I haven’t had much time to play with the crossing lines. I do like their simplicity and the way they can create complex crossing relationships. I’ll be playing around with braids next, and integrating differential equations and chaos of course.
The same equations were used to generate this image as well. The only difference is the starting point. Instead of expanding from the center of the distortion field, the starting point is displaced a little bit towards the bottom. This highlights a number of different things. First while the equations are symmetric, the resulting image is decidedly asymmetric. If you follow the lines to the right vs. the ones to the left, you can see that the paths have decidedly different curves.
This is what you get if you take the parallel stream from the rest of the series and replace them with a circular spray from the center of the distortion. Everything gets bent uniformly when it hits the ring. Moving to second order equations gives you the freedom to have multiple directions from a single point like this without a singularity.
Double twist and wrap. While the last image was on the lackluster side, this one has everything turned up to 11. There is a strong deflection as the particles pass through the first peak and then they are swung back again as they exit the ring on the other side. An important thing to notice is that these are clearly defined and there is little mixing. If these equations were chaotic, then there would be mixing among the lines.
One of the most popular images I’ve developed (I’ve no idea what verb to stick here) is this basic geometric op art style piece. I was just using a very simple mask to test out using differential equations to generate families of curves. It’s grown on me, and I’m starting to put some more images together in a similar style. This is the first of what I hope is a very fruitful series.
The first images I generated used first order differential equations, which have the nice property that the curves never cross. However, it severely limits the shapes that can be produced as each point has one and only one direction through it. Bumping up to second order added multiple directions through a point as an option, and it turns out some of those crossings can end up looking pretty sharp.