Talk:Curve of constant width/GA1

GA Review
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Reviewer: Ovinus (talk · contribs) 21:42, 8 January 2021 (UTC)


 * GA review (see here for what the criteria are, and here for what they are not)


 * 1) It is reasonably well written.
 * a (prose, spelling, and grammar): b (MoS for lead, layout, word choice, fiction, and lists):
 * 1) It is factually accurate and verifiable.
 * a (reference section): b (citations to reliable sources):  c (OR):  d (copyvio and plagiarism):
 * Earwig suggests [], which cites this article itself
 * 1) It is broad in its coverage.
 * a (major aspects): b (focused):
 * 1) It follows the neutral point of view policy.
 * Fair representation without bias:
 * 1) It is stable.
 * No edit wars, etc.:
 * 1) It is illustrated by images and other media, where possible and appropriate.
 * a (images are tagged and non-free content have fair use rationales): b (appropriate use with suitable captions):
 * 1) Overall:
 * Pass/Fail:
 * 1) It is illustrated by images and other media, where possible and appropriate.
 * a (images are tagged and non-free content have fair use rationales): b (appropriate use with suitable captions):
 * 1) Overall:
 * Pass/Fail:
 * Pass/Fail:

Preliminary comments
Happy to take this on. As a math article, I'll try pay extra attention to the balance of understandability to a wide audience and mathematical completeness. Ovinus (talk) 21:42, 8 January 2021 (UTC)

An overarching comment: parts of this article feel dry. It's not a huge problem per the GA criteria, but I would appreciate it if those parts were a bit more motivated. As an example I might write the Examples section like:

The width of a circle is constant—its diameter—making it a curve of constant width. In contrast, a square has varying widths: for a square of side length $$s$$, the width as measured between parallel sides is $$s$$, while the width measured along its diagonal is $$s\sqrt{2}$$. Yet the circle is not the only curve of constant width—in fact, there are infinitely many such curves that are non-circular. A standard example is the Reuleaux triangle, formed from three circular arcs centered on the three vertices of an equilateral triangle, each with the other two vertices as endpoints.

The Reuleaux triangle is not smooth at its three vertices; its 120&deg; angles are actually the sharpest possible for any curve of constant width. But smooth, non-circular curves of constant width are known. The following polynomial equation of degree eight forms such a curve:

...

Eight is the minimum possible degree for a polynomial that defines a non-circular curve of constant width.

Not all of the article feels dry, so I don't think it should be too much work. Ovinus (talk) 02:51, 9 January 2021 (UTC)
 * Um. The relevant Good Article criterion is "the prose is clear, concise, and understandable to an appropriately broad audience; spelling and grammar are correct". You appear to be using the corresponding Featured Article criterion, "its prose is engaging and of a professional standard". So I'm not convinced that this level of review should be necessary for GA. Nevertheless, the feedback is helpful, I thank you for it, and I'll see what can be done to improve the article along these lines. —David Eppstein (talk) 05:14, 9 January 2021 (UTC)
 * You're right, sorry. I won't bug you about it. Ovinus (talk) 05:25, 9 January 2021 (UTC)

