Window's Vista

This week, Futility Closet had a post about the Vista Paradox. The paradox is connected to an effect I discussed before, where the angular nature of our vision causes objects to move in unexpected ways. In this case, we're viewing a distant tower through a window. As we approach the window, the window takes up proportionately more of our vision that the tower, causing it to appear to shrink. I tried making an animation of this to test my understanding:

As expected, the tower (red) changes very little, while the closer window (blue) expands rapidly. Initially, I was concerned with how the window's proportions appear to change, but I checked with my mother Sally, and she gets the same expression I did. There still may be an error in my implementation though.

After reading the Futility Closet post, I was reminded of an effect often used in filmmaking called a Dolly Zoom: The camera moves forward (backward) while zooming out (in), causing the edges of the frame to stay where they are, but the center to contract (expand). The scene that always comes to my mind for this is one from Fellowship of the Ring:

makeagif

Using the same framework as above, I plotted a series of rings at different distances, but the same size. Then I moved the camera toward them, while fixing the edges of the plot to the nearest ring:

This effect makes the center appear to shrink, the opposite of the motion in the film, but exactly the same technique. It's neat to see this artifact of human perception appear in both architecture and filmmaking!

Scents and Scents-Ability

[Yes, it's a homophonous headline!]

We adopted Eros from the Humane Society, which had picked him up as a stray, so we know nothing about his breed and history. He reminds Marika of the beagle she grew up with though, and he has many hound-like traits, including an intense interest in smells. Often during walks he'll plant his feet for several minutes, and swing his head back and forth sniffing.

I was curious what kind of directionality he could get just by moving his head – I imagined that after a good distance, the scents would be too blended to pick out individual sources.

Smells are created by bits of a substance that are picked up by the air. Those bits then spread out according to Fick's Laws of Diffusion. The form of this equation, where the change in time depends on the Laplacian of the current state, is exactly the same as that of the heat equation, which I've discussed before. What's interesting about this case though is that heat has only one type spreading out, while we can have different smells that diffuse and mix.

I decided to adapt my cake-heating simulation to look at how much directional information Eros can get from his head movements. The setup is two sources of scents that emit particles according to Fick's laws. Eros then senses the concentration of each in a cone in front of his nose. I struggled to find a good way to display the concentrations, which drop off rapidly from the sources, but I suppose that's just a testament to the sensitivity of dogs' noses. The first case has the two sources equidistant:

Like I said, you can't see much past the central dots, but if we add up everything in the white cone,

Now we do get directionality. I was curious to see what happens when we put one source closer:

For this case the concentrations are

Because the blue scent is closer, it has a higher concentration in all directions, but comparing the left and right ends of each curve, we see we can still find a direction that gives more scent.

It's a Wash

[This post contains flashing images after the break. Photosensitive folks may want to skip it.]

I recently got this question from my Aunt Linda: I hate my washing machine because it is always off balance and knocks the shit out of itself. So I stare through the clear lid and try and figure out which piece of clothes is causing the problem. It really is trial and error. So I was thinking that the design should include 4 quadrant color distribution and help. Then I thought how fast the rotation is and where to spread out the different colors so they create the illusion of coming together visually by the timing of when the color is matching. Would there be a formula that would work at giving rotation and circumference?

Similar to a previous post on flickering lights, this question brings us into the field of psychophysics. I liked Linda's idea of using a color scale, so I decided to try animating an HSV color wheel with a wobble:

As Linda says, it's difficult to line up the color with the motion. My idea was to try to cut out the extra info by focusing on only a central slit. We can add some scale lines as well to help:

Now we can see pretty clearly that the red points are the heavy ones, pulling the washer out of alignment. Problem solved, right?

Sadly not. The images above are animated GIFs, which have a maximum frame rate of 100 frames per second. Washers on the other hand, tend to spin at over 1000 RPM. For the points I show, that would require 833 frames per second. We can get around the limit by exporting the animation as a giant block of HTML, but the result is pretty dizzying, so I'm going to paste it after the break. Thanks for a great question, Linda, though I wish I could have given you a more satisfying answer.