Talk:Envelope (mathematics)

Cyclic
Interesting, but I can find only one mention of it because "cyclic" (even cyclid/cyclide) is such a common word. take a circle and a curve that does not come into the circle. Take the circles that are centered on the curve and orthogonal to the given circle. The envelope of those circles is the cyclic. 142.177.169.115 19:27, 13 Aug 2004 (UTC)

Problem with Example 2 or definition
It seems to me there is a problem with example 2, the point (u, v) is a solution to the simultaneous equations $$F=F_t=0$$ whether the curvature is 0 or not. The only thing that curvature = 0 implies is that the system of equations has a unique solution for the given t. When the curvature is 0, the equations are dependent and there is a line of solutions corresponding to the tangent line. So the solution to the simultaneous equations is really the union of the curve with all tangent lines at points of inflection. Is there another condition needed to eliminate these extra lines as solutions? Also, if the family is defined by $$F(x,y,t)=x-t^2$$ then the solution to $$F=F_t=0$$ is x = 0, but this does not satisfy the definition given. In this case the family of curves consists of parallel lines and has no envelope. --RDBury (talk) 06:13, 2 April 2008 (UTC)

New example 2
Hello there pma, I hope you're well. Your addition to the envelope article is a very nice piece of mathematics. But I think it might be a bit out of place in the article as it stands. The idea of an envelope is a simple one from differential geometry. When people talk about envelopes they are, by and large, talking about the envelopes of families of smooth submanifolds. Your addition seems very algebraic, and a little out of place. For example: Hölder's inequality? (and in turn Lp-spaces?) It seems to be a very algebraic and topological addition to what was, and ought to be, a differential geometric article. You obviously know a lot about the topic. Why don't you start another article or at least a new section? I think there's a lot of milage to be had in looking at envelopes from this point of view, but at the moment the example looks right out of place. It had me scratching my head, and I have a PhD in singularity thoery (e.g. discriminants, bifurcations sets, families of functions, etc) and differential geometry. If I was left scratching my head then the interested undergraduate (God help the laymen) would be totally lost. Dr Dec ( Talk )    12:50, 10 September 2009 (UTC)

Regularity of F?
Doesn't one usually impose some regularity condition on the smooth function F, e.g., that its Jacobian in the (x,y) variables be nonsingular? This guarantees that the curves cut out by F = 0 are smoothly embedded. In that case, the Whitney theorem appears to be overkill. Sławomir Biały (talk) 13:22, 17 September 2009 (UTC)


 * I do think that some minimal set of conditions should be put on F in the definition. I don't have access to the referenced source, but I strongly suspect that something is left unsaid.   Sławomir Biały  (talk) 22:24, 19 September 2009 (UTC)


 * I've re-written the first paragraph of this section to include a regularity condition. I've also removed Whitney's theorem.  Dr Dec  ( Talk )    10:31, 20 September 2009 (UTC)


 * This looks substantially better. One more thing, though, is that the family of curves Ct is the geometric data, not the function F.  Probably the article should start "Let Ct be a smoothly parameterized family of curves..."  The article can then say that this means that there exists a function F with the indicated properties (there would of course be many such F).   Sławomir Biały  (talk) 12:34, 20 September 2009 (UTC)


 * That's just a matter of taste and point of view: you have a family of curves given by a family of functions. If you define the family of curves first then you need to say why there is a generating function which has these curves as level sets. I think it's fine as it is.  Dr Dec  ( Talk )    13:02, 20 September 2009 (UTC)


 * Yes, that is a problem that I acknowledge: how does one make sense of a "smoothly parameterized family" of curves without a generating function? Of course, there are ways to do it, but none of them is essential for the article.  In any event, I still feel that the article should point out that the construction itself does not depend on the choice of generating function, rather only on the system of curves.   Sławomir Biały  (talk) 13:26, 20 September 2009 (UTC)


 * Actually, I think that overall accessibility of the section could be improved by putting the geometrical definition first. Also, it is possible then to treat both the case when the curves are given parametrically (as is done in most texts I have seen) and implicitly (as is done in the current article) as special cases of the geometrical notion.  I do think this would make things more understandable, and also resolve any potential ambiguity of the kind referred to in my earlier post.  Any thoughts?  Sławomir Biały  (talk) 15:03, 20 September 2009 (UTC)