Lead

 * About the first sentence: In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions, regardless of the slope of the lines. The second "lines" refers to the lines in the parenthetical expression, and "same in all directions" is synonymous with "regardless of the slope of the lines"; this is a bit problematic. I'd suggest something like In geometry, a curve of constant width is a planar curve whose width is the same in all directions. More formally, it is a Jordan curve with the property that the distance between supporting lines (the width) is the same regardless of their slope.
 * I really dislike the "more formally" style of telling readers "this stuff is technical so go away, you're not ready to read it". And I think in the case of Jordan curves, understanding them properly in full generality is very technical, but also very unnecessary for this article (because the technicality only comes into play for things that are far from convex). The reason I said "direction" rather than slope is also technical (but that's ok because it's a technicality that I omitted from the article itself): slope depends on coordinates and, as a number, doesn't even exist for vertical lines, but we don't need coordinates and do need vertical lines. I took out the "regardless" clause, because it was intended to be redundant reinforcement, but judging by your reaction ended up being more confusing than helpful. —David Eppstein (talk) 05:18, 9 January 2021 (UTC)
 * Sounds good. Ovinus (talk) 05:25, 9 January 2021 (UTC)
 * Corresponding to this sentence, the lead picture should be more about the curve of constant width, not the measurement of width itself. We have an intuitive understanding of width, but it's counterintuitive that anything besides a circle could have constant width. I'll try make a sample figure for what I mean. Ovinus (talk) 22:55, 8 January 2021 (UTC)
 * Uh, you did realize that the lead picture depicts a curve of constant width that is not a circle, the Reuleaux triangle, right? I modified the caption to state that more explicitly. —David Eppstein (talk) 04:47, 9 January 2021 (UTC)
 * Of course, but I mean that the lead picture should display that the width is constant by putting two measurements on the Reuleaux triangle. Otherwise it just looks like we're explaining how width is measured, which is pretty clear to most people. Ovinus (talk) 05:25, 9 January 2021 (UTC)
 * It can't really display that the width is constant without showing an infinite family of supporting lines, something that is essentially already done in the rotating-in-a-square image. But my feeling is that showing more than one width will make the image harder to read by making it more cluttered. Also, I really don't think that the technical definition of width used in this context is intuitive enough that we can just take it for granted without stating and illustrating it. You might not have noticed, but it's not actually described at width (which is just a redirect to length), for instance. —David Eppstein (talk) 21:14, 10 January 2021 (UTC)
 * Every curve of constant width must be the boundary of a convex set How about just Every body of constant width is a convex set ?
 * Good simplification, done. —David Eppstein (talk) 04:49, 9 January 2021 (UTC)
 * The minimum area for a given width is obtained by the Reuleaux triangle and the maximum is obtained by the circle. I think it could be phrased a bit better. How about Among all such bodies with a given width, the Reuleaux triangle and circle have minimal and maximal area, respectively. Using "bodies" also corrects the minor distinction between a curve (zero area) and a body (with positive area).
 * If we're being pedantic, a circle is a curve not a body. But I think it's better to deliberately make minor misstatements like this than to confuse readers by being more technical and making them wonder why half the time we call them circles and the other half disks. Anyway, reworded. —David Eppstein (talk) 05:05, 9 January 2021 (UTC)
 * You're right; I'm being too pedantic. Ovinus (talk) 05:25, 9 January 2021 (UTC)
 * The next three sentences feel a bit monotonous, like we're just listing properties. Is there a way to make it flow better?
 * I rearranged and regrouped the sentences to give them some more structure than just "Fact. Fact. Fact." But to some extent this is in conflict with WP:TECHNICAL: the more complicated I make the sentence structure, the more readers have to work to understand the sentence structure before they even get to the mathematics. —David Eppstein (talk) 05:05, 9 January 2021 (UTC)
 * Sure, per what I said earlier I won't bug about it too much going forward. Ovinus (talk) 05:25, 9 January 2021 (UTC)

Definition

 * The definition intrigues me. We start off by talking about compact sets and their supporting lines, and thus the definition of "width" in a given direction. Then given a compact set of constant width, we construct the corresponding body of constant width by taking its convex hull. Is this the textbook definition? Here is why I'm confused: we're talking about all compact sets with constant width. Take $$S=\{(x,y)|1\leq x^2+y^2\leq 2\}$$, for example. Then the boundary of the set is a curve of constant width isn't true of $$S$$?
 * Otherwise all good here. Ovinus (talk) 05:45, 9 January 2021 (UTC)
 * You're right, there was a problem here. I went back to the sources; both of the ones used for this part start with curves rather than bodies (like the lead) and when talking about bodies Rademacher makes the simplifying assumption that they are convex rather than actually proving that convex ones come out of the definition. I rewrote to follow suit. —David Eppstein (talk) 06:13, 9 January 2021 (UTC)

