Talk:Cusp (singularity)

This Article Needs re-writing
The term cusp in the title should be ordinary cusp. The ordinary cusp is given by the equation x2 + y3 = 0. There are many different kinds of cusp, e.g. a rhamphoid cusp (coming from the Greek meaning "Beak-like") given by the equation x2 + y5 = 0. Notice that the ordinary cusp and the rhamphoid cusp are not diffeomorphic to each other, and so from a singularity point of view are quiet different. I shall start making some changes now. Dr Dec ( Talk )    17:25, 28 July 2009 (UTC)


 * The original claim that the Jacobian and the Hessian must be singular is not enough. The cusps are given as zero-level-sets of the A2k-singularities, i.e. with normal forms x2 + y2k+1 where k is a natural number. The A2k+1-singularities, i.e. with normal forms x2 + y2k where k is a natural number are clearly not cusps.  Dr Dec  ( Talk )    18:01, 28 July 2009 (UTC)


 * I've made some pretty wholesale changes. Let me know what you think...  Dr Dec  ( Talk )    19:16, 28 July 2009 (UTC)


 * Dear Dr. Dec, Well, I am not happy with your changes: I cannot understand your definition. I suggest that you keep the original one, fix it, if necessary and then introduce in another section your. TomyDuby (talk) 20:43, 28 July 2009 (UTC)

Hi TomyDuby, I hope you're well. The old definition was wrong! It was trying to say that f needed to have a degenerate singularity, i.e. both the Jacobian matrix and the Hessian matrix need to be singular. This is a necessary, but not sufficient condition. Take f(x,y) = x4 + y4, for example. Here we see that f(0,0) = 0 and both the Jacobian and Hessian matrices are singular at (0,0), i.e. the origin is a degenerate critical point of f. The zero-level-set of f consists of a single point: the origin. A single point is not a cusp! The cusps are given, and trust me on this one, by the family of zero-level-sets x2 - y2k+1 = 0, where k ≥ 1 is an integer.

Now, this doesn't mean that every curve with a cusp is given by one of these equations. Every cusp is locally diffeomorphic to one of these forms. So there will be a diffeomorphism taking a neighbourhood of any old cusp point onto a neighbourhood of one of our standard pictures (x2 - y2k+1 = 0) such that the curve is carried to the curve.

I think that the old article didn't cover enough ground. The ordinary cusp is just one member in a countably infinite family of cusps. To focus on the ordinary cusp (as the old article did) is, in my opinion, a little narrow sighted.

If you didn't understand the definition then that's not a problem with the article per se, maybe we need to give more explanation. You obviously know about the ordinary cusp. What didn't you understand about the definition? Did you understand the diffeomorphism article? (This article is key). We can't talk about singularities of plane curves without talking about diffeomorphisms, they form the basic equivalences, just like homeomorphisms do in topology, and isomorphisms do in group theory.

I know it might seem like over complication, but if you want to get things right then you need to refine things, and the only way to refine things is to use more sophisticated ideas.

As a refernce, see J. W. Bruce and P. J. Giblin's book giving in the main article. The latter was my PhD supervisor and he taught me all about cusps. RSVP  Dr Dec  ( Talk )    23:21, 29 July 2009 (UTC)

Some questions
I'd like to ask several questions about this article:

1. The "Examples" section seems hard to follow for a mathematically savvy non-specialist. Surely there are a lot of examples of geometric objects with cusps, that could be mentioned and linked to. For example, how about cardioid? Or is the cusp of a cardioid only a cusp according to some broader definition of cusp than the one used here? If so, then the article ought to start with the broader definition first.

2. The lede doesn't actually define cusp -- it says: "In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities...." "...[A] type of singular point..." doesn't narrow it down enough. Also, the lede should give an intuitive idea of what a cusp is, in addition to defining it completely.

3. Is it a necessary condition for a cusp that the one-sided derivative at the cusp be the same in both directions (on both branches)?

4. The lede says "The plane curve cusps are all diffeomorphic to one of the following forms: x2 &minus; y2k+1 = 0, where k ≥ 1 is an integer." Yet the image in the lede shows x3–y2=0 as an example. This seems to be inconsistent. Is it actually consistent, or is this evidence that the diffeomorphic comment in the lede is too narrow? Duoduoduo (talk) 14:47, 4 August 2011 (UTC)

Why does "spinode" redirect here?
Not mentioned in article. What does it mean? Equinox (talk) 04:31, 27 November 2015 (UTC)
 * D.Lazard (talk) 10:01, 27 November 2015 (UTC)

