Fractal bodies invisible in 2 and 3 directions

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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

Abstract
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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 and Continuous Dynamical Systems, 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), 1247-1258. doi: 10.1088/0951-7715/22/6/001.
[2]	P. Bachurin, K. Khanin, J. Marklof and A. Plakhov, Perfect retroreflectors and billiard dynamics, J. Modern Dynam., 5 (2011), 33-48.
[3]	D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Birkhäuser (2005).
[4]	G. Buttazzo and B. Kawohl, On Newton's problem of minimal resistance, Math. Intell., 15 (1993), 7-12. doi: 10.1007/BF03024318.
[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), 173-211. doi: 10.1007/PL00009911.
[6]	T. Lachand-Robert and E. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2006), 368-379. doi: 10.1137/040608039.
[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), 153-176. doi: 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.3.CO;2-U.
[8]	I. Newton, Philosophiae naturalis principia mathematica, (1687).
[9]	A. Plakhov, Scattering in billiards and problems of Newtonian aerodynamics, Russ. Math. Surv., 64 (2009), 873-938. doi: 10.1070/RM2009v064n05ABEH004642.
[10]	A. Plakhov, Mathematical retroreflectors, Discr. Contin. Dynam. Syst.-A, 30 (2011), 1211-1235. doi: 10.3934/dcds.2011.30.1211.
[11]	A. Plakhov and A. Aleksenko, The problem of the body of revolution of minimal resistance, ESAIM Control Optim. Calc. Var. 16 (2010), 206-220.
[12]	A. Plakhov and V. Roshchina, Invisibility in billiards, Nonlinearity, 24 (2011), 847-854. doi: 10.1088/0951-7715/24/3/007.
[13]	, "Invisibility,'' Wikipedia article. Available from: http://en.wikipedia.org/wiki/Invisibility.
[14]	, "Unsichtbarkeit,'' Wikipedia article. Available from: http://de.wikipedia.org/wiki/Unsichtbarkeit.

show all references

References:

[1]	A. Aleksenko and A. Plakhov, Bodies of zero resistance and bodies invisible in one direction, Nonlinearity, 22 (2009), 1247-1258. doi: 10.1088/0951-7715/22/6/001.
[2]	P. Bachurin, K. Khanin, J. Marklof and A. Plakhov, Perfect retroreflectors and billiard dynamics, J. Modern Dynam., 5 (2011), 33-48.
[3]	D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Birkhäuser (2005).
[4]	G. Buttazzo and B. Kawohl, On Newton's problem of minimal resistance, Math. Intell., 15 (1993), 7-12. doi: 10.1007/BF03024318.
[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), 173-211. doi: 10.1007/PL00009911.
[6]	T. Lachand-Robert and E. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2006), 368-379. doi: 10.1137/040608039.
[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), 153-176. doi: 10.1002/1522-2616(200106)226:1<153::AID-MANA153>3.3.CO;2-U.
[8]	I. Newton, Philosophiae naturalis principia mathematica, (1687).
[9]	A. Plakhov, Scattering in billiards and problems of Newtonian aerodynamics, Russ. Math. Surv., 64 (2009), 873-938. doi: 10.1070/RM2009v064n05ABEH004642.
[10]	A. Plakhov, Mathematical retroreflectors, Discr. Contin. Dynam. Syst.-A, 30 (2011), 1211-1235. doi: 10.3934/dcds.2011.30.1211.
[11]	A. Plakhov and A. Aleksenko, The problem of the body of revolution of minimal resistance, ESAIM Control Optim. Calc. Var. 16 (2010), 206-220.
[12]	A. Plakhov and V. Roshchina, Invisibility in billiards, Nonlinearity, 24 (2011), 847-854. doi: 10.1088/0951-7715/24/3/007.
[13]	, "Invisibility,'' Wikipedia article. Available from: http://en.wikipedia.org/wiki/Invisibility.
[14]	, "Unsichtbarkeit,'' Wikipedia article. Available from: http://de.wikipedia.org/wiki/Unsichtbarkeit.

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