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Secant Animation 2

The first-ever published animation of a fractal using the secant method.

Although the secant method is quite well-known in the numerical analysis community, very few fractal images have been published using it, and this video, as far as I can tell, contains the first animation ever created based on the secant method.

This is a parameter-roll type of animation created by varying the initial value that is fed into the algorithm. There isn't much zooming, and there's almost no motion at all. Instead, the fractal morphs from a set of little blobs into a swirling multi-armed dance machine.

The details of the math are described below. Additional discussion of the secant method in general is in the technical section. Some examples of this technique applied to solving other equations can be found in the still images section.

Sample frames

The high-quality MP4 file was encoded at a higher bit rate than the high-quality WMV file and is the highest-quality version. The fast-download MP4 file was encoded at a lower bit rate than the WMV file and is the smallest file.

All these files use variable bit rate encoding to minimize the file size for a particular quality level.

Comments

This is the debut of two new elements: the fractal content itself and also the infrastructure in the software that controls the animation.

The Fractal Equation

The fractal equation for this animation comes from applying the well-known but somewhat obscure "method of secants" for numerically solving general equations f(z)=0 for z in the complex number plane. The general technique is described in the technical section page on convergent fractals, and more details can be found in plenty of external references.

The particular equation here is (z+1)(z-1)(z+2)(z-2) = (z²-1)(z²-4) = 0. The roots of this equation are, of course, +1, -1, +2, and -2.

The video shows what happens when the first initial guess z₀ is changed. The second initial guess is always set equal to z₀². The colors indicate the number of iterations required to converge to one of the roots of f(z). Information about which root was reached is not used in this animation.

Motion Control

This animation uses the latest revision to the motion control system. This new version makes it far easier for me overlap key frame segments so that the motion is totally fluid. Although 19 separate key frames were used to specify the paths of all the parameters of this animation, there are only a few moments where the action stops, and those were done by choice. (Yes, I know, there are plenty of places where it gets kind of slow...next time will be better.)

Many other complex, multi-segment animations move to a certain image location, then completely stop, then move to another location, stop, etc. Parameter roll animations like SA2 can also suffer from this move-stop effect as well. By overlapping segments, the motion in SA2 is more fluid, and it's possible to design the video so that something is always moving.