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April  2013, 33(4): 1615-1631. doi: 10.3934/dcds.2013.33.1615

Fractal bodies invisible in 2 and 3 directions

Alexander Plakhov 1, and  Vera Roshchina 2,

1. 

Department of Mathematics, University of Aveiro, Aveiro 3810-193
2. 

Collaborative Research Network, University of Ballarat, VIC 3353, Australia

Received  October 2011 Revised  January 2012 Published  October 2012

We study the problem of invisibility for bodies with a mirror surface in the framework of geometrical optics. We show that for any two given directions it is possible to construct a two-dimensional fractal body invisible in these directions. Moreover, there exists a three-dimensional fractal body invisible in three orthogonal directions. The work continues the previous study in [1,12], where two-dimensional bodies invisible in one direction and three-dimensional bodies invisible in one and two orthogonal directions were constructed.
Keywords: billiards., problems of minimal resistance, Invisibility in geometrical optics.
Mathematics Subject Classification: Primary: 78A05, 37D50; Secondary: 76G2.
Citation: Alexander Plakhov, Vera Roshchina. Fractal bodies invisible in 2 and 3 directions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1615-1631. doi: 10.3934/dcds.2013.33.1615
References:
[1]

A. Aleksenko and A. Plakhov, Bodies of zero resistance and bodies invisible in one direction,, Nonlinearity, 22 (2009), 1247.  doi: 10.1088/0951-7715/22/6/001.  Google Scholar

[2]

P. Bachurin, K. Khanin, J. Marklof and A. Plakhov, Perfect retroreflectors and billiard dynamics,, J. Modern Dynam., 5 (2011), 33.   Google Scholar

[3]

D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems,", Birkhäuser (2005)., (2005).   Google Scholar

[4]

G. Buttazzo and B. Kawohl, On Newton's problem of minimal resistance,, Math. Intell., 15 (1993), 7.  doi: 10.1007/BF03024318.  Google Scholar

[5]

M. Comte and T. Lachand-Robert, Newton's problem of the body of minimal resistance under a single-impact assumption,, Calc. Var. Partial Differ. Equ., 12 (2001), 173.  doi: 10.1007/PL00009911.  Google Scholar

[6]

T. Lachand-Robert and E. Oudet, Minimizing within convex bodies using a convex hull method,, SIAM J. Optim., 16 (2006), 368.  doi: 10.1137/040608039.  Google Scholar

[7]

T. Lachand-Robert and M. A. Peletier, Newton's problem of the body of minimal resistance in the class of convex developable functions,, Math. Nachr., 226 (2001), 153.  doi: 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.3.CO;2-U.  Google Scholar

[8]

I. Newton, Philosophiae naturalis principia mathematica,, (1687)., (1687).   Google Scholar

[9]

A. Plakhov, Scattering in billiards and problems of Newtonian aerodynamics,, Russ. Math. Surv., 64 (2009), 873.  doi: 10.1070/RM2009v064n05ABEH004642.  Google Scholar

[10]

A. Plakhov, Mathematical retroreflectors,, Discr. Contin. Dynam. Syst.-A, 30 (2011), 1211.  doi: 10.3934/dcds.2011.30.1211.  Google Scholar

[11]

A. Plakhov and A. Aleksenko, The problem of the body of revolution of minimal resistance,, ESAIM Control Optim. Calc. Var. 16 (2010), 16 (2010), 206.   Google Scholar

[12]

A. Plakhov and V. Roshchina, Invisibility in billiards,, Nonlinearity, 24 (2011), 847.  doi: 10.1088/0951-7715/24/3/007.  Google Scholar

[13]

, "Invisibility,'', Wikipedia article. Available from: , ().   Google Scholar

[14]

, "Unsichtbarkeit,'', Wikipedia article. Available from: , ().   Google Scholar

show all references

References:
[1]

A. Aleksenko and A. Plakhov, Bodies of zero resistance and bodies invisible in one direction,, Nonlinearity, 22 (2009), 1247.  doi: 10.1088/0951-7715/22/6/001.  Google Scholar

[2]

P. Bachurin, K. Khanin, J. Marklof and A. Plakhov, Perfect retroreflectors and billiard dynamics,, J. Modern Dynam., 5 (2011), 33.   Google Scholar

[3]

D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems,", Birkhäuser (2005)., (2005).   Google Scholar

[4]

G. Buttazzo and B. Kawohl, On Newton's problem of minimal resistance,, Math. Intell., 15 (1993), 7.  doi: 10.1007/BF03024318.  Google Scholar

[5]

M. Comte and T. Lachand-Robert, Newton's problem of the body of minimal resistance under a single-impact assumption,, Calc. Var. Partial Differ. Equ., 12 (2001), 173.  doi: 10.1007/PL00009911.  Google Scholar

[6]

T. Lachand-Robert and E. Oudet, Minimizing within convex bodies using a convex hull method,, SIAM J. Optim., 16 (2006), 368.  doi: 10.1137/040608039.  Google Scholar

[7]

T. Lachand-Robert and M. A. Peletier, Newton's problem of the body of minimal resistance in the class of convex developable functions,, Math. Nachr., 226 (2001), 153.  doi: 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.3.CO;2-U.  Google Scholar

[8]

I. Newton, Philosophiae naturalis principia mathematica,, (1687)., (1687).   Google Scholar

[9]

A. Plakhov, Scattering in billiards and problems of Newtonian aerodynamics,, Russ. Math. Surv., 64 (2009), 873.  doi: 10.1070/RM2009v064n05ABEH004642.  Google Scholar

[10]

A. Plakhov, Mathematical retroreflectors,, Discr. Contin. Dynam. Syst.-A, 30 (2011), 1211.  doi: 10.3934/dcds.2011.30.1211.  Google Scholar

[11]

A. Plakhov and A. Aleksenko, The problem of the body of revolution of minimal resistance,, ESAIM Control Optim. Calc. Var. 16 (2010), 16 (2010), 206.   Google Scholar

[12]

A. Plakhov and V. Roshchina, Invisibility in billiards,, Nonlinearity, 24 (2011), 847.  doi: 10.1088/0951-7715/24/3/007.  Google Scholar

[13]

, "Invisibility,'', Wikipedia article. Available from: , ().   Google Scholar

[14]

, "Unsichtbarkeit,'', Wikipedia article. Available from: , ().   Google Scholar

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