Examples

 * Is it necessary to point out that the ratio is 1 to $\sqrt{2}$ ?
 * I took that out before even seeing this comment. —David Eppstein (talk) 06:14, 9 January 2021 (UTC)
 * I think the polynomial could just be described as a polynomial equation without having to describe the zero set. As an aside, this fact really fascinated me (particularly the degree 8 being the smallest), so thank you for that! Ovinus (talk) 05:45, 9 January 2021 (UTC)
 * How? As a polynomial it's not even a shape in the plane, but an element of some other algebraic space of polynomials. It's the zero set that has the constant width. —David Eppstein (talk) 06:14, 9 January 2021 (UTC)
 * I mean an algebraic equation, like $$x^2 + y^2 = 5$$. Is that not the right term here? Ovinus (talk) 06:22, 9 January 2021 (UTC)
 * You mean put "0=" in front of the rest of the equation? It's not the kind of equation where you can separate y from x and set one of them equal to a polynomial of the other. (Even for the circle, if you try that, you get $$y=\pm\sqrt{1-x^2}$$, and the square root and plus/minus sign makes it non-polynomial.) But then it's a polynomial equation rather than a polynomial, and you still have to talk about its solution set rather than its zero set, so I'm not convinced this would be a simplification. —David Eppstein (talk) 06:27, 9 January 2021 (UTC)
 * How about this: the zero set of the following polynomial (the points $(x,y)$ for which $f(x,y)=0$) forms a non-circular smooth algebraic curve of constant width: The repetition of f(x,y) and formula isn't really needed. Ovinus (talk) 16:52, 9 January 2021 (UTC)
 * Simplified roughly along these lines. —David Eppstein (talk) 22:16, 9 January 2021 (UTC)

