Chapter 6 of the Coding Curves Series

Another physical device which renders complex curves that we can simulate!

This one is called a Pintograph, and I’ve actually built one of these myself.

We’ll start with a video – this is literally the first video that came up when I searched Youtube for “pintograph”, but it does the job.

A pintograph can be considered to be a type of harmonograph, but rather than being based on pendulums, pintographs are usually driven by electric motors (though some are hand-cranked). There are disks attached to the motors and arms attached to the disks and a pen attached to the arms. You can change the size of the disks and where the arms are attached, the length of the arms and where they pivot, and the relative speed and offset of the motors to create a bunch of different types of curves.

For a long time I didn’t know where the word “pintograph” came from. I finally discovered it from the person who coined the term. Actually, their daughter coined the term. It comes from a pantograph, and the idea that the spinning wheels look like a Ford Pinto. Read more here: http://www.fxmtech.com/harmonog.html

In case you are not familiar with a pantograph, it’s a device that is often used for copying drawings. It has a few pivoting arms. You pin one of the pivot points so it doesn’t move, then put a pointer in one of the points and a pen in another. As you move the pointer along the original drawing, it moves the pen along the same shape. You can configure it to copy the drawing at the same size, or scale it up or down.

The Simulation

The pintograph we’ll simulate will be pretty simple. It will have two rotating disks with one arm attached to each. The arms will be attached to each other on the opposite end and that’s where the virtual pen will be.

First of all, we’ll need to simulate the two disks. They’ll each have an x, y position, a radius, a speed and a phase. We’ll make these visible to start with so you can get a feeling for what’s going on. Otherwise it would just be a long complicated formula.

We’ll create two circles and rotate them and show the point where the arms will be attached.

Each disk will be represented by this kind of structure:

disk: {
  x,
  y,
  radius,
  speed,
  phase,
}

Like last time, it doesn’t matter if this is a generic object, a struct, a class or what. I’m going to assume that you’ll also have a function that will create one of these disks like this:

d0 = disk(100, 200, 100, 2, 0.5)
d1 = disk(400, 200, 60, 3, 0.0)

Now we can set up an animation like so:

width = 600
height = 400
canvas = (width, height)
t = 0

d0 = disk(100, 200, 100, 2, 0.5)
d1 = disk(400, 200, 60, 3, 0.0)

function loop() {
  clearScreen()

  circle(d0.x, d0.y, d0.radius)
  stroke()
  x0 = circle.d0.x + cos(t * d0.speed + d0.phase) * d0.radius
  y0 = circle.d0.y + cos(t * d0.speed + d0.phase) * d0.radius
  circle(x0, y0, 4)
  fill()

  circle(d0.x, d0.y, d0.radius)
  stroke()
  x1 = circle.d1.x + cos(t * d1.speed + d1.phase) * d1.radius
  y1 = circle.d1.y + cos(t * d1.speed + d1.phase) * d1.radius
  circle(x1, y1, 4)
  fill()

  t += 0.1
}

Again, loop is a theoritical function that will be fun over and over again so that we get an animation. I’ve set mine up to create the frames for an animated gif, but this can be done as a real time animation as well. Here’s what I got:

This doesn’t smoothly loop, but that’s ok. You can see that we have two disks of different sizes moving at different speeds. The code itself shouldn’t be too complex. We clear the screen and for each disk we draw a circle at its position and with its radius. Then we take an offset from that circle to calculate the point where the arm will be attached. This uses basic trig: x = cos(angle) * radius, y = sin(angle) * radius. Here, the angle is t times the disk’s speed, plus it’s phase. When we get that point, we fill a smaller circle there.

