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*Subject*: Dynamic graphs*From*: sweetser at TheWorld.com (Doug Sweetser)*Date*: Sun, 17 Aug 2003 16:23:28 -0400

Hello: Two months ago at the Cambridge Brewing Co. after a Blu meeting, I had lamented that I wanted a program to make dynamic graphs of the complex plane (OK, it was a strange lament, but I'm like that). The only dynamics graphs I have seen are on oscilloscopes. In all math books, the graphs are static, even ones with time as an axis. I thought a better representation of time would involve animations. Think of scanning, line by line, a graph to generate an animation. That was the goal. Several people made proposals on how to get this sort of thing done. I went with Jabber's idea of using the GD library which works well with Perl, my language of choice. I wanted to make the program command line friendly, because I think the model of small programs that can be combined in a various ways has proven its worth. It also would make writing a cgi script for a web interface direct, should that day ever happen. There are four programs in this alpha-ware set. "dg" creates dynamic graphs if fed real, complex, or quaternion numbers. Quaternions are a set of four numbers that can be added, subtracted, multiplied or divided, like the complex numbers. Three complex numbers actually make up one quaternion, with the three complex numbers all sharing the same real number. If that shared real number is viewed as time, and the three imaginary numbers as x, y, and z, a dynamic graph of a quaternion function may look like a 3D animation. The png format is used for static summaries, mng for animations which can only bee seen with Mozilla-based browsers, ie not IE. "ug" work with unary functions like sine and cosine to generate lots of quaternions that dg can graph. "bsg" takes numbers from both STDIN and ARGV so the binary stream of numbers going into addition or multiplication functions can generate a bunch of quaternions for dg to graph. The module Qlib.pm does the quaternion math. The output of these programs has been fascinating. Even my girlfriend is interested which is exceptionally rare in the obscure math I do for fun, no profit. I have put together a WimpyPoint presentation here: http://sdm.openacs.org/wp/display/1238/ My favorite insight was with the sine function. We all know it has to do with the circle. The long line of camel humps of the standard graph of the sine function doesn't visually say "circle". The reason is that the domain is graphed against the range. With dynamic graphs, it is possible to graph the domain and the range separately. The domain is linear, and thus dull to watch: it either remains still or moves like a steady snail. The range when sine is applied to the steady snail looks like a bunch of circles. This is in 3D, so there is tilt, roll, and yawl, but there are circles none-the-less. How does cosine differ? Cosine is an even function around time. Events appear in the animation in pairs. Sine is an odd function around time, so its animation feels syncopated. At this point, I am not sure if the slides are enough to explain this work, or whether I need to be there to explain these graphs (if you visit, please tell me). If you want to look at the code, a gzipped tarball is available for download here: wget http://theworld.com/~sweetser/quaternions/qemation/dg_tools.tz Although still alpha, it does have -help and perldoc documentation, and "grep # program" will list all the comments. doug

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