 * The problem is that there are three "geometric" ideas of an envelope (given as E1, E2 and E3) and their definitions don't always coincide. If I had to choose a geometric definition then I'd go for the infinitesimal intersection of neighbouring curves. But this, in itself, proves difficult. We see that Ct &cap; Ct+ε comes from solving F = 0 and F(t,(x,y)) = F(t+ε,(x,y)) as ε → 0. Some authors say that this last condition is always
 * $$\lim_{\varepsilon\to 0} \frac{1}{\varepsilon} [F(t,(x,y)) - F(t+\varepsilon,(x,y))] = \frac{\partial F}{\partial t}(t,(x,y)) = 0 \ . $$
 * But what about when F(t,(x,y)) – F(t+ε,(x,y)) is divisable by a higher power of ε, say εn? Shouldn't the limiting intersection be given by solving F = 0 and
 * $$\lim_{\varepsilon\to 0} \frac{1}{\varepsilon^n} [F(t,(x,y)) - F(t+\varepsilon,(x,y))] = 0 \ ? $$  Dr Dec  ( Talk )    15:53, 20 September 2009 (UTC)
 * So basically what I'm saying is that the formal function based definition should go first and then the (non-unique) geometrical interpretations could follow. As for the envelope of parametrised curves: yeah, that's a good idea!  Dr Dec  ( Talk )    15:55, 20 September 2009 (UTC)

(unindent) I see what the issue is. It may be that there is no satisfactory way to address the problem, but it does appear that the current "functional" definition allows for some pathologies that would not ordinarily be considered as part of the envelope. The example given in the text clearly illustrates this: the line y=0 should obviously not be part of the envelope, but there it is. Sławomir Biały (talk) 17:38, 20 September 2009 (UTC)
 * It does indeed introduce some problems, but it also gives some fantastic insight. If the generating family F gives a versal unfolding of a certain singularity type then we know the local structure of the discriminant. In the case of families of smooth plane curves we can prove that the discriminant generically consists of smooth points and ordinary cusp points.  Dr Dec  ( Talk )    18:00, 20 September 2009 (UTC)

Applications section
The new applications section is a nice addition. One of the reasons that envelopes were studies in the past is because they gave singular solutions of differential equations. There are just a couple of things that I think we need to do: Let me know if you'd like me to do either of those. Dr Dec ( Talk )    09:02, 21 September 2009 (UTC)
 * 1) Give a more concrete introduction, say with ODEs, and maybe give an example.
 * 2) Explain how the caustic of reflected parallel light rays relates to differential equations.


 * Okay, I've added a subsection on ODEs. Any thoughts?  Dr Dec  ( Talk )    10:07, 21 September 2009 (UTC)


 * Caustics would make an easy example to segue into a section on the calculus of variations, that can take up conjugate points and the cut locus. A good structure might be: (1) odes, (2) caustics, (3) calculus of variations, (4) pdes.   Sławomir Biały  (talk) 12:13, 21 September 2009 (UTC)


 * Great minds think alike. I seem to have been busy writing a section on caustic when you made this remark. I would be tempted to keep the DEs together for the sake of continuity.  Dr Dec  ( Talk )    13:16, 21 September 2009 (UTC)