Attaching the Arms and Finding the Pen

OK. Now things get a bit more mathy. We’re going to have two arms. each one is going to be attached to one of those spinning points at one and, and they’ll be attached to each other on the other end. At this point, a sketch is in order:

We have our two disks. Through their radii and rotation and position, we know the positions of points p0 and p1. That’s what we just did above. We can also define the lengths of those arms. They’ll be the same for now, but they could be different. We’ll call them a0 and a1. (I know, it looks like 90 and 91 in the sketch. Sorry!) What we need to do is get the position of that top point where the two arms join.

We can easily get the distance between p0 and p1 with the Pythagorean theorem. We can call that d. It’s represented by the dotted line in the diagram.

Now we have a triangle whose sides are a0, a1 and d.

There’s a trigonometric rule called the “law of cosines” that will help us here. With it, if you you know the lengths of all three sides of a triangle, a, b, and c, you can get any angle of that triangle. Usually the way it’s written is as follows though:

c = sqrt(a*a + b*b - 2*a*b*cos(y))

… where y is the angle opposite side c. So if you know the length of two sides and the angle between them, you can find the length of the opposite side. Also useful in some cases, but not what we need here.

But we can rearrange that formula and put the unknown variable by itself on one side, and the other side will be the formula you need to calculate that value.

What I want is to know is one of those angles so I can use it to find the location of the pen. Here’s what I came up with, showing the formula, some hand-wavey algebra and the resulting formula we’ll need.

In our case, the sides we have correspond to the sides in the formula like so:

a = a0
b = d
c = a1

So if we apply this formula, we’ll get the angle between the p1, p0, and the pen.

We can also use atan2 to get the angle from p0 to p1. And if we subtract them, we get the actual angle that goes from p0 to the pen.

Again, a drawing is in order:

The big angle that goes from p1 to p0 to the pen, which we get with the law of cosines, we call p1_p0_pen. The smaller angle we get with atan2, we call p0toP1. Subtract them and we have the angle we have to go towards to get the location of the pen. I’m sure there are different ways to do this, and probably much better ones, but this one works. Once you have all the steps, you can further simplify it to something more concise, but I wanted to show the steps to hopefully have it make some sense.

Anyway, we can now code this up:

width = 600
height = 600
canvas = (width, height)
t = 0

d0 = disk(150, 450, 100, 2, 0.5)
d1 = disk(450, 450, 60, 3, 0.0)

function loop() {
  clearScreen()

  circle(d0.x, d0.y, d0.radius)
  stroke()
  x0 = d0.x + cos(t * d0.speed + d0.phase) * d0.radius
  y0 = d0.y + sin(t * d0.speed + d0.phase) * d0.radius
  circle(x0, y0, 4)
  fill()

  circle(d0.x, d0.y, d0.radius)
  stroke()
  x1 = d1.x + cos(t * d1.speed + d1.phase) * d1.radius
  y1 = d1.y + sin(t * d1.speed + d1.phase) * d1.radius
  circle(x1, y1, 4)
  fill()

  // the length of the arms
  a0 = 350
  a1 = 350

  // get the distance between p0 and p1
  dx = x1 - x0
  dy = y1 - y0
  d = sqrt(dx * dx + dy * dy)

  // find the two key angles and subtract them
  p1_p0_pen = acos((a1 * a1 - a0 * a0 - d * d) / (-2 * a0 * d))
  p0toP1 = atan2(y1 - y0, x1 - x0)
  angle = p0toP1 - p1_p0_pen

  // find the pen point
  pX = x0 + cos(angle) * a0
  pY = y0 + sin(angle) * a0
  
  // draw the arms
  moveTo(x0, y0)
  lineTo(pX, pY)
  lineTo(x1, y1)
  stroke()

  t += 0.1
}

With that, you should be able to render something like what you see below.

Again, this is a non-cleanly looping gif, but shows the idea at work. I’ve commented the code to explain what’s going on in each step. Hopefully that helps. Note that I also changed the size of the canvas and moved the disks towards the bottom to make room for the arms.

