## Making Movies with Processing II

Last week we learned a little about the history of Processing, as well as what the coordinate system is like in Processing (and many other graphics applications as well).  Today I’d like to discuss an idea which I find very helpful in making movies — linear interpolation.  We’ll only have time for one simple example today, but we’ll go through that example very thoroughly.

So here’s the movie we’ll explore today.  Not glamorous, but it’s a start.

You’ll notice what’s happening — the dot slowly turns from magenta to black, while the background does the opposite.  Let’s look at the dot first.  In Processing, RGB values go from 0 to 255.  Magenta is (255, 0, 255) in RGB, and black is (0, 0, 0).  We want the dot to go smoothly from magenta to black.

The way I like to do this is to introduce a parameter p.  I think of p = 0 as my starting point, and p = 1 as my ending point.  In this example, p = 0 corresponds to magenta, and p = 1 corresponds to black.  As p varies from 0 to 1, we want the dot to go from magenta go black.

This is where linear interpolation comes in.  For any start and end values, the expression

(1 – p) * start + p * end

will vary continuously from the start value to the end value as p goes from 0 to 1.  It should be clear that when p = 0, you get the start value, and when p = 1, you get the end value.  Any value of p in between 0 and 1 will be in between the start and end values — closer to the start value when p is near 0, and closer to the end value when p is near 1.

Because this expression is linear in p — no quadratic terms, no square roots, no sines or cosines — we say that this expression gives a linear interpolation between the start and end values.

Let’s see how this works in Processing.  Here’s a screen shot of the code that produced the video shown above.

The basic functions for making a simple movie are setup and draw.  The setup function is executed once before the movie starts.  Then the draw function is called repeatedly — each time the draw function is called, a new frame of the movie is created.

In a typical program, if you wanted to repeat something over and over, you’d need to make a loop.  But in Processing — since its purpose is to create frames for movies — this is built in to how the application works.  No need to do it yourself.

So in Processing, the variable frameCount automatically keeps track of which frame you’re creating.  And it is automatically incremented each time the draw function executes.  You’ll find that this is a very nice feature!  This way, you don’t have to keep track of frames yourself.

Note the size declaration in the setup function.  Last week, I mentioned that you’ve got to set the size of your coordinate system when you start your movie.  The “P2D” means you’re creating a two-dimensional image.  You can make 3D objects in processing, but we won’t look at that today.

Typical applications which make movies (like PhotoShop, or the Movie Maker found in Processing’s “Tools” menu) use about 30 frames per second.  So 360 frames (as above) will be a 12-second movie.  Enough to see the dot and background change colors.

The if/else clause determines the frames produced.  Note that when frameCount gets larger than 360, you skip to the “else” clause, which is “noLoop()”.  This essentially stops Processing from creating new frames.  The saveFrame command saves the individual frames to whatever directory you specify; if you forget the else clause, Processing will just keep generating more and more .tiff files and clutter up your directory.

So the if/else clause basically tells Processing this:  make 360 frames, one at a time, and save then in a directory called “frames.”  Once all 360 frames are made, it’s time to stop.

The makedot function is what actually creates the frame.  Now we’re getting to the linear interpolation!  The frameCount/360. is what creates the value of p.  When frameCount is 1, the value of p is 1/360 (close enough to 0 for the purposes of our movie), and when frameCount is 360, the value of p is 1.  As the value of frameCount increases, p increases from about 0 to 1.

It’s important to use a decimal point after the number 360.  If you don’t, Python will use integer division, and you’ll get 0 every time until frameCount reaches 360, and only then will you get 1.  (I learned this one the hard way…took a few minutes to figure out what went wrong.)

So let’s go through the makedot function line by line.  First, we set the background color.  Note that when p is 0, the color is (0, 0, 0) — black.  As p moves closer to 1, the black turns to magenta.  It is also important to note that when you call the background function, the entire screen is set to the color you specify.  Processing will write over anything else that was previously displayed.  So be careful where you put this command!

Next, the width of lines/points drawn is set with strokeWeight.  Remember that the units are pixels — so the width of the dot is 300 pixels on a screen 500 pixels wide.  Pretty big dot.

The stroke command sets the color of lines/points on the screen.  Notice the (1 – p) here — we want to begin with magenta (when p = 0), and end with black (when p = 1).  So we go the “opposite” direction from the background.

Maybe you noticed that at one point, the dot looked like it “disappeared.”  Not really — note that when p = 1/2, then 1 – p = 1/2 as well!  At this time, the dot and background are exactly the same color.  That’s why it looked like the dot vanished.

And last but not least — the dot.  Remembering the coordinate system, the dot is centered on the screen at (250, 250).

That’s it.  Not a long program, but all the essentials are there.  Now you’ve got a basic idea of how simple movies can me made.  The next post will build on this week’s and look at some more examples of linear interpolation.  Stay tuned!

## Making Movies with Processing I

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 $y=384(x+1).$ Or we might alternatively say

$x_{\rm screen} = 384(x_{\rm user}+1).$

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

$y_{\rm screen} = 384(1-y_{\rm user}).$

Note that the minus sign in front of $y_{\rm user}$ is a result of the fact that the positive y direction is pointing down.

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!