Dubious classification
I have tagged as dubious the last assertion of the lead, as it would imply that the curve defined by the parametric equation
 * $$\begin{align}

x&=t^2\\ y&=t^4+t^5 \end{align} $$ is not a cusp. D.Lazard (talk) 10:48, 28 November 2015 (UTC)


 * From the point of view of singularity theory this is a higher order rhamphoid cusp, discussed lower down in the article. --Salix alba (talk): 11:56, 28 November 2015 (UTC)


 * The article discusses the classification of cusps by using the existence of some diffeomorphisms, and defines a cusp as a singularity that is mapped on $$y^2-x^k=0$$ with odd k. As far as I know the order of the contact of the curve with its tangent is invariant under diffeomorphisms, and this order is k at the cusp. As, in my example, the order is even (4), my example is not a cusp for the authors of this article, although it is a cusp in classical literature. I am not well aware of English terminology, but, in French, such a cusp is called "cusp of second species" (point de rebroussement de seconde espèce). D.Lazard (talk) 14:16, 28 November 2015 (UTC)


 * Possibly "second-order cusp"? I can see a handful of matches in Google Books. Equinox (talk) 00:03, 29 November 2015 (UTC)
 * I do not think so: In general "order" refers to the multiplicity of the singularity, which is two for all cusps considered in the article, as well as for above example. By the way, the article asserts implicitly that a cusp is always a singularity of multiplicity two. I think that singularities of higher multiplicity that look similar are also traditionally called cusps. For example the singularity of
 * $$\begin{align}

x&=t^3\\ y&=t^4 \end{align} $$
 * is probably called a "third order cusp". D.Lazard (talk) 09:17, 29 November 2015 (UTC)

If you do the sums you can show $$(x^2,x^4+x^5)$$ fits in the A4 class with normal form $$(x^2,x^5)$$. You can think of this as the $$(x^2,x^5)$$ form with an inflection at the cusp point. You can see this by drawing the dual curves of the both curves. Inflections are not preserved by diffeomorphism which is why they fall into the same class. Some texts like Rutter's "Geometry of curves" Call the $$(x^2,x^4+x^5)$$ rhamphoid and the $$(x^2,x^5)$$ form a Ceratoid cusp. --Salix alba (talk): 21:09, 29 November 2015 (UTC)
 * More elementarily, the parameterizations $$(t^2,t^4+t^5)$$ and $$(t^2,t^5)$$ are exchanged by the diffeomorphisms $$x\to x, y\to y\pm x^2.$$
 * A problem of terminology remains: this article does not consider as cusps singularities of multiplicity higher than 2, such as $$y^4-x^5=0.$$ Are they called cusps in the literature? I guess so, at least in the classical literature, as they have the same shape. D.Lazard (talk) 10:34, 1 December 2015 (UTC)

Definition of cusp
The Encyclopedia of Mathematics says a cuspidal point is A singular point of a curve, the two branches of which have a common semi-tangent there. Equivalently, Mathworld says a cusp is a point at which two branches of a curve meet such that the tangents of each branch are equal. Our article starts out by saying a cusp is a point on a curve where a moving point on the curve must start to move backward, which I take as implying the same thing, though I think  move backward is vague and should be augmented by the equal tangent feature and the citations, if they are acceptable.

However, our article goes on to say without citation:
 * For a plane curve defined by a differentiable parametric equation
 * $$\begin{align}

x&=f(t)\\ y&=g(t), \end{align} $$
 * a cusp is a point where both derivatives of $f$ and $g$ are zero, and at least one of them changes sign.

I believe this is too broad, based on a superellipse with parameter 1 < n < 2:

\begin{align} x\left(t\right) &= {|\cos t|}^{\frac{2}{n}} \cdot a \sgn(\cos t) \\ y\left(t\right) &= {|\sin t|}^{\frac{2}{n}} \cdot b \sgn(\sin t). \end{align} $$

For a = b = 1,
 * $$ dx/dt= \pm (2/n)|\cos t|^{(2-n)/n}\sin t ,$$
 * $$ dy/dt= \pm (2/n)|\sin t|^{(2-n)/n}\cos t ,$$

(where in each case only one of the ± applies, depending on the quadrant). When 1 < n < 2 these both go to 0 at t = 0, pi/2, pi, 3pi/2, and at each of these points one of them changes sign. But, while the two branches share a common tangent (at each of these points both branches' slopes (dy/dt) / (dx/dt) go to infinity or both go to 0), and hence the two branches share a common tangent, the common tangent is not one where ther point must "move backward".