One thing to be careful of is making sure that the two arms are long enough so that they can always reach between the two connection points on the disks. In the real world, if they were too short, they’t probably just break or jam up the motors. In code, you’ll probably just get a NaN (not a number) error when you go to do the acos and you’ll be sitting there wondering what’s wrong. I speak from experience.

Drawing the Curve

Finally, let’s see what this thing draws. For this, I’m going to abandon the animation and stop drawing the circles and arms. I’ll just track where the pen is on each iteration and use that to draw a long, looping, Lissajous-ish curve.

width = 800
height = 600
canvas = (width, height)

function render() {
  t = 0
  
  d0 = disk(250, 550, 141, 2.741, 0.5)
  d1 = disk(650, 550, 190 0.793, 0.0)

  // the length of the arms
  a0 = 400
  a1 = 400
  
  for (i = 0; i < 50000; i++) {
    x0 = d0.x + cos(t * d0.speed + d0.phase) * d0.radius
    y0 = d0.y + sin(t * d0.speed + d0.phase) * d0.radius

    x1 = d1.x + cos(t * d1.speed + d1.phase) * d1.radius
    y1 = d1.y + sin(t * d1.speed + d1.phase) * d1.radius

    // get the distance between p0 and p1
    dx = x1 - x0
    dy = y1 - y0
    d = sqrt(dx * dx + dy * dy)

    // find the two key angles and subtract them
    p1_p0_pen = acos((a1 * a1 - a0 * a0 - d * d) / (-2 * a0 * d))
    p0toP1 = atan2(y1 - y0, x1 - x0)
    angle = p0toP1 - p1_p0_pen

    // find the pen point
    pX = x0 + cos(angle) * a0
    pY = y0 + sin(angle) * a0
  
    lineTo(pX, pY)
    t += 0.01
  }
  stroke()
}

I’ve left lots of room for you to optimize this, so go for it! Even in its rough form though, this code should draw something like this:

There’s lots of things to play around and experiment with here. But the truth is that this is a pretty simple pintograph, so mostly the shapes are going roughly look like what you see above. But you can build on these principles and make all kinds of more complex devices. This site has several demos to inspire you:

https://michaldudak.github.io/pintograph/demo/

Disks on disks, and rotary pintographs and 3-wheel pintographs, etc.

And if you’re into this stuff, just search “harmonograph”, “pintograph” and “drawing machines” on Youtube to get an endless supply of inspiration – either for coding or actually building. Some of the more interesting ones (in my mind) are the ones that draw the curves on a piece of paper that itself is slowing rotating on a turntable.

Summary

This wraps up our discussion of Lissajous curves and the simulation of physical drawing machines. At least for a while. Next time we’ll go back a bit more to basic, standard geometric curves.

Chapter 5 of the Coding Curves Series

This installment builds on Chapter 4’s discussion of Lissajous curves. Actually, a harmonograph is not a type of curve, it’s a device used to draw Lissajous (and similar) curves. And when I say a device, I mean a real world physical device that has ropes or chains and levers and pen and paper or bottles of sand and pendulums or other mechanics to create these curves.

Real Harmonographs

The first time I saw a harmonograph was at the Museum of Science in Boston on one of my many childhood trips there. It was a pendulum with a container of sand that leaked out and created a trail. The video below is not the exact one I was familiar with, but is essentially the same thing. I didn’t know it was called a harmonograph until many years later.

The video is worth watching and discusses Lissajous patterns in depth. With a pendulum, the time it takes to go back and forth is called its period, but it’s what we called the frequency in the last installment of this series. At around 4:12 in the video, the presenter explains how this pendulum can have two different periods – one on each axis. This is why the patterns it forms look like Lissajous curves – because they are! If each axis had the same period, it would just create circles and ovals and spirals. Still technically Lissajous curves, but not as interesting.

Here’s another version using a pen and paper:

In this case, the pen is stationary and it’s the paper that moves around on a pendulum. But it accomplishes the same thing.

Here’s another video of a similar setup, which appears to be made of cardboard, string and tape!