Constructions
Thank you for the clarifications. Ovinus (talk) 02:31, 11 January 2021 (UTC)
 * I'd note that a Reuleaux triangle is also a Reuleaux polygon, but maybe that's obvious?
 * Ok. Sometimes saying obvious things can be useful. —David Eppstein (talk) 06:40, 9 January 2021 (UTC)
 * irregular Reuleaux polygons are also possible To be more specific, maybe Reuleaux polygons can also be constructed from certain irregular polygons.
 * Done. —David Eppstein (talk) 06:40, 9 January 2021 (UTC)
 * The crossed-lines method description is solid, especially given the method's complexity, but the corresponding diagram could be better. It has a lot of information on the lengths of various segments which isn't reflected in the text. I feel like this is an ideal case for a simple animation—which I'd love to make if you want—but I know some folk don't like excess animations. In the latter case, I think the diagram should be without lengths, or the length calculations added to the article text.
 * I'm not comfortable making animations. But I could try making a cleaner static image instead of just re-using one that was already there. —David Eppstein (talk) 06:29, 9 January 2021 (UTC)
 * Replaced by a new image without all of the unused annotations. —David Eppstein (talk) 00:40, 10 January 2021 (UTC)
 * I don't fully grasp Euler's construction; not all involutes of a deltoid are curves of constant width, right? Only the ones which are convex? A diagram would be helpful but I know we're running out of space.
 * I don't think the article actually says that all involutes work, only that Euler used involutes to construct them. All involutes starting from a sufficiently long segment work but explaining this or detailing how to measure this length seemed like something that would be better to skip over than to try to go into detail about (see WP:GACR 3b). —David Eppstein (talk) 06:40, 9 January 2021 (UTC)
 * Makes sense. So then involutes of the deltoid form smooth curves of constant width should be some involutes of the deltoid ... Ovinus (talk) 16:57, 9 January 2021 (UTC)
 * Any updates with this change? Ovinus (talk) 07:33, 12 January 2021 (UTC)
 * This took some thought, because what we need to say is that all involutes with a sufficiently large distance parameter work, but our article on involutes doesn't provide a clear description of what their parameter means except in either a uselessly technical way as something you add to an integral or a uselessly vague and nontechnical way as the length of a piece of string. I ended up reworking that paragraph to bring in the rolling line segment equivalent version more centrally as a way of describing involutes, so that I could say that it works for all sufficiently long line segments. —David Eppstein (talk) 08:53, 12 January 2021 (UTC)
 * Great! I'll make a simple animation to accompany with your explanation, which I should note is very understandable. Ovinus (talk) 09:04, 12 January 2021 (UTC)
 * Math doesn't display in captions yet. Could we use $\sqrt{3}$/2 here?
 * Those fake-math templates are really really ugly, especially in square roots, which is why they're close to being banned by MOS:MATH. Captions are an exception because of this bug (which isn't happening for me so I didn't notice was a problem). I think maybe just taking the formula out of the caption altogether would be a better solution. —David Eppstein (talk) 06:29, 9 January 2021 (UTC)
 * Cool!
 * It's getting late here so I'll add some more comments in the morning. I wish there was more space for diagrams... it would make things a lot easier to understand. Would a gallery be permissible here? Ovinus (talk) 06:22, 9 January 2021 (UTC)
 * I don't see why not. —David Eppstein (talk) 06:40, 9 January 2021 (UTC)
 * Okay, I'll try to make some simple animations for the Constructions section. Even if we don't end up using them it's good for Commons Ovinus (talk) 16:57, 9 January 2021 (UTC)
 * This construction is universal: all curves of constant width may be constructed in this way. "This" should refer to the general construction for any curve, but it appears to refer to Puiseaux's semi-ellipse construction specifically. Recommend moving this sentence to after "as part of its boundary." Alternatively, if you want to keep it after the semi-ellipse sentence, you might be able to do This construction is universal: all curves of constant width may be constructed with a suitable choice of starting curve. Ovinus (talk) 18:16, 9 January 2021 (UTC)
 * Reordered sentences to make "this construction" clearer. —David Eppstein (talk) 22:16, 9 January 2021 (UTC)
