Plane sets invisible in finitely many directions

Plakhov, Alexander

doi:10.1088/1361-6544/aac63b

Cited by 5 publications

(6 citation statements)

References 14 publications

(36 reference statements)

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“…From this, using (10) and utilizing the double angle formulas for sine and cosine, one obtains the estimate…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…There exist and are described (connected) bodies invisible from one point [3,4] and (infinitely connected) bodies invisible from two points [5]. There exist (connected and even simply connected) bodies invisible in one direction (that is, from an infinitely distant point) [6], (finitely connected) bodies invisible in two directions [7], as well as (infinitely connected) bodies invisible in three [8] and (in the two-dimensional case) in n-directions, where the number n ∈ N of directions is arbitrary [9].…”

Section: Introductionmentioning

confidence: 99%

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The problem of camouflaging via mirror reflections

Plakhov

2017

Proc. R. Soc. A.

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This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is natural to search for bodies that are, in a sense, close to invisible. We introduce a visibility index of a body measuring the mean angle of deviation of incident light rays, and derive a lower estimate to this index. This estimate is a function of the body's volume and of the minimal radius of a ball containing the body. This result is far from being final and opens a possibility for further research. Mathematics subject classifications: 37D50

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“…From this, using (10) and utilizing the double angle formulas for sine and cosine, one obtains the estimate…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The problem of camouflaging via mirror reflections

Plakhov

2017

Proc. R. Soc. A.

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“…Afterwards without loss of generality we assume that j α (S) lies in the regular part of the set M and deduce that S is analytic, as in the above case. Corollary 20. Fix a boundary point x ∈ ∂W and a path ψ ⊂ W going to x such that for every j one has A j (y) = A j (x) for y ∈ ψ arbitrarily close to x.…”

Section: Proposition 513mentioning

confidence: 99%

On 4-Reflective Complex Analytic Planar Billiards

Glutsyuk

2016

J Geom Anal

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The famous conjecture of V.Ya.Ivrii [14] says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study its complex analytic version for quadrilateral orbits in two dimensions, with reflections from holomorphic curves. We present the complete classification of 4-reflective complex analytic counterexamples: billiards formed by four holomorphic curves in the projective plane that have open set of quadrilateral orbits. This extends the previous author's result [7] classifying 4-reflective complex planar algebraic counterexamples. We provide applications to real planar billiards: classification of 4-reflective germs of real planar C 4 -smooth pseudo-billiards; solutions of Tabachnikov's Commuting Billiard Conjecture and the 4-reflective case of Plakhov's Invisibility Conjecture (both in two dimensions; the boundary is required to be piecewise C 4 -smooth). We provide a survey and a small technical result concerning higher number of complex reflections.

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“…The problem for nonconvex bodies is by now well understood [7,8,15,1,17,18]. Generalizations of the problem to the case of rotating bodies have been studied [12,13,22,23], and connections with the phenomena of invisibility, retro-reflection, and Magnus effect in geometric optics and mechanics have been established [21,19,24,2,14,22]. The methods of billiards, Kakeya needle problem, optimal mass transport have been used in these studies.…”

Section: Introductionmentioning

confidence: 99%

A solution to Newton's least resistance problem is uniquely defined by its singular set

Plakhov¹

2021

Preprint

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Let u minimize the functional F (u) = Ω f (∇u(x)) dx in the class of convex functions u : Ω → R satisfying 0 ≤ u ≤ M , where Ω ⊂ R 2 is a compact convex domain with nonempty interior and M > 0, and f : R 2 → R is a C 2 function, with {ξ : det f ′′ (ξ) = 0} being a closed nowhere dense set in R 2 . Let epi(u) denote the epigraph of u. Then any extremal point (x, u(x)) of epi(u) is contained in the closure of the set of singular points of epi(u). As a consequence, an optimal function u is uniquely defined by the set of singular points of epi(u). This result is applicable to the classical Newton's problem, where F (u) = Ω (1 + |∇u(x)| 2 ) −1 dx.

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Plane sets invisible in finitely many directions

Cited by 5 publications

References 14 publications

The problem of camouflaging via mirror reflections

The problem of camouflaging via mirror reflections

On 4-Reflective Complex Analytic Planar Billiards

A solution to Newton's least resistance problem is uniquely defined by its singular set

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