There is a key difference between a pure Lissajous curve and a mechanical, pendulum-based harmonograph though. The pendulum slowly loses energy and the distance it travels on each pass gets smaller and smaller. Eventually it will stop swinging all together and sit stationary in the middle of the drawing.

While this type of harmonograph will produce some interesting drawings, you can make it even more complex by using a double pendulum. A common way to do this is to have the paper or drawing surface moving on one pendulum, and the pen moving on another one.

Here’s one example of this:

Here, both the paper and pen are mounted on top of the pendulums, which are weights swinging below the table. So they move independently and are able to produce more complex curves.

And here’s yet another video of a very fancy double-pendulum, showing some of the amazing drawings it can create. Quite a couple of characters here too.

Simulated Harmonographs

As this is more of a programming focused site, I’m not going to explain how to build a physical harmonograph. But, these devices operate on the principles of physics. And the formulas that control them are known. We can use these formulas to create a virtual harmonograph.

We’ll start with a simple, single pendulum version. But first, let’s revisit our Lissajous curve formula:

x = A * sin(a * t + d)
y = B * sin(b * t)

A and B are the amplitude of the wave on each axis and a and b are the frequencies. And d is the delta, which puts x out of phase with y. And of course t is the parametric time variable. Initially we said that t would go from 0 to 2 * PI, but later we saw how it could increase infinitely.

To start to move towards simulating a harmonograph, we’ll recognize that each axis will have its own phase, rather than thinking one is phased and the other … unphased? So we’ll change the single d to p1 and p2.

x = A * sin(a * t + p1)
y = B * sin(b * t + p2)

This still gives us a Lissajous curve, but with a bit more complex definition. To fully move it to a simulated harmonograph, we’ll need to simulate that loss of energy, or damping. To do this fairly accurately, this will be an additional multiplier that looks like this:

e-d*t

… or “e to the power of minus d times t”

In code, that might be:

pow(e, -d * t)

Where pow is the power function, available in any fine math library.

So what is all this? We have two new variables here: e and d. Actually e is a constant, aka Euler’s number, equal to approximately 2.71828. I’ll let you read up on that on your own, but e is used in all kinds of real world physical formulas. Including, it seems, the damping of pendulums.

By now you might have guessed that d is for the damping factor. We’ll set d to a pretty small number, something like 0.002 is good to start with. Now when t is 0, that will make the exponent 0 and the result of the power calculation will be 1.0.

As t increases, by say 0.01 on each iteration, the value of the exponent will slowly grow negatively. When t is 0.01, the exponent will be -0.00002, and the result of the whole damping equation will be 0.9999800002.

After 100 iterations, t will hit 1.0 and the damping factor will be 0.9980019987. After 1000 iterations, it will be 0.9801986733. So you can see this goes down very slowly. If you increase d, it adds more damping and that number will go towards 0.0 faster. Here’s how we work it into the harmonograph equation:

x = A * sin(a * t + p1) * pow(e, -d1 * t)
y = B * sin(b * t + p2) * pow(e, -d2 * t)

Notice I made it d1 and d2, so you can have separate amplitudes, frequencies, phases and damping for each axis.

To bring it all home, as t increases, that last part of the equation moves closer and closer to 0.0, meaning both x and y will get smaller and smaller and will eventually be 0.0 themselves, simulating the pendulum running down and stopping. The higher d1 and d2 are, the quicker that will happen.

Depending on the math library you use, there may be a shortcut here. Since taking e to some power is a pretty common operation, there’s usually a built in function for doing just that, often called exp. For example in JavaScript, you could say Math.exp(-d1 * t), which would be exactly the same thing as Math.pow(Math.E, -d1 * t), but a bit shorter, and potentially more optimized.

