ISSN 1000-9825, CODEN RUXUEW E-mail jos@iscasaccn Journal of Software, Vol17, Supplement, November 2006, pp70 77 http//wwwjosorgcn 2006 by Journal of Software All rights reserved Tel/Fax +86-10-62562563 1,2+, 1,2,3 1,2, 1 (, 130012) 2 ( ), 130012) 3 (, 150001) Iterated Fractal Based on Distance Ratio ZHANG Xi-Zhe 1,2+, LÜ Tian-Yang 1,2,3, WANG Zheng-Xuan 1,2 1 (College of Computer Science and Technology, Jilin University, Changchun 130012, China) 2 (Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education (Jilin University), Changchun 130012, China) 3 (Department of Computer Science and Technology, Harbin Engineering University, Harbin 150001, China) + Corresponding author Phn +86-24-82072060, E-mail zxzok@163com Zhang XZ, Lü TY, Wang ZX Iterated fractal based on distance ratio Journal of Software, 2006,17(Suppl) 70 77 http//wwwjosorgcn/1000-9825/17/s70htm Abstract The escape time algorithm cannot render the convergence region of mapping, so there are some black regions in escape time fractal In this paper, a novel method is presented to construct fractal image, which is named the distance ratio iteration method This method performs iteration on two points and render fractal image by using their distance ratio convergence times Taking complex mapping z z α +c as example, the generalized Mandelbrot and Julia sets are constructed based on distance ratio and their visual properties are analyzed The result fractal image has complex and self-similarity structure in inner convergence region It is proved that the boundary of distance ratio fractal is the same as M-J set when α>0, and some visual structure of it with various exponent α are discussed When α<0, the generalized Mandelbrot and Julia set based on distance ratio have some complex structures which M-J set does not have Key words fractal; complex mapping; Julia set; Mandelbrot set; distance ratio,, z z α +c M-J M-J, α>0 M-J, α<0 M-J ; ;Julia ;Mandelbrot ; Mandelbrot [1] M, [2] Mandelbrot Julia [3 5] Received 2006-03-15; Accepted 2006-09-11
71 M-J [6,7] [8] ; M-J [9,10] [11] M-J,, 0,,,, M-J,Hooper [12] ε M ;Philip [13], Mandelbrot, [6] M-J [14] ; Lyapunov M [15],,,,,,, M-J, Mandelbrot Julia M J, 1 2, M-J, M-J,,,, z z α +c, J Julia ; M, M Fig1 DRJ for z z 3 +04+05i Fig2 DRM for z z 3 +c 1 z z 3 +04+05i J 2 z z 3 +c M 1 2 J, Julia 3 M, M 4 1 fc C, f n (x)=f o f n 1 (x), f 0 (x)=x,n z 1 z 2 C, L(z 1,z 2 ) Lz (, z) = ( ) ( ) f z1 f z 2 z z z 1,z 2, (1)
72 Journal of Software Vol17, Supplement, November 2006 n L ( z, z ) = n f ( z ) f ( z ) n n 1 n 1 f z1 f z2 ( ) ( ) f, z z,, 1 f z, A(z ), z, z A( z ), lim L k ( z, z ) = f ( ) k z 2 ε, z, z A( z ), k k>k, L k ( z, z ) f ( z ) < ε 1 ; 2 ε,,,,, z Julia, J (distance ratio Julia set, DRJ), k { 1, 2 { lim ( 1, 2) ( )}} k DRJ = z z C L z z f z (3), (3), z, (3) { { lim k (, }} ) ( ) k DRJ = z C L z z f z (4) z,ε, z k, z L k 1 k ( z, z ) f ( z ) > ε, L ( z, z ) f ( z ) ε k k k, k, zone(k), J k DRJ = Uzone( k) 3 f, zone(k+1) z, z zone(k), f 1 ( z ) = k = 1 z zone(k+1), k k L ( z, z ) f ( z ) = L ( f 1 ( z ), z ) f ( z ) = L k 1 ( z, z ) f ( z ) > ε, z (2) (5) (6) L k + 1 ( z, z ) f ( z ) = L k+ 1 ( f 1 ( z ), z ) f ( z ) k = L ( z, z ) f ( z ) ε,z zone(k), z zone(k) 1 f 1 (zone(k))=zone(k+1) 3 zone(k) zone(k+1),, zone(k) zone(k+1) 1, k, 2 J 3 1, zone(k) zone(k+1) zone(1),, zone(2),zone(3),, zone(n),n, J f ( z) f ( z ) (5),zone(1) f ( z ) ε, zone(1) z z, zone(1), zone(k),
