An escape time algorithm for Kleinian group limit sets

[JosLeys]

The 'separation line' that is now in the algorithm works fine for most cases, but there are other!
For certain values of the Klein parameter the limit set becomes 'cusped'. The complement, that is , the part that normally remains empty, now consist of circles also.
In such a case, the separation line has to be very curvy and thus makes it very difficult to determine whether a point is 'under' or 'above' that line.
The image below is for the so-called 33/185 cusp. I got help from Geoff Smith, a Kleinian 'connaisseur' is determining the line.

[DarkBeam]

A fractal separation line!!! Does not look easy to implement it at all ... ouch

[Aexion]

A fractal separation line!!! Does not look easy to implement it at all ... ouch

Hmmm, you can draw the line on a texture (or any bitmap buffer), use a paint algorithm, and then read the transformation from the buffer... It may works for 2d or simple 3d cases.. (it's just an idea)..

[DarkBeam]

Ramiro it is really not helpful ... I don't have access to images normally... And even if, it would mean a dramatic slowdown to read from it!

[JosLeys]

#Darkbeam: if you think reading from an image or texture slows things down, you don't want to know how the points on this goofy separation line are calculated
That will really slow things down...

[DarkBeam]

I do imagine that
The formula must stay as is - "brute" and working for most cases - people simply ignores slow formulas. Sad truth but. 
Little corrections can be done but not really much else. In my opinion.

[JosLeys]

@Darkbeam : I fully agree

[JosLeys]

Here is the result when using the exact separation line. In this example the line consists of about 150 points.
As the image shows, the horizontal 'wrapping' cannot be just by a straight line anymore either.

Not good for a 3D version I'm afraid: taking into account all those points at every iteration and step will be very expensive indeed.

Also: for this level of detail, it required 800 iterations!

[DarkBeam]

A side note: the separation line looks self similar! Can we compute a single segment of it, then "tile" it somewhat?

[JosLeys]

@Darkbeam : I don't think so, I'm afraid....

[JosLeys]

I have a new version ready of my article on the subject: http://www.josleys.com/articles/Kleinian%20escape-time_3.pdf

[Imagyx]

I have to ask again, because it doesn't seem to be working in my program yet.
The Distance Estimation

Code:

float  JosKleinian(vec3 z)
{...}

from one of the posted .frag files
can be inserted into any raymarcher that works well with for example:

Code:

dist = z.getLength() - r;

for tracing a sphere where z is the point the distance estimation is calculated for and r the radius of the sphere.
Right ?

[DarkBeam]

The de is returned into a single float value. Read again the frags for details.

[Crist-JRoger]

Kleinian_test23_zrot.frag from Pupukuusikko, De-Kn based renderer, wanted to see what happens when FudgeFactor over than 1 

[Crist-JRoger]

And GI from Eiffie