Thus we can change the pseudocode to:

x = A * sin(a * t + p1) * exp(-d1 * t)
y = B * sin(b * t + p2) * exp(-d2 * t)

The Function

Let’s make something happen! Here’s our function:

function harmonograph(cx, cy, A, B, a, b, p1, p2, d1, d2, iter) {
  res = 0.01
  t = 0.0
  for (i = 0; t < iter; i += res) {
    x = cx + sin(a * t + p1) * A * exp(-d1 * t)
    y = cy + sin(b * t + p2) * B * exp(-d2 * t)
    lineTo(x, y)
    t += res
  }
  stroke()
}

Whole lotta parameters goin’ on. But you should understand most of this by now. The one I’ll mention is iter. Earlier we were mostly looping from 0 to 2 * PI. Now we want to loop a whole lot more than that, as the curve is going to continue to change as it is dampened and the amount of motion decreases. We’ll use a very high number for iter which simulates the harmonograph running for a long time. Real world harmonographs can take five minutes or longer to complete a single drawing.

Here’s an example of the function in use:

width = 800
height = 800
canvas(width, height)

A = 390
B = 390
a = 2.0
b = 2.01
p1 = 0.3
p2 = 1.7
d1 = 0.001
d2 = 0.001
iter = 100000

harmonograph(width / 2, height / 2, A, B, a, b, p1, p2, d1, d2, iter)

Yeah… 100,000 – a hundred thousand iterations. Might take a second or two. But you should get something like:

Here are some others I came up with by trying random parameters:

I’ve found that it’s best to keep a and b close to whole numbers, and let them vary by a very small amount, like in the first example where they were 2.0 and 2.01. It also works pretty well if the numbers are easily broken down into a relatively simple ratio, like 7.5 and 2.5 which is 3:1. And again, if you add a small amount to one of them it becomes a little more interesting, like 7.5 and 2.501. But totally random numbers like 5.7 and 3.2 will make for rather chaotic drawings.

The d values change how fast the pendulum is dampened, so a very low value will tend to draw more lines away from the center. Here’s the first example with damping on both axes at 0.0003:

And the same with damping of 0.003:

The pendulum decayed more quickly and you get more lines towards the center.

You can play with this endlessly. Try adding color too!

Double Pendulums

The drawings produced in that last video look pretty compelling. To do that, we need to come up with a double pendulum harmonograph simulation. You can consider that we have an x, y pendulum for the pen, and an x, y pendulum for the paper. Both will move independently, and wind up creating much more complex curves. You just need to calculate both x-axis pendulums and add them together for the final x, and the same on the y-axis.

Although the concept is relatively straightforward, this means we’ll be doubling the number of parameters we need.Four each for amplitude, frequency, phase and damping. If we do this naively, we might come up with something like this, which is really tough to manage.

// don't code this!!!
function harmonograph2(cx, cy, a1, a2, a3, a4, f1, f2, f3, f4, p1, p2, p3, p4, d1, d2, d3, d4, iter) {
  res = 0.01
  t = 0.0
  for (i = 0; t < iter; i += res) {
    x = cx + sin(f1 * t + p1) * a1 * exp(-d1 * t) + sin(f2 * t + p2) * a2 * exp(-d2 * t)
    y = cy + sin(f3 * t + p3) * a3 * exp(-d3 * t)+ sin(f4 * t + p4) * a4 * exp(-d4 * t)
    lineTo(x, y)
    t += res
  }
  stroke()
}

I’ve tried this and totally confused myself multiple times trying to remember which frequency value controlled which axis of which pendulum, etc. A better idea (maybe not the best, but better) would be to encapsulate the parameters needed for a single axis pendulum (amplitude, frequency, phase and damping) into a single value object, and pass four of those into the function.

I don’t know if the platform you’re using has classes or structs or plain old generic objects, so I’m just going to say that we have some kind of object with four properties:

pendulum: {
  amp,
  freq,
  phase,
  damp,
}

Don’t worry about the syntax here. Use whatever you need to use that will create such an object.