So: (1) are these point called cusps, in which case the "move backwards" idea is too restrictive, or (2) are they not called cusps, in which case our statement about both derivatives dx/dt and dy/dt going to 0 and at least one changing signs is too broad (and the quotes in the sources need to be more specific about the nature of the shared tangent)? Loraof (talk) 20:32, 3 October 2017 (UTC)
 * Good point. The definition of this article is wrong also because of the example $$x=t^2, y=t^2.$$ It is also wrong if $$y'/x'$$ does not has any limit. On the other hand the quotes in the sources lack of a definition of the tangent in this case: at a regular point, the tangent is the line with the value of $$y'/x'$$ at the point, while, here, $$y'/x'$$ is not defined at the point, and must be replaced by the limit of $$y'/x'$$ when t tends to its value at the point. Note also that your example is a cusp for Mathworld, but not for Encyclopedia of Mathematics. IMO, Mathworld is wrong.
 * The definition may be corrected by replacing "differential parametric equation" by "analytic parametric equation". But this would be too restrictive for the section "Classification in differential geometry". On the other hand this section exclude from the classification of cusps $$x=t^4, y=t^5,$$ and $$x=t^2, y=t^4+t^5$$ (the latter is generally called "cusp of the second kind").
 * These contradictions suggest that there is no universally accepted definition of a cusp, except for ordinary cusps. Even for ordinary cusps the definition in differential geometry differs strongly from that of algebraic geometry. D.Lazard (talk) 04:39, 4 October 2017 (UTC)
 * I have edited the lead, (and added some tags elsewhere), in an attempt to solve the issue. I have restricted the lead to analytic curves, because I have never heard of a definition of a non-analytic curve. Moreover, if one try to generalize definition to smooth functions, many pathological phenomena may occur, which are difficult to take into account in a plausible definition. I have also added an accurate definition for the implicit case.
 * During this, I have found a much simpler example than the superellipse. It is the curve $$y^3-x^4=0,$$ or equivalently $$x=t^3, y=t^4.$$ If the axes are rotated, one obtain the curve $$x=t^3+t^4, y=t^3-t^4,$$ where the singular point looks like a regular although both derivatives are positive near the singular point. Thus, a correct definition must involve a directional derivative; this is the case, after my edits.
 * I hope that, now, the definitions are correct. It remains to find sources. D.Lazard (talk) 14:06, 4 October 2017 (UTC)
 * Thanks. I don't have access to either of the books mentioned in the References section—do you?


 * Loraof (talk) 18:19, 4 October 2017 (UTC)
 * The current stipulation in the parametric case is that the directional derivative, in the direction of the tangent, changes of sign (the direction of the tangent is direction of slope $$ \lim (g'(t)/f'(t))$$). I don't understand this—why is this condition not met by my superellipse example? Loraof (talk) 18:26, 4 October 2017 (UTC)
 * Loraof (talk) 18:19, 4 October 2017 (UTC)
 * The current stipulation in the parametric case is that the directional derivative, in the direction of the tangent, changes of sign (the direction of the tangent is direction of slope $$ \lim (g'(t)/f'(t))$$). I don't understand this—why is this condition not met by my superellipse example? Loraof (talk) 18:26, 4 October 2017 (UTC)


 * Can we just say, in the parametric context, that a point is a cusp if and only if x′ = 0 = y′ at that point and the two branches that meet at the point share a tangent line that crosses the curve? (Never mind, it doesn't always work) Loraof (talk) 20:40, 4 October 2017 (UTC)
 * "where a moving point on the curve must start to move backward" This means that the speed of the point changes of sign, where the speed is a vector in the direction of the tangent; this means that the semi-tangents (that is the tangent rays of the two branches) are equal. In your example, if the drawing is correct, the semi-tangents at, say (1,0), are not in the same half-plane, and thus are not equal. Computation: It seems that, in your formula for the derivative, the $±$ must be replaced by $–$ for $dx/dt$ and by $+$ for $dy/dt$; the directional derivative at t = 0 is $dy/dt$, which is positive for t near from 0, whichever is the sign of t. D.Lazard (talk) 21:52, 4 October 2017 (UTC)

The Giblin and Porteous references are both from the world of singularity theory. The will use the definition of a cusp based of equivalence classes discussed in the "Classification in differential geometry". This definition stems from the work of Thom and Arnold. So a normal cusp is anything which is diffeomorphic to (t^2,t^3). Rhaphoid cusp will be anything diffeomorphic to (t^2,t^5) and there are many higher forms. These definitions work for all function and don't require things to be analytic. The precise definition in these terms is a bit complex and is not really a one liner. --Salix alba (talk): 23:19, 4 October 2017 (UTC)
 * Because of the technicality of this precise definition, it seems better to follow the historical development, that is considering first the analytic (or algebraic) case, and then generalizing to the differentiable case. I have edited the lead accordingly. D.Lazard (talk) 11:00, 5 October 2017 (UTC)