jehovajah


« Reply #360 on: October 26, 2009, 10:54:38 AM » 

This has been a most rewarding experience. This is math as it should be, not hidebound by snobbish conventionality but collaborative and egalitarian and inclusive as well as playful. The use of the processor and algorithms for visualisation, shading , rendering etc show the mathematical toolset benefits by innovation and extension. mathematic c.1380 as singular, replaced by early 17c. by mathematics (1581), from L. mathematica (pl.), from Gk. mathematike tekhne "mathematical science," fem. sing. of mathematikos (adj.) "relating to mathematics, scientific," from mathema (gen. mathematos) "science, knowledge, mathematical knowledge," related to manthanein "to learn," from PIE base *mn/*men/*mon "to think, have one's mind aroused" (cf. Gk. menthere "to care," Lith. mandras "wideawake," O.C.S. madru "wise, sage," Goth. mundonsis "to look at," Ger. munter "awake, lively"). Mathematics (pl.) originally denoted the mathematical sciences collectively, including geometry, astronomy, optics. Math is the Amer.Eng. shortening, attested from 1890; the British preference, maths is attested from 1911. Online Etymology Dictionary free So really! congratulations!! As you may know i think the foundations of maths can now be revised, but not in terms of unifying, an old value and goal.RATHER IN TERMS OF THE ITERATIVE NATURE of all things i perceive. I find you practitioners more stimulating than dry elegant text.



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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!



JosLeys
Iterator
Posts: 199


« Reply #361 on: October 30, 2009, 12:44:03 PM » 

I've been rereading everything that has been posted on the spherical coordinates type 3D fractals, and have been looking for my own way of rendering them. Dave Makin's "delta DE" works fine, but I thought that if we can get the derivative of the transformation, things would be faster. So I've been reading up on Jacobians and all that stuff. Getting the Jacobian matrix for the transformation is no problem, but then I did not know what to do with it :)).. In general one can get a distance estimate from DE= z.log(z)/dz, z and dz being the values after bailout and dz can be calculated iteratively (for z=z^p+c ) as dz=p.z^(p1).dz+1 (starting with dz=1). So I tried (maybe a bit naively, to do something similar with the 'spherical transformation'. Let's do a Mandelbrot of degree p. We have point on the ray x1,y1,z1 or x1=R.cos(ph).cos(th), y1=R.cos(ph).sin(th), z1=R.sin(ph), R=sqrt(x1^2+y1^2+z1^2),ph=atan(y1/x1), th=asin(z1/R) Put dzx=1,dzy=0,dzz=0 . Express this in spherical coordinates also: ph_dz=0, th_dz=0, R_dz=1 We iterate this point: First calculate dz: dzx=p.R^(p1).R_dz.cos((p1).ph+ph_dz).cos((p1).th+th_dz)+1 dzy=p.R^(p1).R_dz.cos((p1).ph+ph_dz).sin((p1).th+th_dz) dzz=p.R^(p1).R_dz.sin((p1).ph+ph_dz) Calculate new R_dz and ph_dz and th_dz. Now the new x,y,z: x=R^p.cos(p.ph).cos(p.th)+x1 y=R^p.cos(p.ph).sin(p.th)+y1 z=R^p.sin(p.ph)+z1 Calculate new R and ph and th. If R>bailout or maxiters reached, calculate DE=R.log(R)/R_dz Move the point along the ray a distance f.DE (f<1) and start over until DE<epsilon. Believe it or not, but this works! Here is the order 12 Mandelbrot : ..and here is a Julia :



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David Makin


« Reply #362 on: October 30, 2009, 10:31:49 PM » 

