Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of $C$, in the sense that the segment $xy$ intersects $S$ at precisely the one point $x$. I am interested in the shortest $C$ with this property. In computational geometry, such paths are called watchman tours, and there are many results concerning polygons in the plane finding such tours.

This question arose at a conference I'm attending, and I was pointed to a paper by V. A. Zalgaller:

"Shortest Inspection Curves for the Sphere" (Journal of Mathematical Sciences, Volume 131, Number 1, 5307-5320; Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 87–108.)

I cannot access the paper from the conference, but from the abstract it appears he focused on open rather than closed curves.

Has anyone heard of this natural question? Can you point me to relevant literature? Thanks!

Addendum. Here is the $4\pi$ saddle / baseball-stitches curve suggested by Gjergji Zaimi: • My guess is $4\pi R$, which I can achieve gluing 4 semicircles of radius $R$ (saddle shape). My attempts at finding a clever way to prove the lower bound using a Crofton formula have failed though... – Gjergji Zaimi Jun 29 '11 at 9:51
• (of course if we allow unions of closed curves, the infimum of length is 0, e.g. C = 6 small circles, each passing through a vertex of the cube $[−1,+1]^3$, or even C = the vertices themselves, as a degenerate case). – Pietro Majer Jun 30 '11 at 8:35
• It is not difficult to see that if the curve is at constant distance from the sphere, then its length is at least $4\pi$. Indeed suppose that it is a constant distance $r$ from the center. The area of the set of points newly seen from a short segment of length $ds$ is $a(r)ds$, with $a(r)=2\sqrt{r^2-1}/r^2$. This is maximal for $r=\sqrt{2}$, then $a=1$. Since each point of the sphere is "newly seen" from at least one point of the curve, the result follows. The curve proposed by Gjergji Zaimi is at constant distance from the center, so it is optimal at least in this restricted sense. – Jean-Marc Schlenker Jul 2 '11 at 6:24
• It's maybe obvious, but let's also remark that there does exist a minimal length (continuous) inspection curve $\gamma:\mathbb{S}^1\to\mathbb{R}^3$, $|\gamma(t)|\ge 1$ (thanks to Ascoli-Arzelà and semicontinuity of the total variation; and because the condition of "seeing the whole sphere" is closed under uniform convergence). A couple of questions: 1: is $\gamma(t) > 1$ for all $t$? (can the minimal inspection trajectory avoid landing?) 2: is $\gamma$ an extreme curve? (= a curve in the boundary of its convex hull). – Pietro Majer Jul 2 '11 at 23:31
• This problem appears in the last lines of a math popularization article by Jean-Baptiste Hiriart-Urruty, Du calcul différentiel au calcul variationnel, in Quadrature 70(2008):8-18, see www.math.univ-toulouse.fr/~jbhu/Fermat_Quadrature.pdf There the problem is presented as open and attributed to Alain Grigis. – Jean-Marc Schlenker Jul 5 '11 at 6:15

I have recently finished a paper called

The length, width, and inradius of space curves

where it is shown that the length $L$ of any closed curve $\gamma\colon[a,b]\to \mathbf{R}^3$ inspecting the unit sphere $\mathbf{S}^2$ must be at least $$6\sqrt{3}\approx 10.3923,$$ which is almost $83$% of the conjectured lower bound $4\pi\approx 12.5664$ by Zalgaller and Gjergji Zaimi. This paper discusses a number of ideas and techniques to study the inspection problem, which I will summarize below. Here inradius is the supremum of the radii of all spheres which are contained in the convex hull of $\gamma$ and are disjoint from $\gamma$. It is easy to see that $\gamma$ inspects $\mathbf{S}^2$, up to a translation, if and only if its inradius $r$ is $1$.

1. The theorem of Wienholtz. The first approach to the inspection problem, which is also called the inradius problem in the paper, is to study the more basic width problem, i.e., minimizing $L$ subject to a constraint on $w$, the infimum of the distances between all pairs of parallel planes which contain $\gamma$ in between them. Since $$w\geq 2r,$$ any lower bound for $L/w$ yields a lower bound for $L/r$. To this end, one can apply a beautiful unpublished result of Daniel Wienholtz from 2000 who used the Borsuk-Ulam theorem to show that any closed space curve $\gamma$ may be positioned between a pair of parallel planes $H_0$, $H_1$, which touch $\gamma$ twice each in alternating fashion as one goes around $\gamma$. Let $L_1$ be the length of the projection of $\gamma$ into a line orthogonal to $H_0$, and $L_2$ be the length of the projection of $\gamma$ into $H_0$. Then, since projections do not reduce width, $$L\geq \sqrt{L_1^2+L_2^2}\geq\sqrt{(4w)^2+(\pi w)^2}=\sqrt{16+\pi ^2}\,w\geq \sqrt{16+\pi ^2}\,2r.$$ So $L/r\geq 2\sqrt{16+\pi ^2}>10.1724$. The first inequality above follows from the Cauchy-Schwartz inequality for integrals, while the second uses Wienholtz's theorem and the Cauchy-Crofton formula, both of which are included in the paper. It is also shown that the inequality $L/w\geq \sqrt{16+\pi ^2}$ is better than $99.43$% sharp. In particular, the length of the shortest closed curve of width $1$ must be approximately $5.1$.

