Stretch by MakinMagic

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Just trying some new options in my 3D quaternion formula for Ultrafractal.

Loki7z Featured By Owner Jun 10, 2012 Hobbyist Artist

Very nice!

lyc Featured By Owner Nov 19, 2006

the fullview/download is night-and-day different from the horrible preview!

quaternionic julias are always a pleasure

how was this rendered? looks like ray tracing to me, and fairly accurate too considering the smoothness of the surface (perhaps the number of iterations were purposefully limited?). i think i read something on fractalforums.com about your having implemented a surface-distance estimation based approach via ultrafractal.

btw, if you have a relatively modern graphics card with shader capabilities, you'll really like [link]

MakinMagic Featured By Owner Nov 19, 2006 Hobbyist Digital Artist

Was ray-traced, the current formula ("Solid-3D Quaternions" in mmf.ufm) uses a smart-step algorithm i.e. it steps smaller distances as the iteration depth increases but the distance is not controlled by a distance estimator algorithm - as Ron Barnett terms it, it's a more brute force approach

It is solid at just 10 iterations to keep it smooth (and to keep rendering time down). Some parts of the middle stages would be less than a pixel in size even at 11 iterations so that's another reason why it's on only 10.
To get the surface really accurately it uses a binary search to find the solid boundary when solid is hit, this is faster than reducing the step distance to rediculously small values as the solid is approached.

MakinMagic Featured By Owner Nov 20, 2006 Hobbyist Digital Artist

I'm really more interested in getting closer and more accurate to the true "inside" and using a distance estimation calculation dramatically improves performance. The new version I'm working on (and Ron Barnetts 3D code in reb.ufm) are comparably 2 to 3 times faster and at least 4 or 5 times more accurate.

I don't know the method of regula falsi - where can I find an example of the algorithm ?

lyc Featured By Owner Nov 19, 2006

i'm very interested in this "smart step" algorithm, as it sounds like a pretty efficient way to bracket roots without a priori knowledge of the surface equation. it would be interesting to compare it to some rudimentary (a priori) spacial partitioning schemes!

btw, one thing i must suggest is the use of regula falsi (the method of false position) rather than bisection; its order of convergence is significantly better - approaching that of newton's method, or more closely the secant method - and it still has the same convergence guarantees and lower error bound as the bisection method. evaluation of the potential function sounds pretty costly and so it's good practise to make best use of those computed samples! it may even be worthwhile to fit something better than a linear equation (a particularly efficient higher order algorithm is brent's method) if the function is smooth, as is the often case when you limit the number of iterations to match the pixel resolution.