Now we can create four of these, maybe call them penX, penY, paperX, paperY. Like this:

penX = pendulum(90.0, 7.5, 1.57, 0.0001)
penY = pendulum(90.0, 4.0, 0.0, 0.0001)
paperX = pendulum(280.0, 1.001, 1.57, 0.0001)
paperY = pendulum(280.0, 2.0, 0.0, 0.0001)

Again, don’t get caught up in the syntax. This could be a factory function, a constructor, or you could just create some kind of object literal with those values – amplitude, frequency, phase and damp – for each pendulum.

Now we can change the harmonograph2 function to look like this:

function harmonograph2(cx, cy, penX, penY, papX, papY, iter) {
  res = 0.01
  t = 0.0
  for (i = 0; t < iter; i += res) {

    x = cx
      + sin(penX.freq * t + penX.phase) * penX.amp * exp(-penX.damp * t)
      + sin(papX.freq * t + papX.phase) * papX.amp * exp(-papX.damp * t)

    y = cy 
      + sin(penY.freq * t + penY.phase) * penY.amp * exp(-penY.damp * t)
      + sin(papY.freq * t + papY.phase) * papY.amp * exp(-papY.damp * t)

    lineTo(x, y)
    t += res
  }
  stroke()
}

There’s still a lot of code duplication there, but at least the method signature is better. I’ve done what I could to make it as readable as possible. There’s probably more you can do to make this whole set up easier, but that’s often the fun part of programming something complex – taking a somewhat janky proof of concept and turning it into an elegantly coded application. I don’t want to take that away from you, so I’ll leave plenty of room for improvement.

Anyway, now you can take those pendulum values we created and feed them into the function like so:

penX = pendulum(90.0, 7.5, 1.57, 0.0001)
penY = pendulum(90.0, 4.0, 0.0, 0.0001)
paperX = pendulum(280.0, 1.001, 1.57, 0.0001)
paperY = pendulum(280.0, 2.0, 0.0, 0.0001)

harmonograph2(width/2, height/2, penX, penY, paperX, paperY, 100000)

And if you’ve done everything right (and if I wrote this all up correctly), you should get an image like this:

penX = pendulum(50.0, 17.5, 1.57, 0.0001)
penY = pendulum(50.0, 11.0, 0.5, 0.0001)
paperX = pendulum(280.0, 0.50, 1.57, 0.0007)
paperY = pendulum(280.0, 1.50, 0.0, 0.0007)

Anyway, play with that for a while. There is an infinity of shapes you can create.

Animation

So far, we’ve just created static images here, but this is ripe for animation. You can do the obvious and show the curve building up over time just as if a real harmonograph were drawing it. I’ll skip over that one as it’s pretty easy and from my viewpoint not all that interesting.

What’s more fun is to animate some of the other properties. The phases are good candidates. Here’s an example where one of the phase values just moves from 0 to 2 * PI. It almost looks three-dimensional.

And here, some of the damp values are going back and forth between 0.001 and 0.0001.

Summary

So that’s harmonographs. Code it up and have some fun with it. There’s an endless ways you can tweak this to create interesting shapes. Heck, you might even decide to go buy some hardware and make a physical harmonograph. I’d love to see it if you do.

Next chapter we’ll be looking at yet another physical device and simulating that.

Chapter 4 in the Coding Curves Series

Make sure you are familiar with at least the first chapter, to understand how the code samples work.

Lissajous curves have always been one of my favorite techniques. They are useful for many things beyond the obvious loopy shapes they create. In this installment, we’ll cover the basics, and as usual, wander off course here and there to look at other ways they can be used.

The Basics

Lissajous curves are also known as Lissajous figures or Bowditch curves. Both names came from the names of men who looked into and wrote about them in the 1800s.

These figures are formed by long looping curves that go back and forth and left and right. In their pure form, they recall the glowing patterns seen on oscilloscopes used for special effects in a vintage scifi movies.