1. The notion of horizon. In order to impove the lower bound for $L/r$ we adopt a more direct integral-geometric approach by developing the notion of horizon. The horizon of a curve $\gamma$ is defined as the measure of all tangent planes of $\mathbf{S}^2$ counted with multiplicity which intersect $\gamma$: $$H(\gamma):=\int_{p\in S^2} \#\big(\gamma^{-1}(T_p S^2)\big)\, dp.$$ Since $\gamma$ is closed and inspects $\mathbf{S}^2$, $H(\gamma)\geq 8\pi$. Next, to derive an upper bound for $H(\gamma)$ we show that for every curve $\gamma$ inspecting $\mathbf{S}^2$ there is another inspection curve $\tilde \gamma$ with length $\tilde L\leq L$ such that the tangent lines of $\tilde\gamma$ do not enter $\mathbf{S}^2$. For these curves one can show that $H(\tilde\gamma)\leq \frac{4\pi}{3\sqrt3}\tilde L$, by studying the following expression for the horizon which we compute using the area formula: $$H(\gamma)=\int_a^b\int_0^{2\pi} \frac{1}{\|\gamma\|^2}\left| \sqrt{\|\gamma\|^2-1}\sin(\alpha)\cos(\theta)+\cos(\alpha) \right|\,d\theta dt.$$ Here $\alpha$ is the angle betweeb $\gamma$ and $\gamma'$ and we are assuming that $\|\gamma'\|=1$. The tangent line property of $\tilde\gamma$ means that $\sin(\tilde\alpha)\geq 1/\|\tilde\gamma\|$. Now putting everying together we have $$8\pi\leq H(\tilde\gamma)\leq \frac{4\pi}{3\sqrt3}\tilde L\leq \frac{4\pi}{3\sqrt3}L,$$ which yields $L\geq 6\sqrt3>10.3923$.

1. Other estimates There are some more things one can do via Crofton's formula, which are included in the paper. For instance if $M:=\max\|\gamma\|$, $m:=\min\|\gamma\|$. Then $$L\geq\frac{2\pi Mm}{\sqrt{M^2-1}}.$$ In particular, when $M=m$, or $M\leq 2/\sqrt3$, then $L\geq 4\pi$. This generalizes the obsevartion by Jean-Marc Schlenker above on inspection curves of constant height. Further, in the paper we study the above integral formula for the horizon via a computer which shows that if $m\geq 1.6$, then $L\geq 4\pi$. Thus if a counterexample to the $4\pi$ conjecture exists, then for some $t\in[a,b]$ we must have $$1.15 \leq\|\gamma(t)\|\leq 1.6.$$ Finally the paper describes how the Wienholtz's theorem may be used to extend the above estimates to higher dimensions, but I do not know how sharp these are.

In conclusion let me point out that it is very tempting to think that the inspection problem should have a slick or elegant solution via some kind of integral formula in the style of Crofton-Blaschke-Santalo, and someone might yet see that; but even the proof of the inradius problem for curve segments in the plane (see the paper for references) remains quite long and complicated. So it might turn out that the complete solution to the inspection problem will require substantial effort. Hopefully the ideas and techniques developed in the above paper will help.

The baseball stitches curve suggested by Gjergji Zaimi appears in another paper of Zalgaller:

V. A. Zalgaller. Extremal problems on the convex hull of a space curve. Algebra i Analiz, 8(3):1–13, 1996.

Here Zalgaller also conjectures that this curve should be the minimizer.

• Nice, Mohammad, I didn't know about that paper. Thanks! – Joseph O'Rourke Apr 18 '16 at 23:03

I am quite sure that the problem is open, but we can play math-sport --- who makes a better constant.

Let me make a long remark, mostly based on the Zalgaller's paper. I will describe a family of examples, which includes the $4{\cdot}\pi$-example of Gjergji Zaimi, but I can not see if $4{\cdot}\pi$ is the best constant in this family. (It seems that you are a friend of computer and it would be easy for you to check.)

The curve can be viewed a $S^1$-family of circles on the sphere. In the example of Gjergji Zaimi, each circle in the family has exactly one point on one half-equator and one on the opposite meridian. Instead of half-equator and meridian one can choose two curves and consider corresponding $S^1$-family of circles; such a pair of curves is described by few parameters. (The centers of the circles in this family also can be described as envelop-line for circles of half-radius in the Zalgaller's family)

• Does math-sport include free generalizations? I would then ask the best (=minimum) constant $C$ such that, for any curve $\gamma:\mathbb{S}^1\to\mathbb{R}^3$ with $\|\gamma(t)\|\ge 1$, the area of the portion of the unit sphere which is visible from $\gamma$ is at most $2\pi+ C \mathrm{length}(\gamma)$. A nice guess is $C=1/2$, that would produce $\mathrm{length}(\gamma)\ge4\pi$ for any inspecting curve. – Pietro Majer Jun 30 '11 at 15:08