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Author Topic:  Constructing polynomials whose Julia set resemble a desired shape  (Read 347 times)

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Offline marcm200

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Placing roots in word shape was straightforward forum link and the Quaternion paper is currently too complicated for me (forum link (a lot of algorithms and libraries that are unknown to me), I settled with 2D polynomial Julia sets constructed from an input image.

Code: [Select]
Lindsey, Kathryn & Younsi, Malik. (2016).
Fekete polynomials and shapes of Julia sets.
Transactions of the American Mathematical Society
full text https://arxiv.org/abs/1607.05055

The article is very mathematical. I focused on chapter 8 ("numerical examples"), especially on the bottom of page 20 where a three-step algorithm is formulated, where I filled in the mathematical terms as needed.

The author's method works by identifying the boundary of the filled-in set of interest, then they iteratively construct a set of Leja points (never heard of them before) that are used as roots in the resulting polynomial and calculating some coefficient. Then the filled-in Julia set resembles the input image.

The implementation was quite simple (except for some indexing in lemma 8.2 where I got confused between n and n+1).

Below is the first example.

Upper left is the desired shape (drawn in paint, should be a bat as in fig. 4 of the article)
Upper right is the filled-in Julia set of a result polynomial of degree 128+1 (128 Leja points), black is interior, white is exterior, red is the boundary of the desired shape, afterwards included for comparison.
Lower left is the filled-in Julia set of a degree 256+1 polynomial.
Lower right: 512+1 degree

Next I'll try to write a sentence (the authors constructed a polynomial with their initials, quite impressive, fig. 5).

Technical details
  • All calculations were done in double precision.
  • Polynomial evaluation was done in the root form in a binary tree manner to (hopefully) multiply values of similar magnitude.
  • The target set was arbitrarily placed in the 2-cube, max it was 100 and escape radius was set to 2.
  • Input image was 512x512 pixels (the authors use 5000x5000, but that's not tractable in my current implementation).



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Offline gerrit

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Re: Constructing polynomials whose Julia set resemble a desired shape
« Reply #1 on: December 19, 2019, 06:36:59 PM »
Very interesting!


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