|Table of Contents|

Efficiently Generating Hyperbolic Symmetries of Dynamic Systems(PDF)

南京师范大学学报（工程技术版）[ISSN:1006-6977/CN:61-1281/TN]

Issue:
2014年01期
Page:
76-81
Research Field:
Publishing date:

Info

Title:
Efficiently Generating Hyperbolic Symmetries of Dynamic Systems
Author(s):
Chen Lu
School of Educational Sciences,Nanjing Normal University,Nanjing 210097,China
Keywords:
hyperbolic patterns; dynamical systems; equivariant mappings; color scheme
PACS:
TP391
DOI:
-
Abstract:
Exploration on automatic generation of symmetric patterns of fractals has been an important branch in the study of dynamical systems.In this paper,a new and fast algorithm is presented for automatically generating hyperbolic patterns using dynamical systems method.By employing the construction of equivariant mappings under the dynamical system groups,the method maps points in the unit circle to a fundamental region.We can get the entire space without overlapping except at the boundaries with the fundamental region and its transformed copies.Colors of the points is decided based on the hyperbolic distance between two consecutive points on the same orbit.It not only reflects the convergence rate of after a specific number of iterations,but also makes the symmetrical points having the same color and causing the picture to symmetry,thus enhancing the artistic appeal of the generated image.The method can generate a series of hyperbolic model patterns.

References:

[1] Trudeau R J.The Non-Euclidean Revolution[M].Boston:Birkhauser,1987.
[2]Escher M C.Escher on Escher:Exploring the Infinity[M].New York:Harry N Abrams,1989.
[3]Dunhan D.Hyperbolic symmetry[J].Computers and Mathematics with Applications,1986,12B(1/2):139-153.
[4]King G,Stewart I.Nonlinear Equations in the Applied Sciences[M].New York:Academic Press,1992:257-315.
[5]Chung K W,Chan H S.Symmetrical patterns from dynamics[J].Computer Graphics Forum,1993,12(1):33-40.
[6]Field M,Golubitsky M.Symmetry in Chaos[M].New York:Oxford University Press,1992.
[7]Brisson G F,Gartz K M,McCune B J.Symmetric attractors in three-dimentional space[J].Chaos,Solitons and Fractals,1996,7(7):1 033-1 051.
[8]Chung K W,Chan H S,Wang B N.Hyperbolic symmetries from dynamics[J].Computers and Mathematics with Applications,1996,30(2):33-47.
[9]Chan H S Y,Chung K W,Wang B N.Computer generation of symmetrical patterns from dynamical systems[C]//Proceedings of the International Conference of Chinagraph.Hangzhou,1996(96):497-501.
[10]Chung K W,Chan H S Y,Wang B N.Tessellation with the modular group from dynamics[J].Computer and Graphics,1997,21(4):523-534.
[11]Carter K W,Eagles R L,Grimes S M.Chaotic attractors with discrete planar symmetry[J].Chaos,Solitons and Fractals,1998,9(12):2 031-2 054.
[12]Chung K W,Chan H S,Wang B N.Tessellation in three-dimentional hyperbolic space from dynamics and the quaternions[J].Chaos,Solitons and Fractals,2001,12(10):1 941-1 945.
[13]Adcock B,Jones K C,Reiter C A.Iterated function systems with symmetry in the hyperbolic symmetry[J].Computer and Graphics,2000,24(5):791-796.
[14]叶瑞松,欧阳培昌.动力系统结合扩张模群自动生成彩色对称图形[J].汕头大学学报:自然科学版,2005,20(4):14-19.
Ye Ruisong,Ouyang Peichang.Automatic generation of colored symmetric patterns by dynamical systems with the extended modular group[J].Journal of Shantou University:Natural Science Edition,2005,20(4):14-19.(in Chinese)
[15]Coxeter H S M,Moser W O J.Generators and Relations for Discrete Groups[M].4th ed.New York:Springer,1980.
[16]Beardon A F.The Geometry of Discrete Groups[M].New York:Springer,1983.

Memo

Memo:
-
Last Update: 2014-03-31