Category: Digital Art

The last post in this series will address how I used Processing in the classroom this past semester.  Although my experience has been limited so far, the response has been great!

In the Spring 2016 semester, I used Processing in my Linear Algebra and Probability course, which is a four-credit course specifically for CS majors.  The idea is to learn linear algebra first from a geometrical perspective, and then say that a matrix is simply a representation of a geometrical transformation.

I make it a point to generalize and include affine transformations as well, as they are the building blocks for creating fractals using iterated function systems (IFS).  The more intuition  students have about affine transformations, the easier it is for them to work with IFS.  (To learn more about IFS, see Day034, Day035, and Day036 of my blog.)

If you’ve noticed, many of the movies I’ve discussed deal with IFS.  Within the first three weeks of the class (here is the link to the course website), I assign a project to create a single fractal image using an IFS.  I use the Sage platform as it supports Python and is open source, and all of my students were supposed to have taken a Python course already.  All links, prompts, and handouts for this project may be found on Days 5 and 6 of the course website.

This sets up the class for a Processing project later on in the semester.  The timing is not critical; as it turns out the project was due during the Probability section of the course (the last third) as most students had other large projects due a bit earlier.  A sample movie provided to the students may be found on Day 22 of the course website, and the project prompt may be found on Day 33.

The basic idea was to use a parameter to vary the numbers in a series of affine transformations used to create a fractal using an IFS.  As the parameter varied, the fractal image varied as well.  This allowed  for an animation from an initial fractal image to a final fractal image.

My grading rubric was fairly simple:  each feature a student added beyond my bare-bones movie bumped their grade up.  Roughly speaking, four additional features resulted in an A for the assignment.  These might include use of color, background image, music, text, etc.

I was inspired by how eagerly students took on this assignment!  They really got into it.  They appreciated using mathematics — specifically linear algebra — in a hands-on application.  I do feel that it is important for CS majors to understand mathematics from an applied viewpoint as well as a theoretical one.  Very few CS majors will go on to become theoretical computer scientists.

We did take some class time to watch movies which students uploaded to my Google drive.  All had their interesting features, but some were particularly creative.  Let’s take a look at a few of them.

Here is Monica’s description of her movie:

My fractal movie consists of a Phoenix, morphing into a shell, morphing into the yin-yang symbol, and then morphing apart. I chose my background color to be pure black to create contrast between the orange and yellow colors of my fractal. The top fractal, which starts out as yellow, shifts downward the whole time. On the other hand, the bottom fractal, which starts out orange, shifts upward. In both cases, a small number of the points from each fractal get stuck in the opposite fractal and begin to shift with it. This leaves the two fractals at the end of the movie intertwined. I created text at the bottom of the fractal movie, which says “Phoenix.” I wanted to enhance the overall movie and give it a name. Lastly, I added music to my fractal movie. I picked the song “Sweet Caroline” by Neil Diamond.

Ethan says this about his movie:

The inspiration for this fractal came from a process of trial and error.  I knew I wanted to have symmetry and bright color, but everything else was undecided.  After creating the shape of the fractal, I decided to create a complete copy of the fractal and rotate one copy on top of the other.  After seeing what it looked like with a simple rotation, I decided something was missing so I had the copied image rotate and either shrink or grow, depending on a random variable.  In this movie the image shrinks.  I used transitional gradients because I wanted to add more color without it looking too busy or cluttered.

Finally, this is how Mohamed describes his video:

This video starts as a set of four squares scaled by 0.45, and each square has either x increased by 0.5, y increased by 0.5, both increased by 0.5, or neither increased by 0.5. The grays and blacks as the video starts show the random points plotted as the numbers fed into the function are increasing, while the blues and whites show the points as the numbers fed into the function are decreasing. I chose to do this because we often see growth of functions in videos, but we do not see the regression back to its original form too often….

I was very pleased how creative students got with the project, and how enthusiastic they were about their final videos.  I have another project underway where I use Processing — a Mathematics and Digital Art course I’ll be teaching this Fall semester.  I’ll be talking about this course soon, so be sure to follow along!