I've been rereading everything that has been posted on the spherical coordinates type 3D fractals, and have been looking for my own way of rendering them. Dave Makin's "delta DE" works fine, but I thought that if we can get the derivative of the transformation, things would be faster.
So I've been reading up on Jacobians and all that stuff. Getting the Jacobian matrix for the transformation is no problem, but then I did not know what to do with it :))..
In general one can get a distance estimate from DE= z.log(z)/dz, z and dz being the values after bailout and dz can be calculated iteratively (for z=z^p+c ) as dz=p.z^(p1).dz+1 (starting with dz=1).
So I tried (maybe a bit naively, to do something similar with the 'spherical transformation'. Let's do a Mandelbrot of degree p.
We have point on the ray x1,y1,z1 or x1=R.cos(ph).cos(th), y1=R.cos(ph).sin(th), z1=R.sin(ph), R=sqrt(x1^2+y1^2+z1^2),ph=atan(y1/x1), th=asin(z1/R) Put dzx=1,dzy=0,dzz=0 . Express this in spherical coordinates also: ph_dz=0, th_dz=0, R_dz=1 We iterate this point: First calculate dz: dzx=p.R^(p1).R_dz.cos((p1).ph+ph_dz).cos((p1).th+th_dz)+1 dzy=p.R^(p1).R_dz.cos((p1).ph+ph_dz).sin((p1).th+th_dz) dzz=p.R^(p1).R_dz.sin((p1).ph+ph_dz) Calculate new R_dz and ph_dz and th_dz. Now the new x,y,z: x=R^p.cos(p.ph).cos(p.th)+x1 y=R^p.cos(p.ph).sin(p.th)+y1 z=R^p.sin(p.ph)+z1 Calculate new R and ph and th.
If R>bailout or maxiters reached, calculate DE=R.log(R)/R_dz
Move the point along the ray a distance f.DE (f<1) and start over until DE<epsilon.
Thanks for that Jos, I think I know where I was going wrong when I tried using the Jacobian to get the derivative for DE when I tried it on the 4D "true 3D" formula that I suggested a while ago  I'm going to try again now......



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David Makin


« Reply #363 on: October 31, 2009, 03:53:59 AM » 

A degree 5 White/Nylander Julia. This shows that the difference in renders using the delta DE and analytical DE is sometimes not that great especially with a formula such as the White/Nylander where calculating the derivative is as complicated as calculating the normal iterate The time differnce is not so great  note that how often the viewing rays approach the surface without striking it also has a bearing on the timing, this generally causes more slow down in the analytical method than the delta method. You should note that the final analytical DE calculation was modified to correct for scale differences for different powers/degrees, the UF calculation used (after trial and error based on getting the normals correct  the normals being calculated from adjacent DE values with no extra raytracing) was: dist = 0.5*sqrt(@mpwr1.0)*log(magn)*sqrt(magn/(dzri+sqr(dzj))) i.e. the usual calculation but scaled by the square root of the power/degree minus one.


« Last Edit: October 31, 2009, 04:30:00 AM by David Makin »

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Buddhi
Fractal Molossus
Posts: 655


« Reply #364 on: October 31, 2009, 11:48:27 AM » 

Hi I wrote completely new program for rendering 3D fractals. I resigned with rendering slices and fixed grid. Now fractal is calculated directly in 3D space using scanline algorithm. DE is not implemented yet but for accurate searching of fractal boundary I use binary search algorithm. I included all shading algorithms which I used in old program: global illumination, hard shadows and normal vector calculation (angle of incidence of light). Without volumetric fog (opacity proportional to number of iterations) rendering is much more faster than in old program and quality is higher. Details are sharper because I didn't have to use any interpolation algorithms. This image was rendered 25 minutes in 2560x2560 resolution and max. 20 iterations (rendered on Intel Core 2 Duo Quad 8200) http://www.fractalforums.com/gallery/?sa=view;id=1029



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twinbee


« Reply #365 on: October 31, 2009, 03:38:10 PM » 

Congrats on the renderer speedup  that looks really ace! (nice golden brot before too). Here is an animation showing cross sections of the old Mandelbulb gateau I did:


« Last Edit: October 31, 2009, 03:47:25 PM by twinbee »

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David Makin


« Reply #366 on: November 03, 2009, 12:56:44 AM » 

Hi all, you may remember I had a 3D based suggestion for a "true 3D" Mandy using the following: *  r i j  r  r i j i  i r j j  j j r Which gives a square of (x,y,z): new x = x^2  y^2  z^2 new y = 2*x*y new z = 2*z*(xy) I finally got around to investigating further and realised why noone had commented on it, here's a rotation of the top of the Mandy: http://www.fractalgallery.co.uk/FlatMandy.movHowever it wasn't a dead loss because here's the main minibrot:



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cKleinhuis


« Reply #367 on: November 03, 2009, 03:38:20 AM » 




