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Author Topic: Convergent Distance Estimation  (Read 5263 times)
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David Makin
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« on: April 07, 2009, 01:48:47 AM »

Hi all,

Just wondering if any mathematicians out there can shed light on how/why the method I found for distance estimation on a Newton fractal works from a math theory standpoint ?
Basically it's the same method as normal divergent distance estimation but obviously the derivative involved is that of the Newton formula and in the final calculation of the distance (z-zold) and (dz-dzold) are used instead of z and dz (where dz is the derivative).

I'm asking as it has a flaw in the Newton render in that there are static points at the roots and "root echoes" that I would like to remove and also the method does not generalise to all other convergent scenarios (e.g. to inside the main cardoid of the standard Mandelbrot).
It does produce reasonable results in the interesting areas of the Magnet and Nova fractals however (but again with static points).
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« Reply #1 on: April 07, 2009, 09:02:09 PM »

It sounds like you're holding back your erroneous method?
 :smiley
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David Makin
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« Reply #2 on: April 08, 2009, 01:28:35 AM »

It sounds like you're holding back your erroneous method?
 :smiley

No smiley I've already released colourings using it - it may not be mathematically perfect but it still provides a very useful colouring from an artistic standpoint.
The colourings are class colourings in mmf.ulb for the Newton, the Magnets and the Nova - I also added convergent distance estimation based colouring to the standard distance estimator colouring for z^p+c in mmf.ulb but as noted that does not give a true distance estimate for the convergent areas, it's just a useful colouring alternative and does understand periodic attraction (of course in the divergent areas you get "correct" distance estimation).
The distance estimation colourings in mmf.ulb can be plugged into the "Field Estimator" colouring also in mmf.ulb - this allows you to map textures/images into your fractal using the 2D coordinate system made up of either field lines or distance estimation angles and smooth iteration value or distance estimate value (as yet I haven't tried getting accurate field lines for convergent areas).
« Last Edit: April 08, 2009, 01:40:50 AM by David Makin » Logged

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« Reply #3 on: April 08, 2009, 09:47:14 PM »

I had a keyboard smashing time trying to learn UF version 5 HOW TO OPEN A CLASS
and now I've done it, I saw a blurry Newton set among few other alternatives. I realize this has artistic potential to the general extent of choosing light *beneath* the fractal.

Only now I realized my notebooks never had contents attached in any way... but, in file mt.ucl/Newton basin is the traditional outside coloring for Newton fractals.


color_newton.ucl
Code:
init:
 <snip>
loop:
 <nothing>
final:
  offset = atan2(#z)
  if offset < 0
    offset = offset + 2 * #pi
  endif
  offset = offset / (2 * #pi)
  #index = offset + @density * 0.01 * #numiter

This ancient method depends from atan2 function to calculate the angular position of #z. The density is where fun is, because gradient contains colors of spectrum or anything else it can contain. Since colors go in circles, density is crude. It is used only to leave more room for all yellow colors for one root, all green colors for another... We can map the gradient discretely or continuously to one particular root. That function hold the beauty.
Normally, colors go in circles within gradient and there are too many fractal contours (if not even too many roots) so after a small number of contours AWAY from roots, colors start to spiral due to "delay" (shift) that occurred between roots (e.g. gradient/5 for 5 roots). If you count the contours away from root and remember the number of roots, you got some control over the design of DENSITY MAPPING (fun function).
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David Makin
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« Reply #4 on: April 08, 2009, 10:07:14 PM »

Here are some examples using the convergent distance estimator colourings:

The first test image after I corrected iteration breaks in the Newton DE algorithm:

http://makinmagic.deviantart.com/art/Fixed-Newton-DE-104288692

Some later images:

http://makinmagic.deviantart.com/art/Tapestry-105590257
http://makinmagic.deviantart.com/art/Leather-and-Lace-105576927
http://makinmagic.deviantart.com/art/Magneticed-105568242
http://makinmagic.deviantart.com/art/Scorpion-Tales-105092657
http://makinmagic.deviantart.com/art/Mandelbrot-Nebula-104603840
http://makinmagic.deviantart.com/art/Mandelbrot-Rocks-104598715
http://makinmagic.deviantart.com/art/Magnetic-Blooms-104516876

Mandelbrot Rocks is pure distance estimator angles and Magnetic Blooms uses the DE angles and DE distance to map images into the fractal.
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http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
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