Koch Curves as Attractors and Repellers

Przemyslaw Prusinkiewicz and Glen Sandness

Abstract

This paper presents two methods for generating Koch curves, analogous to the commonly used iterative methods for producing images of Julia sets. The attracting 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. This characterization was first studied by Hutchinson. The repelling method is in principle dual to the attracting 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.

Reference

P. Prusinkiewicz and G. Sandness: Koch curves as attractors and repellers. IEEE Computer Graphics and Applications 8(6), pp. 26-40, 1988.

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