Last week, we saw how to use linear interpolation to rotate a series of fractal images.  It was not unusually difficult, but it was important to call functions in the right sequence in order to make the code as simple as possible.

This week we’ll explore different ways to use color.  Using color in digital art is a broad and complex topic, so we’ll only scratch the surface today.

The first movie shows how the different parts of the Sierpinski triangles corresponding to different affine transformations of the iterated function system can be drawn in different colors.

Recall that the points variable stored all the points to be drawn in any given fractal image.  Since the points are drawn all at once, it is difficult to say which of the three transformations generated a particular point.  But this is important because we want the color of a point to correspond to the transformation used to generate it.

One idea is to use three variables to store the points, and have these variables correspond to the three affine transformations.  Here is the code — we’ll discuss it in detail in a moment.

We use variables points1, points2, and points3 to store the points generated by each of the three affine transformations.  Note that the use of append is now within each if statement, not after all the if statments.  This is because we want to remember which points are generated by which transformation, so we can plot them all the same color.

As a result, we now need three separate calls to the stroke and point routines.  Recall that in Processing, a call to the stroke command changes the color of everything drawn after that stroke command is called.  So if we want to draw using three different colors, we need three calls to the stroke command.

Of course it follows that we need three calls to the point routine, since once we change the color of what is drawn, we need to make sure the correct set of points is that color.  In this case, all the points in points1 are yellow, those in points2 are red, and those in points3 are blue.

Again, not unusually complicated.  You just have to make sure you know how each function in Processing works, and the appropriate order in which to call the functions you use.

On to the next color experiment!  It’s been a few weeks since we used linear interpolation with color.  You’ll see in the movie below that the yellow triangle gradually turns to red, the blue triangle changes to yellow, and the red triangle becomes blue.

Let’s see how we’d use linear intepolation to accomplish this.  Below is the only code which needs to be altered — the stroke and point commands.  Also, I left out the rotate function so the changing of the colors would be easier to follow.

We’ll focus on how to change the red triangle to blue in this example, which occurs for the points in the variable points2.  The other color changes are handled similarly.  All we need to do is use linear interpolation on each of the RGB values of the colors we are looking at.

For example, red has an R value of 255, but blue has an R value of 0.  Now when p = 0 the triangle is red, and when p = 1, the triangle is blue.  So we need to start (p, R) at (0, 255) and end at (1, 0).  Creating a line between these two points results in the equation

R = (1 – p) * 255.

You can see the right-hand side of this equation as the first argument to the stroke command used to change the color for points2.

Working with the G values is easy.  Since both red and blue have a G value of 0, we don’t need linear interpolation at all!  Just leave the G value 0, and everything will be fine.

Finally, we look at the B value.  For red, the B value is 0, but it’s 255 for blue.  So we need to start (p, R) at (0, 0) and end at (1, 255).  This is not difficult to do; we get the line

R = p * 255.

You’ll see the right-hand side of this equation as the third argument to the stroke command which changes the color for points2.

Just linear interpolation at work again!  It’s not too difficult, once you look at it the right way.

For our last example, we’ll let the triangles “fade out,” as shown in the following movie.

Can you figure out how this is done?  Linear interpolation again, but this time in the strokeWeight function.  Here are the changes:

Let’s what this if-else clause does.  If the parameter p is less than 0.5, leave the stroke weight as 2.  Otherwise, calculate the stroke weight to be (1 – p)*4.

What does this accomplish?  Well, at p = 0.5, the stroke weight is (1 – 0.5)*4, which is 2.  And at p = 1, the stroke weight is 0.  This means that the stroke weight is 2 for the first half of the movie, then gradually diminishes to 0 by the end of the movie.  In other words, a fade out.

Of course when you write the code, you have to reverse engineer it.  If I call my stroke weight W, I want to start (p, W) at (0.5, 2) and end at (1, 0).  Creating a line between these two points gives the equation

W = (1 – p) * 4.

That’s all there is to it!

I hope you’ve seen how linear interpolation is a handy tool you can use to create all types of special effects.  The neat thing is that it can be applied to any function which takes numerical parameters — and those parameters can correspond to color values, angles of rotation, location, or stroke weight.  The only limit to how you can incorporate linear interpolation into your movies is your imagination!

