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-- Mike Condron

Once again, this past month has been very busy and the demands of real life have limited how much time I have available for fractal work. Still, some great stuff has been accomplished, and a lot of software infrastructure has been put in place to create some really nice animations in the near future.

Secant Animation 2

Secant Animation 2 is the first published animation of a fractal generated by the secant method (more on this below, and on the web site) that I know of. It is also a major deviation from my usual genre, being neither a Mandelbrot set fractal nor a deep zoom.

Why 2?

It's number 2 because there's another related project that I started before this one. SA2 turned out to be a better tool for debugging some revisions to the software, so I went ahead and finalized it before SA1. I'll get SA1 done soon.

Math Details

The fractal is the result of applying the secant method for numerically solving equations to a particular fourth-order polynomial. The secant method is similar to Newton's method, which has been used extensively to create lots of great still images and animations. The secant method is not as efficient as Newton's method for actually solving equations in real-world applications, but it generates much more interesting fractals.


This animation is quite different from my usual style -- it doesn't move around or zoom in much. Instead, the parameters that control the underlying equation change with time.

There are a few places where the action slows down a bit, I know. Overall the motion is very nicely fluid and has a decent balance between action and calm, but I think a higher proportion of action would have made this better.

Historical Perspective

As far as I can tell, the first visualizations of fractals generated by the secant method were created by Szyszkowicz in 1990. An UltraFractal formula file exists from about that same time, but not much seems to have ever been done with it.


After the trial run of CanyonDeep, I've decided to revise the plan for that project. It's not practical to render a full-size version of that animation right now. I will make something that goes moderately deeply into the crevice, then zooms more deeply into the structures at the edge of the walls. This should happen in late March or early April.

New Music Loops

I bought a couple of new sets of loops for Acid. Secant Animation 2 is my first effort using one of them.


I have posted a few new blooper images on the web site. These images are the result of hours of tedious work that had to be followed by still more hours of tedious work.

Software Improvements

Motion Path Control

To create an animation, all the details of the location and magnification of the images, as well as the parameters of the mathematical function that generate the fractal, have to be carefully scripted in a series of key frames. The key frames are images and parameter values that define what points the video will pass through as it progresses. The software interpolates all the parameters between each of the key frames to generate the animation.

Most of my previous animations have been just deep-zooms, which don't have much complicated motion. Typically they just move over to the zoom point and then start zooming. So I have not had much incentive to develop a more sophisticated motion control system. SA2 changed this.

The simplest way of interpolating between the key frame values results in an animation that is a series of independent segments strung together, with abrupt starts and stops at the juntions. It works, but visually sort of lurches from one set of parameters to another. Most software that produces fractal animations, including mine, has the ability to smoothly transition the position, magnification, etc., from one segment to the next, which looks a lot better. But the transitions at the key frames are often still evident as moments in the video where everything seems to stand still, then start up again. I have wanted for a long time to be able to easily get more fluid transitions between key frame segments.

I've reworked the way transitions across key frames are done, and now I can make it so you really can't tell where the key frames are anymore.

For an example of the jerky sort of motion I've finally been able to eliminate, watch the last few seconds of SA2. The final zoom segment has no smooth transition. I did this deliberately, for artistic effect. Most of the rest of it is very fluid -- you would not know this by looking at it, but there are 19 separate key frames defining distinct starting and stopping values for all the parameters in SA2. I've overlaid them so almost nothing is ever static and it all flows together very nicely.

Adaptive Animation Coloring

Not much has been done with this component this month. This remains a priority, but it got pre-empted by my desire to get some of the secant fractals done and to upgrade the motion path stuff. Secant Animation 1 will probably require this technique. We'll see.


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