Wikipedia:Reference desk/Archives/Mathematics/2023 August 8

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August 8

Closed plane curve questions

Got some questions about plane curves. In particular, I'm pretty sure that the answer to all of these questions is yes, but proofs elude me for the moment.

1. Does every closed plane curve admit some line that touches the curve at exactly one point? Similarly, do they all admit some line that touches the curve at exactly two points?

2. Does every concave closed plane curve admit some line, lying entirely outside of the open region enclosed by the curve, that touches the curve at exactly two points? Essentially, an exactly two-point tangent line?

3. In the isometric problem, there's a trick where a "dent" in the boundary of a concave closed plane curve can be reflected across a two-point tangent line to increase the enclosed area while maintaining boundary length. Can every concave closed $C^{1}$ plane curve be turned convex by a finite number of such reflections? GalacticShoe (talk) 03:27, 8 August 2023 (UTC)

For the purpose of this discussion, let a tangent of a closed plane curve be a line that intersects the curve, but not the interior of the region enclosed by the curve. So the line given by the equation $x+y=0$ in the Cartesian plane is a tangent of an axes-oriented square in the first quadrant with the origin as one of its corners.

1. While I do not immediately see a proof, I think that indeed every closed plane curve has some tangent that intersects the curve solely at one point. No tangent of a circle touches it at more than one point, and each tangent of a square either touches it at a single point or at an infinte collection of points, never exactly two, so the second part of 1 should be answered with no.

2. By "concave plane curve", do you mean one enclosing a non-convex region? Take a square and make a dent halfway one of its sides. This is a counterexample.

3. You can put infinitely many smaller and smaller dents in the side of a square, like each next one half the size of the previous one. Then you'll need infinitely many reflections. You can also make a smaller out-dent on the first in-dent, which becomes an in-dent after reflection. In that smaller out-dent you can make an even smaller in-dent and repeat the process, alternating in and out, ad infinitum.

 --Lambiam 09:04, 8 August 2023 (UTC)

I think there may some confusion about what is meant by a "closed curve". If you insist that it's the image of a one-one differentiable map from the circle to the plane, and that the inverse map is differentiable, then a square would not be allowed. On the other hand if you only insist that the map be continuous then you allow space filling curves and come of the questions become pretty meaningless. Even if you add the conditions that the map is one-one and it's inverse is continuous, you'd still allow curves which have no tangent line at all in the usual sense. --RDBury (talk) 14:00, 8 August 2023 (UTC)

That's an excellent point; for the purposes of this question, I take a "closed curve" by the continuity definition. Since this allows for space-filling curves, I think this now means that the second half of the first question is proven false, since there appear to be closed space-filling curves, like the Moore curve. This does leave the question open for Jordan curves, however, since no space-filling curves are simple. GalacticShoe (talk) 14:19, 8 August 2023 (UTC)

Hey Lambiam, thanks for the response!

1. I should clarify for this question that in terms of lines touching the curve at exactly two points, I also mean lines that can intersect the shape rather than lying tangent to it; I imagine that every closed plane curve has some one-point tangent line that can be translated to intersect the curve at exactly two points, but I don't know how one would prove this.

2. That's exactly right, thanks for the counterexample. I was thinking too much of smooth curves when I was coming up with this question, and upon reflection even with smoothness there would be counterexamples (e.g. by smoothing out the corners and only the corners of the shape you described.) For bookkeeping purposes I'm just going to write a bold note here saying that this is resolved in the negative.

3. I was wondering, when you mention the denting process, are the dents made "smoothly"? For the purposes of the question I'm looking for a $C^{1}$ curve, and I'm curious whether an infinite amount of "smooth" denting can create a shape that is still $C^{1}$. GalacticShoe (talk) 14:10, 8 August 2023 (UTC)

Re 3, I think this is even possible while resulting in a $C^{\infty }$ curve. Define function $\rho$ on the interval $(-1,1]$ by:

$\rho (x)=0$ for $x=0,1$;

$\rho (x)=\sin({\frac {1}{x}})\exp \left({\frac {-1}{x^{2}(1-x^{2})}}\right)$ otherwise.