The Formula

In this case, a useful parametric formula was one of the first things I found in researching this subject.

x = A * sin(a * t + d)
y = B * sin(b * t)

Basically what we have here is a sine wave on the x-axis, and another sine wave on the y-axis. So instead of going off infinitely in any direction, it keeps looping back in on itself.

We have a bunch of variables here. Let’s break them down.

A and B wind up being the width and height of the curve on the x- and y-axes. Or technically, half the width and height because the curve will extend that distance in each direction.

The t is a parametric variable that will range form 0 to 2 * PI. Although it can actually go beyond that in either direction, it will effectively just loop back on itself when it does. t is multiplied by a for the x component, and by b for the y component, and the sine is taken of the result on each axis. In addition, the x component has a d or delta variable added to it to move it out of phase.

This formula might look a bit familiar from the previous chapter on circles. If we set A and B equal to each other and call them r, and set a and b both equal to 1, then remove the d and use cosine instead of sine for the x component, then we get:

x = r * cos(t)
y = r * sin(t)

This is the parametric formula for a circle (with its center at 0, 0). Since cosine is the same as sine but 90 degrees out of phase, we could also say:

x = r * sin(t + d)
y = r * sin(t)

… where d equals 90 degrees, or PI / 2 radians.

You can also see the relation to the ellipse formula if you keep A and B separate, where A is what we called the “x radius” and B is the “y radius”:

x = A * sin(t + d)
y = B * sin(t)

So when everything is in sync like this, we get a circle or ellipse, but when we start changing these parameters, we get much more interesting curves.

OK, enough talk, let’s code. We can jump right into making a Lissajous function. I’ll abbreviate a bit.

function liss(cx, cy, A, B, a, b, d) {
  res = 0.01
  for (t = 0; t < 2 * PI; t += res) {
    x = cx + sin(a * t + d) * A
    y = cy + sin(b * t) * B
    lineTo(x, y)
  }
  closePath()
}

Here, the resolution value is not as straightforward as in circles and ellipses. For now, I’m just going to keep the res value very small, even if it means we’re doing to much work for simpler, smaller curves. Since we’re going from 0 to 2 * PI (6.28…), incrementing by 0.01 will give us 628 line segments, which should be enough for most cases. If it starts getting blocky, you can increase it, but I’m not going to go into how to best predict it.

Now we can use this in a sketch like this:

width = 600
height = 600
canvas(600, 600)

liss(300, 300, 250, 250, 1, 1, PI / 2)
stroke()

Here, I did what I described above, setting A and B equal to each other, a and b to 1, and d to PI / 2. This should give us a circles, and in fact…

… it does.

Let’s set d to 0 and mess with a and b for a while. Here, a and b are 2 and 1 (again, with d at 0):

And now they are 2 and 3:

Let’s crank them up to 11 and 8.

In all these cases, with d at 0, the waves on each axis are in phase with each other. Here, I kept 11 and 8 and set d to 0.5, moving them out of phase:

Here’s an animation with a at 6, b at 7, and d varying.

It’s worth noting that a and b should be whole numbers if you want the curve to join its start and end points together smoothly. Here’s what happens if you set a to 6.4 and b to 7.3:

You can see even more clearly what is happening, if you remove the closePath call from the function:

Now you can see the curve starts and ends in random locations, rather than joining up smoothly as it does when a and b are whole numbers.

Of course, when you use fractional numbers here, you can now extend your t way past the 0 to 2 * PI range and continue to fill the space. Here, t goes from 0 to all the way up to 20 * PI:

The paths will eventually join up again if you use rational numbers for a and b. But it might take a while. Here it looks like they came pretty close.

We can of course, change A and B to get wide figures:

or tall figures:

That’s about it in terms of drawing these figures directly. Check out the Wikipedia article on the subject to learn more about the various properties of this type of curve. I was thinking about creating an interactive demo here, but it turns out there are a bunch of them on line already. A few are linked near the bottom of the Wikipedia article I just mentioned.