How to write more clearly
Honestly, I'm sick of being reverted after good faith attempts to improve the attrocious writing style of this article. Does anyone here (besides User:Declan Davis) honestly agree that the version of the writing in the definition section is clearer after this revert? Do other editors here really believe that an enormous profusion of the term "i.e." is the hallmark of good prose style? I don't wish to get into an edit war over this, already having been told off once before (and then, apparently, given free license to make such copyedits&mdash;evidently that has been rescinded). If there is consensus to make the changes, I would also like to clarify some of the text throughout the article, but I have a feeling that unless I seek such consensus here, Dr. Dec will just continue to revert all of my improvements. Sławomir Biały (talk) 14:58, 21 September 2009 (UTC)
 * I've changed it twice, and both times left an edit summary to explain my reasoning. After the first change you failed to comment on my edit summary, you just changed it back, without any reasoning. I've left edit summaries and notes on your talk page. You don't seem to have returned the favour. There's no edit war here. In fact the problem would go away if you left notes on my talk page and tried to discuss conflicts, as I have repeatedly done today. This post you have just made is antagonistic. You should have come to my talk page first and explained your reasoning, not just reverted it and then placed a thinly veiled insult on a general talk page.  Dr Dec  ( Talk )    15:05, 21 September 2009 (UTC)
 * I didn't realize that you are the WP:OWNer of the article. Is there anyone else whose permission I need to ask before making minor copyedits to the article?  Sławomir Biały  (talk) 15:08, 21 September 2009 (UTC)
 * I have not suggested that I am the owner. As I said on your talk page: "It's becoming very frustrating: we've both spent a lot of time over the last couple of days working on this article. While I take the time to suggest changes to you, and then explain changes, you just undo my work without any warning, let alone explanation. Please be more considerate." And your sarcasm isn't very constructive. If you want to make changes then explain them to people and leave an edit summary. Being frustrated by you going around making changes without any discussion is not claiming ownership; it's simply natural. Wikipedia works much better when editors collaborate.  Dr Dec  ( Talk )    15:21, 21 September 2009 (UTC)

Specific problems with the writing: -- Sławomir Biały (talk) 15:17, 21 September 2009 (UTC)
 * It is loaded with the first person plural. This is not encyclopedic language.  One of the purposes of the copyedit that was reverted was to clean up some of this in the definition section.
 * The Definition section needs to give context early on to the function F. The lead talks about curves and manifolds, but the definition kicks off with a discussion of functions and only later (vaguely) makes the connection with curves.  A segue was needed.
 * The article has an enormous profusion of the term "i.e.", which is generally poor style.


 * I agree with all of these points. I just recommend a multilateral resolution instead of what has been the so far unilateral one.  Dr Dec  ( Talk )    15:25, 21 September 2009 (UTC)


 * Good. I've tried to clarify the first section, without the term "cut out" that you seem to disagree with (though I would argue that it is quite a standard term throughout mathematics for the locus of an equation).  Sławomir Biały  (talk) 15:37, 21 September 2009 (UTC)

Possible sample image
Does this image (used in Timbre) fulfill the definition of mathematical envelope? I'm considering adding it as a sample to the article. unsigned comment by 128.138.158.200, 23:47 (UTC), 28 September 2009.

This does not fulfill the definition of a mathematical envelope. As the article clearly states: "In mathematics, an envelope of a family of manifolds (especially a family of curves) is a manifold that is tangent to each member of the family at some point." Your picture is just a single curve, and not a family of curves. The red curve in the picture is tangent to the black curve at some of its points, but that's nothing special: there are uncountably many curves tangent to any given curve, not least the curve itself! Dr Dec ( Talk )    22:47, 28 September 2009 (UTC)

Thanks for your input. This notion of envelope (or something close to it) is widely used in signal processing. Let me try to formalize it. Let $$V = A cos(\phi)$$ be a signal; all its parameters are time-dependent: $$V(t), A(t), \phi(t)$$. The signal $$V(t)$$ is said to oscillate within the envelope defined by $$\pm A(t)$$. What do you think? (There is another illustration at .) unsigned comment by 128.138.158.200, 00:25 (UTC), 29 September 2009.


 * It's a nice idea, but it's just not what this article is about. You have a single curve, and not a family! and the "envelope" in this sense is just a curve that that bounds the region filled by this one curve. You could write a new article, say Envelope (sound) or something like that and then link to it from this page via a disambiguation. Secondly, please sign your posts by putting ~ at the end of your messages. It'll authomatically be changed into your user details, the time and date so that people know who wrote what and when.  Dr Dec  ( Talk )    09:52, 29 September 2009 (UTC)
 * There is already an article called Envelope detector with most of what you're talking about. It has a red link for Envelope (waves) but it's hard to see why you'd need two articles so maybe a move would be better. Looks like there's some work to be done but it's not my area of expertise.--RDBury (talk) 12:25, 29 September 2009 (UTC)
 * You're right, it's there, right under my nose. Thanks! 128.138.43.211 (talk) 21:59, 1 October 2009 (UTC)
 * Envelope detector should be about the circuits that have envelope (waves) as their output. But it is not, it is largely about envelope (waves) themselves.  Sp in ni  ng  Spark  21:48, 15 May 2010 (UTC)

Confusing definition
"In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point."