Now take the curve with polar representation $r(\varphi )=1+\rho \left({\frac {\varphi }{\pi }}\right),-\pi <\varphi \leq \pi .$  --Lambiam 16:00, 8 August 2023 (UTC)

This is a great example but I was wondering how you can determine whether the resultant curve is indeed concave? It would seem that the nature of polar curves allows for some 'leeway' in whether perturbations to the curve make concave sections. For example, $r(\theta )=1+{\frac {1}{6}}\sin(2\theta )$ is a convex curve despite the fluctuations in ${\frac {1}{6}}\sin(2\theta )$. Similarly, I'm having a hard time determining whether $1+\rho ({\frac {\theta }{\pi }})$ becomes concave in infinitely many places. Thanks again! GalacticShoe (talk) 23:08, 8 August 2023 (UTC)

I had some doubts myself and should not have posted this without checking. I am fairly convinced, though, a provably correct counterexample can be constructed.  --Lambiam 10:32, 9 August 2023 (UTC)

After thinking more about it, I'm even fairlier convinced. The basic step would be to have $r(\varphi )$ behave in a neighbourhood of $0$ like $\left(1+\rho (\varphi /\pi )\right)/\cos \varphi .$ In the mapping $r(\varphi )\mapsto (x,y)$ we then have $x-1=\rho (\varphi /\pi ),$ which switches from positive to negative and back infinitely often in any neighbourhood of $0.$ Function $r$ can always be glued, with preservation of $C^{\infty }$-hood, to a function making it periodic with period $2\pi$.  --Lambiam 18:57, 9 August 2023 (UTC)

Re 1. The Latin word tangens from which we have our word tangent literally means "touching", so the use of the word "touch" for lines suggests that tangent lines are meant. So is it correct to say that you are interested in constraints on curves such that for each curve $C$ meeting certain conditions there exists some line $L$ such that the cardinality of $C\cap L$ is $1$ (or $2$)?  --Lambiam 20:27, 8 August 2023 (UTC)

That is indeed what I was interested in. In particular, whether all closed plane curves $C$ admit some line $L$ such that $\#(C\cap L)=1$, and whether all Jordan curves $C$ admit some line $L$ such that $\#(C\cap L)=2$. GalacticShoe (talk) 23:09, 8 August 2023 (UTC)

The dragon curve consists of an infinite sequence of similar space-filling segments joined at each junction by one point. Its boundary consists of an infinite sequence of Jordan curves, joined likewise. I suspect this dragon-segment boundary is a counterexample to the existence of a two-point intersection; in fact, if there is more than one point of intersection, I think there are countably infinitely many.  --Lambiam 10:45, 9 August 2023 (UTC)

It does seem likely that all lines intersecting the dragon curve do so in either one or more than two points, but I think that, as a space-filling curve, it can't be Jordan. At least, that's what I'm reading from this StackExchange post; it seems that although space-filling curves are indeed continuous, and can very well be closed, they can't be injective. That being said, I get the feeling that some non-space-filling fractal might work as a counterexample, although I struggle to imagine what such a fractal would look like. GalacticShoe (talk) 13:12, 9 August 2023 (UTC)

I was talking about the boundary, which is a chain of similar Jordan curves. An approximation of this Jordan curve, only a few iterations but hopefully conveying the idea, can be seen here.  --Lambiam 19:42, 9 August 2023 (UTC)

Ah, I see what you mean; the boundary of the dragon curve certainly does seem as good a counterexample as any, and I can't think of any way a line could intersect the curve in exactly/only two points. I will have to look into whether there are explicit ways of representing the boundary so as to formalize this, but as it stands it seems likely that this works as a counterexample. GalacticShoe (talk) 21:07, 9 August 2023 (UTC)

Just for posterity's sake, I should note here that for no. 2, I think it's possible for the counterexample rounded square to be $C^{\infty }$ as well. I haven't constructed any explicit examples, but I imagine one could use flat functions to keep most of the sides perfectly straight while still rounding off the corners. GalacticShoe (talk) 13:30, 9 August 2023 (UTC)

Define function $\gamma$ by

$\gamma (x)=0~$ if $~x\leq -{\tfrac {1}{2}},$

$\gamma (x)=1~$ if $~x\geq +{\tfrac {1}{2}},$

$\gamma (x)={\tfrac {1}{2}}\left(1+\tanh \,\tan(\pi x)\right)~$ otherwise.

Then, if $f$ is $C^{\infty }$ on $[a,+{\tfrac {1}{2}}],$ where $a<-{\tfrac {1}{2}},$ and $g$ is $C^{\infty }$ on $[-{\tfrac {1}{2}},b],$ where $+{\tfrac {1}{2}}<b,$ so is function $h$ defined by

$h(x)=(1{-}\gamma (x))f(x)+\gamma (x)g(x).$

on the interval $[a,b].$ So such gluing to round the corners is possible.  --Lambiam 20:15, 9 August 2023 (UTC)

On reflection, I think the first half of #1 may be simpler than I expected. I suspect that if one can prove that every convex curve admits such a line (which is almost certainly the case), then for any concave curve $C$, one can just take the line corresponding to the convex hull of $C$. GalacticShoe (talk) 21:40, 9 August 2023 (UTC)

Daniel Mathias on Math StackExchange makes an even simpler argument: rather than the convex hull of $C$, consider an arbitrary circle circumscribed around $C$. At a point of intersection $P$, the tangent line of the circle suffices. GalacticShoe (talk) 22:39, 9 August 2023 (UTC)