If you’ve been following my Twitter (@cre8math), you’ll have noticed I posted quite a few videos last fall. At the time I was writing a lot about creating fractal images, and I really wanted to learn how to make animations using these images. I had heard about Processing from a lot of friends, and decided that it was time to dive in! Here is one of earlier movies, but still one of my favorites.
Processing has been around since 2001. From the very beginning, Processing was used in the classroom as a tool to allow students to quickly create graphical images and movies. It is now used widely by educators, artist, designers, architects, and researchers. Many companies use Processing for data visualization. Click on the link to read more!
One of the great things about Processing is that it’s open source — a specific intention of the developers. Now, there’s even a Processing Foundation. Its purpose is to “promote software literacy within the visual arts, and visual literacy within technology-related fields — and to make these fields accessible to diverse communities.” The idea is to provide everyone access to software you can use to create a wide range of visual media. Not just those who can afford it.
I support open-source initiatives as well. Most of the digital art I’ve talked about on my blog was written in Mathematica, a powerful programming language, but expensive to buy. I rewrote all the algorithms you’ve seen on the Sage worksheets in Python because I wanted them to be available to anyone who has access to the internet. Not just those who can afford pricey software….
Now there won’t be time to go into everything about Processing — that’s what the internet is for. I often google phrases like “How do you change the background color in Processing?”, and usually get the information I need pretty quickly.
What I want to focus on today is the coordinate system used in Processing. If you’re new to computer graphics, it might be a little unfamiliar — the coordinate system is based on pixels. For example, the screen on my Mac is 1440 x 900 pixels. But the origin (0, 0) is in the upper left corner of the screen, so that the lower right corner has coordinates (1440, 900). In other words, while the positive x-axis is still to the right, the positive y-axis is now pointing down. This is sometimes called a screen coordinate system, or screen space.
This convention has an interesting history. Older television screens used cathode ray tubes, and literally updated the screen by refreshing one line at a time, from top to bottom and from left to right. After updating the pixel at the lower right, refreshing began again at the upper left. This process occurred several times each second to give the illusion of continuous motion. Because the refresh was done from top to bottom, the y-coordinates increased from top to bottom as well.
Such a coordinate system is not unusual. The PostScript programming language (which I told you about two weeks ago) has a coordinate system based on points. The definition of a point has changed over time, but PostScript uses a coordinate system where there are 72 points per inch, and the lower left corner has coordinates (0,0).
PostScript is used for printing physical documents. Because copier paper typically has dimensions 8.5″ x 11″ in the United States, the upper right corner of your document would have coordinates (612, 792).
In Processing, you need to specify the size of the screen you want for your movie. In the example above, I chose 768 x 768 pixels. This may sound like a strange number, but a commonly used screen size is 1024 x 768. Because of what my image looked like, though, I wanted the screen to be square so there wouldn’t be extra space on the left and right.
Now the objects you see in the movie have rotational symmetry. So it would make sense for the origin (0,0) to be at the center of the movie screen, not the upper left corner. A convenient coordinate system for these objects might have the upper left corner be (-1, 1), and the lower right corner be (1, -1), just like the usual Cartesian coordinate system. This system of coordinates is sometimes called local space or user space. In other words, it’s the space the user is thinking in when designing various images.
So now the issue is converting from user space to screen space. This isn’t difficult, so we’ll just work through this example to see how it’s done. All you need to remember is how to find the equation of a line.
Why a line? Let’s look at x-coordinates first. In user space, x-coordinates range from -1 to 1. In screen space, they range from 0 to 768. If you went 1/4 of the way from -1 to 1 in user space — which would put you at -0.5 — then you’d expect the corresponding point in screen space to be 1/4 of the way from 0 to 768, which would be 192. You’d expect this for any ratio, not just 1/4. This is precisely what it means for the transformation of coordinates from user space to screen space to be linear.
Since -1 in user space corresponds to 0 in screen space, and 1 corresponds to 768, all we need to do is find the equation of the line between the points (-1,0) and (1,768). In Cartesian coordinates, this would be 
Once we know how to do this for the x-coordinates, we can proceed the same way for the y-coordinates. You should be able to figure out that
Note that the minus sign in front of 
This is a very general idea which will allow you to convert from one space to another fairly easily. And it’s definitely necessary in Processing to set up the screen for your movie.
Next week we’ll look at linear interpolation — another important concept which will help us make movies. Until then!
Creativity in Mathematics 