This week, we begin a discussion of creating movies consisting of animated fractals.  Last week’s post about the dot changing colors was at a beginning level as far as Processing goes.  This week’s post will be a little more involved, but will assume a knowledge of Iterated Function Systems.  I talked about IFS on Day034, Day035, and Day036.  Feel free to look back for a refresher….

Today, we’ll see how to create the following movie.  You’ll notice that both the beginning and final Sierpinski triangles are fractals discussed on Day034.

As a reminder, these are the three transformations which produce the initial Sierpinksi triangle:

Also, recall that to get the modified Sierpinski triangle at the end of the video, all we did was change the first transformation to

We’ll how to use linear interpolation to create the animation.  But first, let’s look at the Python code for creating a fractal using an iterated function system.

The parameter p is for the linear interpolation (which we’ll discuss later), and n is the number of points to plot.  First, import the library for generating random integers — since each transformation will be weighted equally, it’s simpler just to choose a random integer from 1, 2, and 3.  The variable points keeps track of all the points, while last keeps track of the most recently plotted point.  Recall from earlier posts that you only need the last point in order to get the next one.

Next, the for loop just creates new points, one at a time, and appends them to points.  Once an affine transformation is randomly chosen by selecting a randint in the range from 1 to 3, it is applied to the last point generated.  For the purpose of writing Python code, it’s easier to use the notation

rather than matrix notation.  In order to use vector and matrix notation, you’d need to indicate that (1,2) is a vector by writing

v = vector(1,2),

and similarly for matrices.  Since we’re doing some fairly simple calculations, just writing out the individual terms of the result is easier and requires less code.

Once the points are all created, it’s time to plot them.  You’ll recognize the background, stroke, and strokeWeight functions from last week.  Nothing fancy here, since we’re focusing on algorithms today.  Just a black background and small orange dots.

The last line plots the points, and is an example of what is called list comprehension in Python.  First, note that the iterated function system would create a fractal which would fit inside a triangle with vertices (0,2), (0,0), and (2,0).  So we need to suitably scale the fractal — in this case by a factor of 225 so it will be large enough to see.  Remember that units are in pixels in Processing.

Then we need to compensate for Processing’s coordinate system.  You’ll notice a similarity to what we did a few weeks ago.

What the last line does is essentially this:  for every point x in the list points, adjust the coordinates for screen space, and then plot x with the point function.  List comprehension is convenient because you don’t have to make a loop or other iterative construct — it’s done automatically for you.

Of course that doesn’t mean you never need a for loop.  It wouldn’t be easy to replace the for loop above with a list comprehension as each new point depends on the previous one.  But for plotting a bunch of points, for example, it doesn’t matter which one you plot first.

Now for the linear interpolation!  We want the first frame to be the usual Sierpinski triangle, and the last frame to be our modified triangle.  The only difference is that one of the constants in the first function changes from 0.5 to 0.25.

This is perfect for using linear interpolation.  We’d like our parameter p to be 0.5 when p = 0, and 0.25 when p = 1.  So we just need to create a linear function of p which passes through the points (0, 0.5) and (1, 0.25).  This isn’t hard to do; you should easily be able to get

The effect of using the parameter p in this way is to create a series of fractals, each one slightly different from the one before.  Since we’re taking 360 steps to vary from 0.5 to 0.25, there is very little difference from one fractal to the next, and so when strung together, the fractals make a convincing animation.

I should point out the dots look like they’re “dancing” because each time a fractal image is made, a different series of random affine transformations is chosen.  So while the points in each successive fractal will be close to each other, most of them will actually be different.

For completeness, here is the code which comes before the sierpinski function is defined.

It should look pretty familiar from last week.  Similar setup, creating the parameter p, writing out the frames, etc.  You’ll find this a general type of setup which you can use over and over again.

So that’s all there is to it!  Now that you’ve got a basic grasp of Processing’s screen space and a few different ways to use linear interpolation, you can start making movies on your own.

Of course there are lots of cool effects you can add by using linear interpolation in more creative ways.  We’ll start to take a look at some of those next week!