a Julia study

[bh]

Hi, this is my first post.

I am doing a study on a particular region of a Julia set (c = 0.4+0.3*I), with the aim of making many different renderings.  I currently have made 11, and here are the first four.  They are essentially different colorings based on the triangle inequality average.  Click to enlarge to 1000px versions.


http://i12.tinypic.com/4d4epvr.jpg


http://i17.tinypic.com/4ify4qr.jpg


http://i10.tinypic.com/2iho0v6.jpg


http://i14.tinypic.com/4bzm688.jpg

[heneganj]

There you go, instant Warhol!

[lycium]

those are really nice renders, i'm going to have to try and work out how the triangle ineq avg trap works (perhaps someone can enlighten me?)!

welcome to the forums

[Unhooked]

Great julia. The designs look fantastic with the TIA formula. 

[matera]

It's like breeding flowers - anyway, those make me think of chrysanthemums. It's fun to be surprised by the transformations.

I had a nice color-change adventure recently. Deep in a fractal that was mostly peacock-like, I found a dragon. I made some changes that looked interesting, and wandered off to another area to find the dragon's golden hoard. That kept me going for another thirty or forty - so far. Terrible addiction...

[bh]

Thank for the comments. lycium, The tia coloring method is quite easy to use. It's not really a trap, it is a sum over the orbit:
sum (|z_i| - | |z_i-z₀| - |z₀| |) / (|z_i-z₀| + |z₀| - | |z_i-z₀| - |z₀| |)
with a partial summation for the last iteration, and dividing by the smooth number of iterations. An important parameter to play on is the bailout value.

Anyway, here comes the second batch. Those are based mainly on the angle of the last iteration, and lighting effects.


http://i10.tinypic.com/34r9iyf.jpg


http://i11.tinypic.com/44kdlao.jpg


http://i10.tinypic.com/2ihnvr6.jpg

[lycium]

wow, those images are really really nice the antialiasing is particularly good too!

the formula for TIA is much appreciated; it seems there's some room for experimentation with various metrics (choice of norm), though unfortunately it depends on a continuous escape time which makes applying it to other types of fractals rather difficult

edit: you know, i really can't get over how good the antialiasing in those images is. i'm pretty sure you're taking special (mathematical?) care to do it well - correct? the bandlimiting of the black/white stripes towards the spiral is near-analytic, very impressive!

[bh]

the formula for TIA is much appreciated; it seems there's some room for experimentation with various metrics (choice of norm), though unfortunately it depends on a continuous escape time which makes applying it to other types of fractals rather difficult :/

I have tried variations, like:
sum z_i / |z_i|
sum (z_i-z_i-1)/|z_i-z_i-1|
sum |arg((z_i-1-z_i)/(z_i-1-z_i-2))|
(the first two give complex numbers, and then you can take the real part or the imaginary part or whatever). The results for these and the tia have some similarities.

edit: you know, i really can't get over how good the antialiasing in those images is. i'm pretty sure you're taking special (mathematical?) care to do it well - correct? the bandlimiting of the black/white stripes towards the spiral is near-analytic, very impressive!

I have taken good care yes, but not with mathematical filters like I think you are using. I have rendered them initially at 32000x32000, and downscaled to sizes from 8000x8000 to 1000x1000. Perhaps I could have done better or quicker with appropriate filters...? Anyway the renders were still reasonably fast (less than one hour).

There is a point in my antialiasing that might be original though: I am taking screen gamma correction into account when downscaling. That is, I am not computing 1/n*sum x_i (with x_i=red, green or blue channel in the [0,1] range), but (1/n*sum x_i^gamma])^1/gamma. I assume a gamma value of 2.2. This method of downscaling seems logical to me, but I have never seen it addressed anywhere. Sometimes it makes little difference, and sometimes the difference is dramatic. Here is the same image downscaled from 32000x32000 with the first formula, and with the second:


http://i11.tinypic.com/4g45yz9.jpg

http://i11.tinypic.com/44kdlao.jpg

[Nahee_Enterprises]

Bernard Helmstetter wrote:
>
>    Anyway, here comes the second batch.  Those are based
>    mainly on the angle of the last iteration, and lighting effects.

Greetings, and welcome to this Forum!!     

These are some very interesting images.  Thank you for sharing them with us.

[bh]

Two more images and I'm ready for the 3x3 Warhol.

The first is a variation on the TIA, by computing the sums:
sum (z_i-z_i-1)/|z_i-z_i-1|


http://i10.tinypic.com/2v13tdw.jpg

The second is based on lighting effects, though what we perceive is different from the physical model behind (the white parts are in fact the flat ones, and the others are strong slopes). Two lights from different directions.


http://i13.tinypic.com/2qv88yc.jpg

[lycium]

damn, those are really really goodlooking pics i esp like the 2nd new one!

[alan2here]

I esspecally like the first one posted in this thread.

[GFWorld]

bh wrote >Two more images and I'm ready for the 3x3 Warhol ...

Thats some wonderful work I saw now !
Especially these ones   ...tinypic.com/34r9iyf.jpg &.tinypic.com/44kdlao.jpg 
Margit

[bh]

Just to say that I won't post new images here for a while. I'm drawing fractals with my own software (mainly because I'm a linux user), and I'm planning to make orbit traps variations, so I need to modify my program first.

[bh]

Three more images with orbit traps, and some blending:


http://i16.tinypic.com/3yxhjrl.jpg


http://i10.tinypic.com/2gxhi60.jpg


http://i12.tinypic.com/451gr4x.jpg