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Quaternion Julia Set Shape Optimization
Author(s) -
Kim Theodore
Publication year - 2015
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.12705
Subject(s) - quaternion , iterated function , computer science , invariant (physics) , shape optimization , julia set , algorithm , energy minimization , convergence (economics) , set (abstract data type) , translation (biology) , mathematics , mathematical optimization , geometry , pure mathematics , mathematical analysis , chemistry , computational chemistry , finite element method , programming language , economic growth , biochemistry , messenger rna , gene , physics , economics , mathematical physics , thermodynamics
We present the first 3D algorithm capable of answering the question : what would a Mandelbrot‐like set in the shape of a bunny look like? More concretely, can we find an iterated quaternion rational map whose potential field contains an isocontour with a desired shape? We show that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. The problem can be written as an energy minimization, the optimization can be made practical by using an efficient method for gradient evaluation, and convergence can be accelerated by using a variety of multi‐resolution strategies. The resulting shapes are not invariant under common operations such as translation, and instead undergo intricate, non‐linear transformations .