lines in CP1 aren't algebraic curves anyway, and I agree that it's annoying to have to special case two parallel lines. Do you have a reference for the definition you're proposing? If so then I think we could cite that reference for the definition. Ozob (talk) 20:06, 30 May 2008 (UTC)

Grace
I'd like to bring the article to FAC soon, maybe as early as next week. I'm conscious, though, that the article could be improved in many ways, if only by finding the article titles for all those 19th century references. (Unfortunately, I don't have access to the Correspondance sur l'Ecole polytechnique. The devil offered to exchange it for my soul, but so far I'm holding out for a better bargain. ;)  There's the Poncelet solution (which I haven't been able to track down; it might be the same as a solution already there) and we need a reference for the connection between Apollonius' problem and the Hardy-Littlewood circle method.

One immediate need is to see whether the article is even intelligible to its intended readers; I might be able to ask some friends to help with that. But more importantly, the article would benefit from the touch of someone knowledgeable, someone selfless and with ample reserves of patience and grace. Each of us should try in our own way to reach such a white knight or lady. :)

I'm also conscious that I might be part of the problem. I'll gladly take a little vacation from the àarticle for the good of the article. Willow (talk) 19:59, 30 May 2008 (UTC)