As far as I understand english, this definition is wrong. I would prefere something like that:

"In geometry, an envelope of a family of curves in the plane is a curve that is tangent to some member of the family at each point." (i.e. at each point of the envelope). It is not clear if the word "point" in the original definition is related to "point of the envelope" or "point of the member of the family". Perhaps it is more clear for native speakers, but other wikipedians are often confused by this definition (and, moreover, try to "correct" proper definitions in their local languages... Could you please write the definition more precisely? Tescobar (talk) 21:21, 15 September 2010 (UTC)


 * Not just confusing but still wrong (now two years later).


 * Consider for example the family of curves $$(x-a)^3$$. Now consider the $$x$$-axis. Clearly this axis is "tangent to each member of the family at some point" and even is "tangent to some member of the family at each point". Yet every curve in the family crosses onto both sides of that axis, in total disagreement with the intuitive concept of it being an envelope (although I'm less sure about the concept of a caustic).


 * A more sensible definition would be something like: an envelope is (a continuous section of) the (tightest possible) boundary which encloses (the image of) a family of curves (or at least of curve segments). (Or maybe need something about being a closed boundary of open curve segments..?) At least, there needs to be a greater discussion of different ways authors have defined the concept and their motivations. (The text about higher dimensions is also unbalanced. And I think explaining the relation to signal processing, where the signal is the envelope of the family of modulated carrier waves that can be transmitted to represent the signal, is warranted.) Cesiumfrog (talk) 02:41, 8 November 2012 (UTC)

$\mathcal{D}$ in Alternative definitions
In section Alternative definitions a set $\mathcal{D}$ makes its appearance, which is then explained as being "the set of curves given by the first definition at the beginning of this document".

In fact, $\mathcal{D}$ refers to a differently defined set  on September 10, 2009; the definition was  on August 16, 2010, leaving dangling references to an undefined $\mathcal{D}$. The original definition was sourced to. It is hard to understand and I do not recommend reinstating it here.

It seems to me that alternative definition #2 ("tangent to all of the Ct") is the same as the definition given at the beginning of the article ("tangent to each member of the family at some point"). What is missing in either case, I believe, is that each point of the envelope serves as a tangent point. (To see this is needed, take some arbitrary circle C and let L be some tangent of C. Now consider the family of curves F defined by Ft = C for all t. Is L an envelope of F? According to the definition it is: L is tangent to each member of F at some point.) Additionally, the envelope should be defined as the largest set having that property. (For consider again the same family F as before. Without the maximality requirement, any proper segment of C is also an envelope of F.) --Lambiam 16:18, 7 January 2018 (UTC)

The last issue is in fact the same as identified by Tescobar and Cesiumfrog in the preceding section. It has not been addressed for more than seven years. --Lambiam 16:43, 7 January 2018 (UTC)


 * Part of the problem seems to be ambiguity about what is and isn't an envelope.


 * The boundary set of the union of images of a set of curves
 * The set of points that each intersect exactly one of the curves.
 * A curve that intersects every member of the set.
 * A curve that at every point is tangential to some member of the set.
 * Zeros of the first derivative with respect to the family parameter.
 * Combinations of the above, etc.


 * It's fairly easy to think of examples that are envelopes according to some definitions, and are not according to other definitions. Cesiumfrog (talk) 23:26, 8 January 2018 (UTC)

Faulty Proof?
The derivation of the condition $$\partial F/\partial t = 0$$ seems to be wrong since the point of intersection $$(x,y)$$ of the curves $$F(t,x,y)=F(u,x,y)=0$$ changes with t and u. The limit as u tends to t does not give the partial derivative since that would assume x and y are fixed. — Preceding unsigned comment added by 193.188.47.48 (talk) 11:01, 16 January 2019 (UTC)

Bad picture of astroid
The astroid should appear symmetrical about the line x = y, but it does not. It would be much better to replace it with a good picture of an astroid.50.205.142.50 (talk) 01:52, 13 May 2020 (UTC)