Talk:Mrs. Miniver's problem

General presentation notes
I come to this page expecting to see a mathematical presentation, which would include an exposition of the problem and its solution. The original problem is adequately presented, but what would be apropos is an expanded presentation of the general problem given two circles of specified radii and a specified distance between their centers, what is the area of overlap (or inversely, given an area of overlap, what is the distance between their centers)? Its solution would include as a special case, the solution for the particular overlap in Mrs. Miniver's problem.

But the solution here is missing, except for one special case. There are other readily computable special cases. Other sources, including those cited, invariably present both a derivation as well as the final transcendental equation. Why not here? This problem is readily solvable, though extremely messy, requiring only analytic geometry. A senior high school mathematics major should be able to solve it, if producing the transcendental equation is considered "solving it".

Beyond that, the putative final equation, which the text correctly notes is not a closed form, "but it can be solved numerically", beggars the question - what does it look like? The numeric solution is a sequence of messy trial and error iterations of a recursion relationship until it converges to the desired precision. That it can be solved is specious; no one would compute with that equation. It's more reliable and verifiable to step thru the original equalities (see below). If one needs multiple data points, to plot the curve that relates distance between centers to the ratio of their radii for example, computation becomes intractable. There is an urgent need to have a practical solution, if one is out in the field and encounters this or a related problem and needs a limited precision numeric answer on the spot. I don't know if there is one, but it would be the kind of useful info I'd want to see in the article. That transcendental relation isn't close to being linear or even quadratic, but some higher degree polynomial or a rational functional approximation of modest degree might suffice. The sources simply end with that equation. What do you do if you actually need an answer? To be continued... Sbalfour (talk) 17:52, 28 May 2022 (UTC)

The diagram
The circles in the diagram aren't atypical, and don't represent at least visually, a solution to Mrs. Miniver's problem. An atypical solution would be circles of different radii in juxtaposition which conforms to the area constraint of the problem. I think a visual representation of what the solution looks like for some pair of circles helps the reader to grasp the problem. It should be easy enough for someone with a graphics app to produce such a diagram. Sbalfour (talk) 18:04, 28 May 2022 (UTC)
 * ✅ Replaced by an accurate image with several choices of radii. It took more than a graphics app to produce the diagram, though — it is also necessary to calculate the overlap (that is, to numerically solve the problem described by this article), which involves grinding through some messy formulas to set up a binary search to solve it numerically. —David Eppstein (talk) 06:56, 29 May 2022 (UTC)
 * That was a wonderful enhancement to the article (and a chunk of gruntwork, I know). Tnx. Sbalfour (talk) 14:54, 30 May 2022 (UTC)

Plot of the relationship
A useful visual aid here would be a graph of the relationship between the ratio of the radii and the relative distance between the centers. It may be assumed without loss of generality that the radius of the smaller circle is 1. Mathematica could probably do it, but I don't have a command of Mathematica. If there's not an app, it's a hellacious piece of gruntwork to plot points. Furthermore, on the same diagram, it'd be enlightening to graph polynomial regression lines for some set of low degree polynomials, for example, 1,3,5, to demonstrate the futility of approximating the transcendental function with a computationally convenient polynomial. What one would probably do is use a spline, but general procedures for approximating transcendental functions is well beyond the scope of this article. What would be apropos is to derive some computationally expeditious approximation expression, possibly transcendental, uniquely associated with the geometry. My experience with similar messes like this suggests that none will be found, but I'm looking because I need one. Sbalfour (talk) 15:26, 30 May 2022 (UTC)
 * "derive some computationally expeditious approximation expression" is heading into original research territory. If we want to report on a formula that provides an accurate approximation, we need to find a reliably published source for that formula. (If that paper does not already exist, it would also be possible to write it, publish it, and then cite it here, but that's a slow process.) —David Eppstein (talk) 16:34, 30 May 2022 (UTC)

Solution section
I have a bit of an issue with the solution section. I can verify the equation, but I am mathematically adept. I wager than the average editor/reader does not have the applicative knowledge of algebra, geometry and trigonometry necessary to derive that equation from the problem statement, nor know how the equation is solved if presented to him. A person may even try plugging angles in degrees into the equation and obtain a nonsensical answer. The accepted way is in fact trial and error, but most people won't know that. Solving it to the 6-7 significant digits of precision required to verify the stated distance between centers is a non-trivial effort. To make the article more accessible to ordinary folk, I propose modifying the latter portion of the solution section thus: In the case of two circles of equal size, these equations can be simplified somewhat. The lens formed by the overlap of the circles is two identical circular segments each with central angle $$\theta$$, radius $$r \doteq 1$$, and area $$\tfrac{r^2}{2}(\theta - \sin \theta)$$. If $$A_s$$ is the area of one circular segment, then the area constraint of the problem has equation $$2A_s=2(\pi r^2-2A_s)$$. Substituting for $$A_s$$ and reducing yields the transcendental equation $$\theta-\sin\theta=\frac{2\pi}{3}$$. That equation is solved by trial and error, and yields $$\theta\approx 2.605$$ radians. The ratio of the distance between their centers to their radius is then $$2\cos\tfrac\theta2\approx0.529864$$.

That a rhombus can be constructed in the geometrical figure is rather oblique to the problem - it's relating the areas of the circular segments to the radius and areas of the circles that's the fundamental paradigm. Sbalfour (talk) 19:53, 7 June 2022 (UTC)
 * It is incorrect that it is solved by trial and error. If you had to guess randomly until you found a number accurate to that many digits of precision it would take far too long. Numerical techniques as simple as bisection search or more complex methods such as Newton iteration are called for. But we already say "but it can be solved numerically"; going into detail about numerical methods for transcendental equations would make this material more technical and confusing, not less, and would risk original research unless we can find sources for the application of specific methods to this specific problem. —David Eppstein (talk) 21:52, 7 June 2022 (UTC)
 * "trial and error" is to be taken broadly - it's not a random process, but a guess followed by a convergence process, whether systematic binary, or guided by progressive knowledge about where the convergence is headed. The main point here isn't the numerical process, but the info on how the equation is derived from the geometry. I don't have all the sources, but they probably have some derivation like the above, and it's almost self-evident (if one has a bit of expertise - this is for the others who don't). Sbalfour (talk) 22:43, 7 June 2022 (UTC)