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divide and conquer  iterate and rule  chaos is No random!



bugman


« Reply #368 on: November 03, 2009, 06:22:44 AM » 

Hi
I wrote completely new program for rendering 3D fractals. I resigned with rendering slices and fixed grid. Now fractal is calculated directly in 3D space using scanline algorithm. DE is not implemented yet but for accurate searching of fractal boundary I use binary search algorithm. I included all shading algorithms which I used in old program: global illumination, hard shadows and normal vector calculation (angle of incidence of light). Without volumetric fog (opacity proportional to number of iterations) rendering is much more faster than in old program and quality is higher. Details are sharper because I didn't have to use any interpolation algorithms. This image was rendered 25 minutes in 2560x2560 resolution and max. 20 iterations (rendered on Intel Core 2 Duo Quad 8200)
Wow, 24 minutes without any distance estimation? That's quite impressive for such quality.



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twinbee


« Reply #369 on: November 05, 2009, 12:58:46 PM » 

The article I'm working on is almost finished. I want to leave some things as a surprise, but couldn't resist these..... Hope you enjoy, the second Cave piece is available at aprx. 4000x2000 resolution here (downscaled from 8000x4000!). "Ice Cream From Uranus""Cave of Lost Secrets"


« Last Edit: November 14, 2009, 09:31:13 AM by twinbee »

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jehovajah


« Reply #370 on: November 05, 2009, 01:37:22 PM » 

Hi all, you may remember I had a 3D based suggestion for a "true 3D" Mandy using the following: *  r i j  r  r i j i  i r j j  j j r Which gives a square of (x,y,z): new x = x^2  y^2  z^2 new y = 2*x*y new z = 2*z*(xy) I finally got around to investigating further and realised why noone had commented on it, here's a rotation of the top of the Mandy: http://www.fractalgallery.co.uk/FlatMandy.mov
I would like to see the results of the table you discarded as uninteresting where j x i transforms to i. If you have time could you render that please.


« Last Edit: November 05, 2009, 02:24:06 PM by jehovajah, Reason: remove you tube ref. »

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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!



jehovajah


« Reply #371 on: November 05, 2009, 03:18:11 PM » 

: r i j  ...................  r : rr ri rj 
i : ir ii ij  j : jr ji jj  ....................... 
Define rr as r ^{2} , ii as i ^{2} = 1 , jj as j ^{2} = 1 all of which are real values. Define ri = ir which is an i value and rj = jr which is a j value. In the case of commutativity ij = ji so define ij = i or ij = j in the case where the operators are non commutative ij / ji define ij = j and ji = i or ij = i and ji= j If you have explored all these alternatives i would love to see the results.



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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!



bugman


« Reply #372 on: November 05, 2009, 05:26:10 PM » 

The article I'm working on is almost finished. I want to leave some things as a surprise, but couldn't resist these..... Hope you enjoy, the second Cave piece is available at aprx. 4000x2000 resolution here (downscaled from 8000x4000!). Twinbee, your deep zoom renderings are still my favorite. I suspect that your "Ice Cream From Uranus" rendering is hiding some beautiful spirals in it, but we cannot see them because the iteration depth is not deep enough. I wonder if it is possible to make iteration depth a function of the cumulative derivative or Cauchy method in such a way that we could render the fine detail in the spirals without going into too much detail in other regions.



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jehovajah


« Reply #373 on: November 05, 2009, 05:42:13 PM » 

Just in passing ; it occurs that using the non commutative forms model left handed and righr handed forms in 3D so called. Combining the forms in this way may produce relevant results to natural forms.



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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!



David Makin


« Reply #374 on: November 06, 2009, 02:18:52 AM » 

Twinbee, your deep zoom renderings are still my favorite. I suspect that your "Ice Cream From Uranus" rendering is hiding some beautiful spirals in it, but we cannot see them because the iteration depth is not deep enough. I wonder if it is possible to make iteration depth a function of the cumulative derivative or Cauchy method in such a way that we could render the fine detail in the spirals without going into too much detail in other regions.
Just solid based solely on distance estimate threshold would probably achieve the desired result in terms of detail i.e. use distance estimation with maxiter set higher than is ever used in the distance estimation.



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