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Issue No.06 - November/December (1988 vol.8)
pp: 26-40
<p>Two methods are presented for generating Koch curves, analogous to the commonly used iterative methods for producing images of Julia sets. The attractive method is based on a characterization of Koch curves as the smallest nonempty sets closed with respect to a union of similarities on the plane. The repelling method is in principle dual to the attractive one but involves a nontrivial problem of selecting the appropriate transformation to be applied at each iteration step. Both methods are illustrated with a number of computer-generated images. The mathematical presentation emphasizes the relationship between Koch construction and formal languages theory.</p>
Przemyslaw Prusinkiewcz, Glen Sandness, "Koch Curves as Attractors and Repellers", IEEE Computer Graphics and Applications, vol.8, no. 6, pp. 26-40, November/December 1988, doi:10.1109/38.20316
1. J.E. Hutchinson, "Fractals and Self-Similarity,"Indiana Univ. Mathematics J., Vol. 30, No. 5, Sept./Oct. 1981, pp. 713-747.
2. B.B. Mandelbrot,The Fractal Geometry of Nature, W.H. Freeman, San Francisco, 1982.
3. H. von Koch, "An Elementary Geometric Method for Studying Some Questions in the Theory of Planar Curves" (in French),Acta MathematicaVol. 30, 1906, pp. 145-174.
4. A.R. Smith, "Plants, Fractals and Formal Languages,"Computer Graphics(Proc. SIGGRAPH), Vol. 18, No. 3, July 1984, pp. 1-10.
5. H. O. Peitgen and P.H. Richter,The Beauty of Fractals, Springer-Verlag, Berlin, 1986.
6. A. Norton et al., "Dynamics of eiθx(1 - x)" (film),SIGGRAPH Video Review, Issue 25, ACM, New York, 1986.
7. A. Lindenmayer, "Mathematical Models for Cellular Interaction in Development," Parts I and II,J. Theoretical Biology, Vol. 18, No. 3, Mar. 1968, pp. 280-315.
8. G. Rozenberg and A. Salomaa,The Mathematical Theory of L Systems, Academic Press, New York, 1980.
9. A.L. Szilard and R.E. Quinton, "An Interpretation for DOL Systems by Computer Graphics,"The Science Terrapin, Vol. 4, 1979, pp. 8-13.
10. F.M. Dekking, "Recurrent Sets,"Advances in Mathematics, Vol. 44, No. 1, Apr. 1982, pp. 78-104.
11. F.M. Dekking, "Recurrent Sets: A Fractal Formalism," Report 82-32, Dept. of Mathematics and Informatics, Delft. Univ. of Technology, The Netherlands, 1982.
12. P. Prusinkiewicz, "Graphical Applications of L-Systems,"Proc. Graphics Interface, Vision Interface 86, Canadian Information Processing Soc., Toronto, 1986, pp. 247-253.
13. B. Grunbaum and G. Shephard,Tilings and Patterns, W.H. Freeman, New York, 1987.
14. K. Kuratowski,Introduction to Set Theory and Topology, Pergamon Press, Oxford, UK, and Polish Scientific Publishers, Warsaw, 1972.
15. S. Demko, L. Hodges, and B. Naylor, "Construction of Fractal Objects with Iterated Function Systems,"Computer Graphics(Proc. SIGGRAPH), Vol. 19, No. 3, July 1985, pp. 271-278.
16. M.F. Barnsley and S. Demko, "Iterative Function Systems and the Global Construction of Fractals,"Proc. Royal Soc. London A, Vol. 399, No. 1817, June 1985, pp. 243-275.
17. J. Levy-Vehel and A. Gagalowicz, "Shape Approximation by a Fractal Model,"Eurographics 87, North-Holland, Amsterdam, 1987, pp. 159-180, 572.
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