This is what I was aiming for with the post from the other day.
It’s interesting, but doesn’t have quite the same punch as the “error” version.
Happy Accident #2
I had an error with the length function, but I really like the results.
Related Images:
Chaos #1
I’m starting a new series based on chaotic dynamical systems.
Related Images:
HTML5 Canvas Image Effects
It helps when you actually upload the page for your demos. Â One of the down sides of scheduling posts before finishing them is that you might publish prematurely.
This is the first page I put together to play around with the combining the canvas and file upload controls. Â The page works by loading up the image and generating random circles using the center point of the circle to sample the color. Â Playing with the alpha level of the circle led to some interesting effects, but I didn’t prefer one setting over any of the others, so I set it to vary with time.
I’m trying to get it to preload an image from the site, but the canvas security and random errors I’m getting are making it not worth the effort. Â Long story short, you have to have an image to upload for this page to work, so when you navigate to it, click the button and upload an image.
Related Images:
Nothing fancy: Winter bubbles
My inspiration board on pinterest has been filling up, so I wanted to take a few hours to whip up something quick. I really liked the simplicity of winter trees. I didn’t want to do a direct copy, so this is my riff on the composition.
Winter Bubbles
Related Images:
Gravitational slingshots
In my last few posts, I’ve been trying to characterize different potentials through the shapes of their orbits (gravitational, harmonic, lorentz). In the middle of it, I came across a post on bad astronomy about gravitational slingshots. I figured it would be a perfect opportunity to use these images to show the effect in a different way.
To start, I’m sending a beam of particles directly towards the planet in question, just like my previous demos. This time I slowly increase the speed of the planet and the orbit shapes change accordingly. It was pretty hard to see what was going on with the first few renderings, so I started by increasing the intensity and color of the orbit with the speed of the particle. Things were starting to look better, but they didn’t really start to illuminate the dynamics until I subtracted off the initial velocity of the test particles and only showed particles that were moving faster than their initial speed.
This one shows the default case of a stationary target. The particles accelerate as they get close and then slow down to their initial speed as they move a way.
This is the first shot of a moving target. You can see there are now a few particles that are being shot back at a higher speed, extending past the first line of acceleration from the last image.
There are a few more features visible now as the speed picks up. Since these are lorentz potentials, you can see a more pronounced central core of paths that pass right by the target and are only slightly deflected. The same thing happens with particles further out, but there is a sweet spot that generates two beams. The faster the target moves, the less deflection is seen. With a slow target, the final trajectory is almost a 180, but the angle decreases more and more as the target speed increases, though the final boost in speed increases as well.
I wanted to finish up with one image that captured the gravitational result. There are many similarities, but some significant differences. The infinity at dead center means that there isn’t a max deflection angle like there is in the lorentz case. As you you approach dead center, there will always be a set of bound states. This leads to simpler images, and really those trajectories aren’t all that interesting as far as the slingshot goes. Hyperbolic orbits are the only ones that get launched somewhere.
Anyway I hope these images illustrate some of the features of this process. I’ve learned a lot putting them together, both about these processes, and how to illustrate them in a way that makes the physics visible to the naked eye.
Related Images:
Central force images #2
This is the second in a series profiling the solutions to common central force fields from physics. Check out the the first one on the Newtonian gravitational force field.
These are even more plain than the gravitational well. That’s because these are for the harmonic potential. One of the things that distinguishes this potential is that all orbits are bound and elliptical in shape.
Contrasting that with the gravitational potential, there are 3 orbit shapes for that force: elliptical, parabolic, and hyperbolic. You can see the effects in the image. There is a relatively dark area bounded by the parabolic orbit, with a diffuse bright spot near the tip of the parabola. This is due to the tightly curved hyperbolic orbits nearby. The bright spot fades out into a dim patch further out as the trajectories become straighter the further they are from the center of the field.
Both of these fields have infinities which means that they valid only as approximations to actual forces. The harmonic field has the most severe, it extends out to infinity and no particle can escape from it’s pull. It doesn’t matter how fast our test particles are moving, they will never escape the grasp of the central pull.
Newtonian gravity has the opposite problem. The force becomes infinitely strong as you approach the central point. This causes problems for the differential equation solver and leads to the lower two images with kinked up trajectories and the rays radiating from the central point.
In our next installment I’ll be showing a field without those infinities and see how it stacks up.
Related Images:
These aren’t all that pretty
Any guesses as to what these are?
Rather than focus on chaotic dynamics I wanted to see how I could explore the differences between the classical central forces. This one is newtonian gravity.
Related Images:
Sensitive dependence on initial conditions
This next demo highlights the butterfly effect. These curves show the same pendulum as before, this version has the time axis wrapped around the center of the screen and the exponential of the angle as the radius. All the pendulums start at a very similar initial position and for some driving functions they diverge wildly and others they all stay pretty close.
Related Images:
Another one I just like
The swarm of cracks just looks nice to me.