 * Caption An irregular Reuleaux polygon: maybe make it An irregular Reuleaux polygon constructed by the crossed-lines method.
 * That would be redundant (all Reuleaux polygons can be constructed in this way), overly specific (even though they may be constructed in this way, who's to say that this one was?), and confusing (likely to cause readers to think that the crossed-lines method always constructs Reuleaux polygons, but it doesn't). —David Eppstein (talk) 04:51, 10 January 2021 (UTC)
 * I see. I just thought the gray lines were purposely to show the underlying regular star polygon. Would mentioning that be worthwhile? Ovinus (talk) 07:48, 10 January 2021 (UTC)
 * The gray line segments are the underlying star polygon, but it is not regular. If those segments were extended to lines, they could be used for the crossed lines construction. But showing that would confuse the two constructions, one of which is more general than the other. (The crossed line example in the other illustration is not a Reuleaux polygon because it does not have sharp corners where its circular arcs meet, and because the arcs do not all have the same radius.)
 * smooth curves of constant width, not formed from circular arcs To be clear, this means that deltoid involutes can produce curves which contain some circular arcs, but are not entirely made up of circular arcs? Or is no subset of a deltoid involute an arc?
 * No subset is a circular arc. Changed to "not containing any circular arcs" to be less ambiguous (and also because "formed from" was overused). —David Eppstein (talk) 05:08, 10 January 2021 (UTC)
 * it can be applied to the arc -> it is applied to the arc
 * More careful wording was needed here, because Puiseaux found these shapes but used a different construction for them. —David Eppstein (talk) 05:08, 10 January 2021 (UTC)
 * obtained by a calculation involving an alternating sum of the lengths of arcs of the starting curve between its cusps I think this may be a bit too in-depth while unhelpful someone who wants to know the details.
 * Ok, simplified. —David Eppstein (talk) 05:08, 10 January 2021 (UTC)
 * convex curved arc -> convex curve I think this is specific enough?
 * The point is that it is not a closed curve. It has endpoints, like the arc of a circle. It is the arc of a convex curve, not the whole curve. Unless you have some other meaning in mind for "convex curve" that is not "boundary of a convex set"? —David Eppstein (talk) 05:08, 10 January 2021 (UTC)
 * Good point, never mind.
 * The arc must have the property (required of a curve of constant width) that each of its supporting lines is tangent to a circle of radius w containing the entire arc I feel like this is trying to jam together two properties: that the arc is tangent to the supporting lines, and that it is contained within a circle. Also we need to make it clear that the supporting lines are the aforementioned parallel lines.
 * No, there are two things wrong with what you wrote here that indicate that you are misreading this sentence. First, a supporting line is not always a tangent line; for instance, supporting lines at the vertices of polygons are generally not tangent. Second, it is the supporting line and the circle that must be tangent. It is not adequate to surround the arc by a single circle of radius w, nor even a collection of circles that touch it at all of its points (or are tangent at the points where it is possible to be tangent); we really need a different circle for each supporting line. —David Eppstein (talk) 05:12, 10 January 2021 (UTC)
 * Indeed I'm puzzled. I didn't realize you were not talking about a single circle tangent to both lines and containing the arc, but two circles . In general I'm having a lot of trouble understanding this construction because it's super hard to visualize. I think it would be better served in list form and with a corresponding step-by-step diagram/animation. I'd be willing to make the latter, but I still don't understand the construction. Ovinus (talk) 07:48, 10 January 2021 (UTC)
 * You didn't notice that there's a visualization already present in the article? It's the one with all the blue circles centered on a semi-ellipse, and with one of the tangent-to-supporting-line circles shown in red. Also, we have a way of expressing things step by step and separating the steps from each other in English prose (and note that WP:USEPROSE is one of the guidelines included in the GA criteria): the things being separated are called sentences and the things separating them are called periods or full stops. So there already is a step-by-step description of the construction as well as a visualization of it. —David Eppstein (talk) 08:29, 10 January 2021 (UTC)
