Once you get the hang of what's going on in the mapping function, it's pretty easy to explore and come up with some wild and amazing images. The custom mapping function for the top layer is similar, but I changed the complex numbers to real plane coordinates with ReIm. This one has two layers also, the bottom layer's custom mapping is based on the complex distance to the corner coordinates of the image. Then I'd better turn my attention to trimming the hedges and other things my wife thinks are important. Finance, Statistics & Business Analysis. ![]() Wolfram Knowledgebase Curated computable knowledge powering Wolfram|Alpha. Wolfram Universal Deployment System Instant deployment across cloud, desktop, mobile, and more. If any software does this kind of change of variables automatically, I'd be impressed.Wolfram Data Framework Semantic framework for real-world data. Using perturbation has a slight issue of converting from delta-r in the image plane to delta-s in the calculation plane, but I think it's not too hard (just do this once-per-pixel calculation in high precision). Practically speaking, you would convert from the unusual polynomial to the standard polynomial, then iterate that, converting back to the unusual polynomial for escape tests to get the right iteration count. Iterations of w -> w 2 + s should now be equivalent to iterations of z -> (rz) 2 - z - r, where w = A z + B z = (w - B)/A. ![]() We can solve for the parameters by conjugating:(Az+B) 2 + s = A((rz) 2 - z - r) + B gives A = r 2, B = -1/2, s = -3/4 - r 3 Not quite an answer you might hope for, but I think it's possible by affine conjugacy (equivalence of quadratic polynomials) with a change of coordinates z -> Az + B: Author: Topic: 3D Fractals with Ultrafractal (Read 6395 times) Description: A variety of 3D fractals created with Ultrafractal 0 Members and 1 Guest are viewing this topic. After your purchase, you can upgrade to a different edition at any time: simply click Upgrade Edition on the Help menu (Windows) or Ultra Fractal menu. The table below shows the included features in each edition: The evaluation version works in Extended Edition mode so you can try all features. So what can be done with perturbation is probably not much more than what stardust4ever already exploited and I implemented in KF. Ultra Fractal 6 is available in three editions. ![]() Replace z with z+d and r with r+c, then minus the original formula. ![]() I received a formula, (z*r)^2-(z+r) which I didn't succeed render with perturbation. Since r is an high precision variable, which very probably is out of bounds for hardware datatypes, it is as important to be removed as the high power of z! Perturbation of the standard mandelbrot is (z+d)^2+(c+r) - (z^2+r) => 2*z*d + d^2 + cĪs you can see, both z^2 and the start of the reference r are eliminated. Set d as delta and c is the start point of delta, z is the reference and r is the start point of the reference: Take the standard mandelbrot formula as example. I have reasons to doubt that UF6 would be able to take any formula and make perturbation of it.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |