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Author Topic: A degree 4/2 rational function  (Read 1902 times)
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BradC
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« on: December 05, 2009, 05:16:36 AM »

Here's the Mandelbrot set of z\leftarrow \frac{z^4+0.36 z^3+0.56 z^2+0.31 z-0.46}{z^2+0.29 z-0.67}+c, where division is defined as here http://www.fractalforums.com/theory/triplex-algebra/. I evaluated the polynomials as written, as a sum of powers (rather than putting them in Horner form).

http://www.fractalforums.com/gallery/?sa=view;id=1146

Edit: Oops, looking at my code I just realized that I used c as the starting value for the iteration rather than 0 as I had intended. Hopefully that didn't make too much difference heh...
« Last Edit: December 05, 2009, 05:24:12 AM by BradC » Logged
jehovajah
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« Reply #1 on: December 05, 2009, 03:16:08 PM »

This is fantastic brad! Have you experimented with the order of calculation? Does doing the powers and summing then doing the division before each iteration produce different results to doing after each iteration?

What iteration level is this? And what view are you presenting?

Changing c might affect whether this is a julia or a mandelbrot.

Have you tried z^4\over z^2  + c ?
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
kram1032
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« Reply #2 on: December 05, 2009, 03:28:14 PM »

nice and colourful fractal looking whipped cream cheesy
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BradC
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« Reply #3 on: December 14, 2009, 10:15:30 PM »

Have you experimented with the order of calculation? Does doing the powers and summing then doing the division before each iteration produce different results to doing after each iteration?
I haven't experimented with different orders of operations.

What iteration level is this? And what view are you presenting?
I plotted the surface based on distance estimate and pixel size and distance to camera, rather than on iteration level. Although I did use a maximum iteration cutoff of 64, most points wouldn't have ever gotten that high. The view is looking in the positive y direction, with the z-axis pointing up.

Have you tried <Quoted Image Removed><Quoted Image Removed> ?
I didn't try z^4/z^2 + c, but I did try one other rational function, this time of degree 8/2. These rational functions render so slooow I think I'm going to look into something else for a while... tongue stuck out



* Rational82.jpg (55.07 KB, 512x512 - viewed 490 times.)
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BradC
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« Reply #4 on: December 14, 2009, 10:18:25 PM »

A zoom into the original fractal above:
http://www.fractalforums.com/gallery/?sa=view;id=1185
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jehovajah
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« Reply #5 on: December 16, 2009, 10:54:54 AM »

Is the centre of the view the xy plane, and so is the symmetry a type of reflection in xy? Looks very interesting and vortex like and like a nebula surrounding a black hole in a cosmic ray wind!1
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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
BradC
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« Reply #6 on: December 16, 2009, 03:28:35 PM »

The camera is located on the negative y-axis, looking at the origin, with the z-axis pointing up. So yeah the reflection is across the xy-plane.
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kram1032
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« Reply #7 on: December 16, 2009, 03:59:57 PM »

looks cool cheesy
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gaston3d
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« Reply #8 on: December 17, 2009, 07:52:47 PM »


what coloring function you use?
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BradC
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« Reply #9 on: December 17, 2009, 08:43:31 PM »

The coloring just comes from different colored lights at different locations. I don't have shadows implemented yet, but I am using the origin point orbit trap idea for fake ambient occlusion.

...but I'm also currently making a simplification in my lighting calculations that might not be valid here. The surface I'm rendering is based on a distance estimate, but the surface normals I'm using are based on a numerical gradient of the smoothed iteration count, because that's easier than actually computing the gradient of the distance estimate. (I use the iteration gradient to compute my DE.) So the surface normals I'm using in my lighting calculations aren't really guaranteed to be truly normal to the surface I'm plotting. I was hoping it would be a close enough approximation, but this lighting looks somehow too unrealistic to me so I suspect that in an unusual fractal like this it's not really that close.
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