 * Yes, I saw the visualization, but call me thick, I didn't understand it either. I think I represent an average or slightly above average reader so my problems won't be unique. Here's my conundrum:
 * The arc must have the property (required of a curve of constant width) that each of its supporting lines is tangent to a circle of radius w containing the entire arc Interpretation one: "supporting lines" refers to the two parallel lines from earlier. In this case, there is a circle of radius w for each of the two lines which encompasses the arc (lets call them C1 and C2). Interpretation two: "supporting lines" refers to the infinite number of supporting lines of the arc. But this doesn't make sense for geometric reasons: the supporting line through the two endpoints of the arc, for example, cannot be tangent to a circle of radius w which encompasses the arc.
 * The next step is to intersect an infinite family of circular disks of radius w, both the ones tangent to the supporting lines and additional disks centered at each point of the arc. Under interpretation one, C1 and C2 are already part of the latter group, but that basically forms a lens, which doesn't make sense. Under interpretation two, there are two "families" of disks. The first family kind of "rolls around" the starting arc, and the second family has its center along the arc. This makes more sense.
 * So my educated guess is that interpretation two is correct, but it needs the added caveat that only certain supporting lines—those which aren't vertical or pass through an endpoint—are subject to the curvature condition. Am I understanding it now? Ovinus (talk) 08:56, 10 January 2021 (UTC)
 * Rewritten, also avoiding the transitive use of "intersect". It's closer to interpretation two (how do you get infinitely many circles from interpretation one??) but we only use one of each pair of parallel supporting lines (the one on the convex side of the curve). —David Eppstein (talk) 21:21, 10 January 2021 (UTC)
 * As long as it meets this condition, it can be used in the construction. Can be removed
 * You do understand the distinction between necessary and sufficient conditions, right? The preceding sentences were saying what is necessary. This one is saying that the same conditions are also sufficient. —David Eppstein (talk) 05:12, 10 January 2021 (UTC)
 * Well of course, but it's implicit in the fact that you continue the construction with no qualms. Ovinus (talk) 07:48, 10 January 2021 (UTC)
 * Rewritten; see above. —David Eppstein (talk) 21:21, 10 January 2021 (UTC)
 * Noted.
 * both the ones tangent to the supporting lines I think this is an unnecessary clarification. How about The next step is to consider the infinite family of circular disks of radius w centered at points along the arc. Their common intersection forms ... My misunderstanding
 * That would be incorrect. We also need the disks tangent to the supporting lines. They're the ones that cause the given arc to be part of the boundary of the intersection. —David Eppstein (talk) 06:44, 10 January 2021 (UTC)
 * intersect an infinite family I think a lot of people don't understand this meaning of "intersect", hence my proposed amendment above.
 * To me that seems wordy and therefore harder to read, for no gain in meaning. What other meaning could "intersect" as a verb have in this context? —David Eppstein (talk) 06:46, 10 January 2021 (UTC)
 * It depends on the reader. I understand what "intersect" means here, but I don't think most people without knowing about set theory will understand the idea of "intersecting" an infinite number of circles. Note that Merriam Webster considers "intersect" in the context of areas to be an intransitive verb. In other words, we can say the squares intersect and form a rectangle but not we intersect the squares to form a rectangle. Ovinus (talk) 07:48, 10 January 2021 (UTC)
 * Rewritten; see above. —David Eppstein (talk) 21:21, 10 January 2021 (UTC)
 * Cool.
 * it can be applied to the arc formed by half of an ellipse between the ends of its two semi-major axes, as long as its eccentricity is at most ..., low enough to meet the curvature condition. This is a bit verbose. How about it is applied to half of an ellipse with eccentricity e, spanning its major axis. The construction only works if $e \leq \frac{1}{2}\sqrt{3}$ (low enough to meet the curvature condition).
 * This part has now been reworded as part of other earlier changes. —David Eppstein (talk) 06:47, 10 January 2021 (UTC)
 * Noted.