73, (1) k=1,n=10 000, zone(k); (2) zone(k) zone(k+1); (3) k<n, k=k+1, (2); (4) zone(1),zone(2),,zone(n 1),,zone(n), J 3(c) z z 2 1001 005i, 3(a) 3(b) 5(a) Julia,DRJ Julia, Julia, (a) (b) Fig3 (c) DRJ for complex mapping f (z)=z 2 1001 005i 3 f (z)=z 2 1001 005i J 3,DRJ,,, 3(b),, 3(c), zone(k), k 1, DRJ Julia, 4 α>0,drj f, Julia,zone( )=J( f ),,,,,,DRJ Julia, DRJ Julia,DRJ,, α>0, Julia [12 14], Julia α<0, z z α +c, [16,17] α<0 Julia,,, Julia α<0, 4 z z 2 +0001+03i J 5(b) Julia,, DRJ,, Julia,DRJ Julia
74 Journal of Software Vol17, Supplement, November 2006, Julia,, Fig4 DRJ for z z 2 +0001+03i and zoomed image 4 z z 2 +0001+03i J b 3 M (a) z z 2 1001 005i (b) z z 2 +0001+03i Fig5 Julia sets generated by escape time algorithm 5 Julia f c, c f f c c, M (distance ratio Mandelbrot set, DRM), { { lim k (0, }} ) ( ) c k DRM = c C L z f z (7) (7) (0,0) z, (0,0) f c (z)=z α +c, z, M, DRM { {, k+ }} 1 (0, ) k (0, ) c c DRM = c C k N L z L z ε (8) (8) DRM,, k, k, D, 1 n=10000; ε=00000001;k=1 2 D c, f(z,c); z 1 =(0,0),z 2 f(z,c) z 3 k<n, f k (z 1 )= f ( f k 1 (0,c)), f k (z 2 )= f ( f k 1 (z,c)), 6 k+ 1 k+ 1 f (0) f ( z ) k 4 L ( z1, z2 ) = k k f (0) f ( z ) 5 L k (0,z ) L k 1 (0,z ) >ε, k=k+1, 3 6 k, c 7 2~6, D, f M 6(a) z z 2 +c M, 6(b)
75 6(c) 6(d) α=5 α=14, M,,,,, 6(a), ;, DRM,DRM 6 [18] M,, α>1, M DRM M, M,, DRM, [19,20] (a) α=2 (b) Zoomed image for α=2 (c) α=5 (d) α=14 (a) α=2 (b) α=2 (c) α=5 (d) α=14 Fig6 DRM for z z α +c 6 z z α +c M α<0, DRM, DRM, 7 α= 2 DRM DRM 1 DRM, 2, (a) (b) (c) (d) Fig7 Distance ratio fractal image for complex mapping z z 2 +c 7 α= 2 DRM z z 2 +c,, n, n=1+ α n, n,,,
76 Journal of Software Vol17, Supplement, November 2006 8 α= 2 M, 7(a), DRM M DRM, 9,, M M, DRM M,, α<0, M, Fig8 Mandelbrot set for α= 2 Fig9 DRM and M-set for α= 2 8 α= 2 M 9 α= 2 M DRM α<0,drm α +1,, ; α +1, ; α +1,, ;DRM M, 4, M-J,,,, M-J M-J α>1, M-J, M-J ; α<0, M M, M,, Julia,, References [1] Mandelbrot BB The Fractal Geometry of Nature WH Freeman, 1982 5 47 [2] Spehar B, Clifford C, Newell B, Taylor R Universal aesthetic of fractals Computers & Graphics, 2003,27(5)813 820 [3] Liaw SA Find the mandelbrot-like sets in any mapping Fractals, 2002,10(2)137 146 [4] Buchanan W, Gomatam J, Steves B Generalized Mandelbrot sets for meromorphic complex and quaternionic maps Int l Journal of Bifurcation and Chaos, 2002,12(8)1755 1777 [5] Leys J Sphere inversion fractals Computers & Graphics, 2005,29(3)463 466 [6] Wang XY Researches on some internal structures of the general Mandelbrot sets for negative real index number Journal of Computer-Aided Design & Computer Graphics, 2001,13(10)868 872 (in Chinese with English abstract) [7] Wang XY, Liu XD, Zhu WY, Gu SS Analysis of c-plane fractal images from z z α +c for (α<0) Fractals, 2000,8(3)307 314 [8] Wang XY, Shi QJ The generalized M-J sets of complex exponent mapping Progress in Natural Science, 2004,14(8)934 940 (in Chinese with English abstract) [9] Chen N, Zhu WYM-Fractal image from complex mapping z z w +c (w=α+iβ) in the complex C-plane Journal of Computer Research and Development, 1997,34(12)899 907 (in Chinese with English abstract)
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