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

Fractal bodies invisible in 2 and 3 directions

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.
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

[1]

Hongyu Liu, Ting Zhou. Two dimensional invisibility cloaking via transformation optics. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 525-543. doi: 10.3934/dcds.2011.31.525

[2]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems & Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991

[3]

Andrea Cianchi, Vladimir Maz'ya. Global gradient estimates in elliptic problems under minimal data and domain regularity. Communications on Pure & Applied Analysis, 2015, 14 (1) : 285-311. doi: 10.3934/cpaa.2015.14.285

[4]

Manuel Gutiérrez. Lorentz geometry technique in nonimaging optics. Conference Publications, 2003, 2003 (Special) : 386-392. doi: 10.3934/proc.2003.2003.386

[5]

Gang Bao. Mathematical modeling of nonlinear diffracvtive optics. Conference Publications, 1998, 1998 (Special) : 89-99. doi: 10.3934/proc.1998.1998.89

[6]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621

[7]

Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649

[8]

Mathieu Molitor. On the relation between geometrical quantum mechanics and information geometry. Journal of Geometric Mechanics, 2015, 7 (2) : 169-202. doi: 10.3934/jgm.2015.7.169

[9]

Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055

[10]

Timothy C. Reluga, Jan Medlock. Resistance mechanisms matter in SIR models. Mathematical Biosciences & Engineering, 2007, 4 (3) : 553-563. doi: 10.3934/mbe.2007.4.553

[11]

Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2227-2272. doi: 10.3934/dcds.2015.35.2227

[12]

W. Patrick Hooper, Richard Evan Schwartz. Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2009, 3 (2) : 159-231. doi: 10.3934/jmd.2009.3.159

[13]

Serge Tabachnikov. Birkhoff billiards are insecure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035

[14]

Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893

[15]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255

[16]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048

[17]

Qian Liu, Shuang Liu, King-Yeung Lam. Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020050

[18]

Urszula Ledzewicz, Heinz Schättler. Drug resistance in cancer chemotherapy as an optimal control problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 129-150. doi: 10.3934/dcdsb.2006.6.129

[19]

Sebastian Bonhoeffer, Pia Abel zur Wiesch, Roger D. Kouyos. Rotating antibiotics does not minimize selection for resistance. Mathematical Biosciences & Engineering, 2010, 7 (4) : 919-922. doi: 10.3934/mbe.2010.7.919

[20]

Cristian Tomasetti, Doron Levy. An elementary approach to modeling drug resistance in cancer. Mathematical Biosciences & Engineering, 2010, 7 (4) : 905-918. doi: 10.3934/mbe.2010.7.905

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]