Properties

 * I think it might be good to mention that the center (er, centroid) of a body of constant width moves when it is rotated between two supporting lines. Ovinus (talk) 19:44, 9 January 2021 (UTC)
 * Do you have a suggested source for that claim? This request fits under GACR 3a but it must also obey 2c. —David Eppstein (talk) 20:19, 9 January 2021 (UTC)
 * Hm... could probably find it for the Reuleaux triangle for some article on the Wankel engine. But otherwise Didn't realize the Wankel engine wasn't a true Reuleaux triangle. So no, sorry. Ovinus (talk) 21:01, 9 January 2021 (UTC)
 * On second thought, you cited that "The center of the roller moves up and down as it rolls" later. So I think that's sufficient information already. Ovinus (talk) 01:38, 10 January 2021 (UTC)
 * This sequence of rotations of the curve can be obtained by keeping the curve fixed in place and rotating two supporting lines around it, and then applying rotations of the whole plane that instead keep the lines in place and cause the curve to rotate between them. Is this description necessary? I think it's pretty clear without it
 * This is the explanation for why these curves can be used as rollers. Why do you think it is so obvious as to be superfluous? I think to many readers, the jump from keeping the shape fixed and rotating lines around it to keeping lines fixed and rotating the shape between them will be an actual jump. We can't just say that rotating the set and rotating the lines are the same thing, because they're not the same thing. Anyway, rewritten. —David Eppstein (talk) 00:49, 11 January 2021 (UTC)
 * The rewrite is clear, thanks.
 * In the same way, a curve of constant width can be rotated between two pairs of parallel lines with the same separation. Ah, so as stated this requires the previous observation. But the conclusion isn't so straightforward as saying "we rotate two pairs of lines and get this result"; it requires the observation that pairs of parallel lines, each separated by a width $$w$$, form the same rhombic intersection no matter their position, unlike the three pairs of a regular hexagon. I recommend for brevity to say that "A curve of constant width can be rotated in a rhombus, including a square, with its four sides acting as supporting lines."
 * But it's incorrect that pairs of parallel lines of separation w form the same intersection. It's actually incorrect in three different ways: (1) you need to add a requirement that the lines keep the same angle with respect to each other, (2) the shape you're talking about is not the intersection of all the lines (the empty set), nor even the union of intersections of pairs of lines (four points), but the intersection of the slabs of the plane between the pairs of lines, and (3) even if you correct points (1) and (2) you don't get the same intersection, you get two different intersections that are congruent to each other. Anyway, rewritten, as above. —David Eppstein (talk) 00:49, 11 January 2021 (UTC)
 * Yes, I was being imprecise too. The new wording is clear.
 * for every non-convex I'd prefer "since" here
 * Done. —David Eppstein (talk) 00:49, 11 January 2021 (UTC)
 * These two circles again touch the curve in at least three pairs of opposite points, but these touching points might not be vertices This is vague, because the pairs are between circles (not within the points of each circle as my first literal interpretation was).
 * How vague? The minimum enclosing circle touches the curve in at least three points. The largest enclosed circle touches the curve in at least three points. That makes a total of at least six points (because, as you probably know, 3+3=6), which have to alternate between the two circles. So far, this looks a lot like the vertices (points of locally minimum or maximum curvature), for which there are also at least six points alternating between local minima and local maxima. Because of this similarity, I thought it would be important to point out that the six circle-touching-points are not actually the same six points as the six vertices. I have no idea what "the pairs are between circles" is supposed to mean. —David Eppstein (talk) 00:57, 11 January 2021 (UTC)
 * The first clause could mean that each individual circle has at least six points (coming in opposite pairs) at which it touches the curve—which is absurd. How about something like: The two circles together touch the curve in at least three pairs of diametrically opposite points, although these touching points are not necessarily vertices. Ovinus (talk) 02:54, 11 January 2021 (UTC)
 * Rewritten to use "two circles together" instead of "two circles again" to avoid this ambiguity. —David Eppstein (talk) 05:45, 12 January 2021 (UTC)
 * Several countries have coins shaped as non-circular curves of constant width; examples include the British 20p and 50p coins. Their heptagonal shape with curved sides means that the currency detector in an automated coin machine will always measure the same width, no matter which angle it takes its measurement from. The same is true of the 11-sided loonie (Canadian dollar coin). How about Several coins are non-circular bodies of constant width, such as the heptagonal 20p and 50p coins of the UK and 11-sided loonie (Canadian dollar coin). Regardless of such a coin's rotation, automated currency detectors will always measure the same width.
 * Your rewrite omits the punchline. Why is it important that currency detectors always measure the same width? What does this measurement have to do with the actual goal of these machines? Anyway, rewritten, but not exactly with your proposed wording. —David Eppstein (talk) 05:45, 12 January 2021 (UTC)
 * Cool.
 * its central reflection How about "its reflection about its center" ? I'd note that we never directly talk about the curve's center until section Applications, so some explicitness is needed. Ovinus (talk) 07:33, 12 January 2021 (UTC)
 * The central reflection is well-defined up to translation without specifying the center. But when you say "reflection about its center" you need to know what the center is. So that seems a lot more complicated to me, for no particular benefit. —David Eppstein (talk) 07:50, 12 January 2021 (UTC)
 * I get the Minkowski sum doesn't care about translation but I've never heard the term central reflection before. Why not "point reflection" or "reflection about a point" then, wikilinked? Ovinus (talk) 08:00, 12 January 2021 (UTC)
 * Because, again, what point? We could call it the 180° rotation if you think that would be clearer. As for adding a wikilink: the same link already exists three lines up. See WP:OVERLINK. —David Eppstein (talk) 08:01, 12 January 2021 (UTC)
 * 180° rotation is fine. It's more an issue of terminology; I've never heard "central reflection" but I have heard "point reflection" and "180&deg; rotation" so I don't see why we can't use the latter. If it's for parallelism with the earlier mentioning of central symmetry than I find "reflection about its center" to be clearer. Ovinus (talk) 08:08, 12 January 2021 (UTC)
 * I'm pretty sure the original intent was parallelism, and you can find plenty of hits for "central reflection" in Google Scholar, but whatever, clarity is more important than parallelism. Changed. —David Eppstein (talk) 08:17, 12 January 2021 (UTC)

