HPDZ.NET News

Two months have passed since the last update, partly because of my having much less time available for this work and partly because the latest video project was much more complex than any prior animation. Some technical challenges with the web site and file downloading also took some time to resolve.

What's New

Secant Animation 1 ("Wessel")

Secant Animation 1 has been completed, which is a major milestone, the culmination of multiple significant improvements in the fractal software. It is one of the most complex fractal animations ever created, both in terms of motion complexity and colorizing. It is the second published animation of a fractal generated by the secant method, the first, of course, being Secant Animation 2 published in March.

This video is named in honor of Caspar Wessel (1745-1818), a Danish-Norwegian mathematician who was the first to recognize the geometrical representation of complex numbers that helped them gain widespread acceptance. All fractal animations on HPDZ.NET owe their existence to the mathematicians who developed complex numbers.

File Playback Problems

Some users have reported that MP4 files are not playing back as they download. The problem has been identified (actually it is multiple problems) and fixed. SA1 is the first set of videos that support fast start. Other files on this site will eventually be modified and re-uploaded to support fast start playback. This only affects MP4 files, not WMV files played with Windows Media Player. For more details, see this page.

Details

Secant Animation 1

This newest animation is what I call a "tour" video -- it moves around the fractal, zooming in to certain areas, then zooming out and moving over to a new area, to give a sense of the entire structure. This kind of animation is enormously complicated to plan and requires careful storyboarding and lots of trial and error to get all the motion correct.

This kind of video is different from the style of SA2, which was a parameter roll type of animation (that is, the viewpoint stayed essentially fixed while the numerical values controlling the fractal shape changed). This is also different from most of the other animations I've made, which have mainly been deep-zooms that zoom into a specific point on the fractal without moving around at all.

Why 1?

It's number 1 because there's another related project that I started after this one. SA1 turned out to be much more difficult to render due to the elaborate motion paths that had to be carefully tuned, and it was also very difficult to colorize properly. SA2 was easier to make and also useful for some software testing issues, so it got published first.

Math Details

The fractal is the result of applying the secant method for numerically solving equations to finding the zeros of the cosine function. More details can be found here.

Comments

The colorizing was done manually for the first time in this video. Overall, I am pleased, but there is one span around time 4:10 in the final video where the colors get kind of simplified and too monotone. This could have been changed, but the cost would have been that the ten seconds or so leading up to that point would have the color palette changing at an unacceptably fast and annoying pace.

Software Improvements

Motion

As mentioned in the last newsletter, the motion/zooming system has been greatly improved, and SA1 is the first video to take full advantage of it. Eighteen key frames were used, each of which required painstaking adjustments to the lead-in and lead-out for both the motion and zooming.

Adaptive Animation Coloring

Colorizing continues to be an area requiring major work. As anticipated in the last newsletter, SA1 required some significant enhancements to the colorizing technology.

Many, if not most, nontrivial fractal animations have to deal with enormous ranges of numerical values in the raw fractal data that is turned into colors to make the videos and images. One of the major continuing projects at HPDZ.NET is developing new ways of smoothing out the noisy data from the fractals to provide attractive colorizing of the fractals.

Previously, I had been relying on various types of digital filters to help with this, and generally the results were adequate. SA1 could not be colorized with any of the existing tools I had created, so I decided to take control and create a way to manually specify how this raw data gets converted to colors. It is very tedious and requires a lot of trial and error, but ultimately yielded superior results in this very difficult video.