Generalizations

 * convex bodies in R^3 Could just be 3-dimensional convex bodies
 * and their boundaries leads could just be leads—as you said we don't have to be too pedantic
 * There is also a concept of space curves of constant width Could just be Space curves of constant width are defined
 * Copyedited this part. —David Eppstein (talk) 05:52, 12 January 2021 (UTC)
 * but small modifications of it, the Meissner bodies "modifications" is referring to two things here: the Meissner bodies, and the "small" process of obtaining them. How about but slightly modified variants of it, the Meissner bodies ? Ovinus (talk) 07:33, 12 January 2021 (UTC)
 * Is your problem here that "small" could either mean that the modifications are small or that the bodies are small? I've changed it to "minor". That makes the same point less ambiguously, without all the extra verbiage of your suggested rewording. —David Eppstein (talk) 21:07, 12 January 2021 (UTC)
 * It's not a matter of ambiguity/clarity but of grammar. "Modifications" still refers simultaneously to the process of modification and the end product. "Minor modification" means a small change, which isn't a Meissner body. Another option: but small modifications of it produce the Meissner bodies, which do. Ovinus (talk) 21:14, 12 January 2021 (UTC)
 * Oh. That's not grammar, it's semantics. I was using modification in the sense of wikt:modification #3: "The result of modifying something". You appear to be thinking of it with a later meaning, #4: "The act of making a change". But I reworded it to use "change" instead of modification since that was simpler. —David Eppstein (talk) 22:14, 12 January 2021 (UTC)
 * I knew you were using that sense of the word, but a "minor result" or "minor body" doesn't make sense. Oh well, I'd rather not debate this endlessly and it's clear now. Ovinus (talk) 22:23, 12 January 2021 (UTC)
 * Sure it makes sense. A small modification is a thing that is the result of modifying something else in a small way. Although grammatically an adjective, small has its adverbial meaning, of modifying the action rather than modifying the thing. —David Eppstein (talk) 23:39, 12 January 2021 (UTC)

Concluding
There are three items left here ("some involutes" clarification, "central reflection", and "modifications"). Once I upload some images and after your advice appropriate ones are put in the article to illustrate the more complex elements, I'm inclined to promote. Ovinus (talk) 07:33, 12 January 2021 (UTC)
 * David: Sorry for the enormous delay. I got stuck trying to make the animations for a few days and then promptly forgot about it. I'll be working on it again, but I hope they will ultimately make the article easier to understand. Cheers, Ovinus (talk) 01:43, 28 January 2021 (UTC)
 * You realize that, again, this is well beyond what a GA review is supposed to do? It is neither expected nor at all typical for the reviewer to supply major pieces of content. See WP:GACR footnote 7 "The presence of media is not, in itself, a requirement. However, if media with acceptable copyright status is appropriate and readily available, then such media should be provided." — it should never be a requirement for GA that new not-readily-available media be created and added. See also WP:GAI: "Review timeframes vary from one nomination to the next, but a responsive nominator and reviewer can complete a review in about seven days." —David Eppstein (talk) 02:20, 28 January 2021 (UTC)
 * I see. Promoting, then. Ovinus (talk) 08:34, 28 January 2021 (UTC)
 * Thank you! And thanks also for the many constructive suggestions you made in this review. It's much more helpful (despite being more effort for me) than a review that just passes everything without comment. —David Eppstein (talk) 08:40, 28 January 2021 (UTC)
 * And thank you for bearing with me! This is the first GA review of a technical article that I've ever done, hence my general stumbling about. Cheers, Ovinus (talk) 08:44, 28 January 2021 (UTC)