Instead we can look at some other applications here.

Animation

Here, I’m not talking about animating the curve itself, like I did above, but using the Lissajous curve results to animate an object. But first let’s talk about a simpler level of animation.

Oscillation

Often you want to animate an object but have it remain on screen. You just want it to go back and forth or up and down, or twist one way or the other or even grow and shrink. You might recognize that these are some of the more basic 2D transformations – translation, rotation and scaling. You can use a sine or cosine function to generate values for these transformations, and just change the input to that function over time.

We’ll have to expand our pseudocode to include a loop function that runs repeatedly forever and can draw something differently each time it’s run to create an animation. Processing has something like this built in, as may other graphics systems. But you might have to put something together yourself. If you’re using HTML and JavaScript you can use requestAnimationFrame.

And we’ll postulate into existence a clearCanvas function too. Why not? We’ll also include a circle function that can be the one we created in the last installment, or one that comes with your drawing api. We’ll start something like this:

width = 400
height = 400
t = 0
canvas(width, height)

function loop() {
  clearScreen()
  circle(width / 2 + sin(t) * 100, height / 2, 20)
  fill()
  t += 0.1
}

Which should give you something close to this:

This takes the sine of t and multiplies that by 100 to get an offset position for the circle on the x-axis. As t is continually incremented, the circle keeps going back and forth. You could easily do the same thing on the y axis, or you could apply the sine to the radius:

width = 400
height = 400
t = 0
canvas(width, height)

function loop() {
  clearScreen()
  circle(width / 2, height / 2, 50 + sin(t) * 20)
  fill()
  t += 0.1
}

Here, we’re taking 50 as the base radius, and using the sine of t times 20 to that. Because sine will go from -1 to +1, we’ll be adding -20 up to +20 to the radius, making it oscillate from 30 to 70.

We could do the same thing with rotation, but then we’d have to switch to something other than a circle so we could actually perceive it rotating. I’ll leave that to you to try.

But going back to moving the circle around. Say we want to have it move all around the canvas. We could of course use sine and cosine to make it move in a circle:

width = 400
height = 400
t = 0
canvas(width, height)

function loop() {
  clearScreen()
  circle(width / 2 + cos(t) * 100, height / 2 + sin(t) * 100, 20)
  fill()
  t += 0.1
}

But what if we wanted something more organic? We could start adding random motion, but then you run into the problem of how to constrain it so it doesn’t wander off screen. Not insurmountable, but we can do a pretty good trick using the Lissajous formula, like so:

width = 400
height = 400
t = 0
a = 13
b = 11
canvas(width, height)

function loop() {
  clearScreen()
  circle(width / 2 + sin(a * t) * 100, height / 2 + sin(b * t) * 100, 20)
  fill()
  t += 0.02
}

This is an imperfectly looping gif, but you get the idea. The circle just seems like it’s randomly moving around the screen, but it’s actually following the path of a Lissajous curve. To me it looks like a fly buzzing around. In fact, if you add several of these with different parameters, it gives you a pretty convincing swarm of flies. High numbers for a and b, that don’t have common denominators, make the most random looking path.

Lissajous Webs and Random Lissajous Webs

This isn’t anything remotely “official” that you’ll find documented anywhere (well, maybe one place I’ll mention below), but I think it’s always good to cross-pollinate different ideas. You often come up with something very unique. So I’m just throwing these out there as something I came up with.

Some years ago I had been playing about with drawing Lissajous curves and was thinking about different ways of rendering them. Rather than just rendering the paths themselves, I decided to try capturing each point that made up the curve and then connecting nearby points with lines. The results were pretty cool, and I dubbed them “Lissajous Webs”. Here’s a few examples:

You can find a few more examples here: http://www.artfromcode.com/?p=657

I thought these were pretty damn cool looking. The technique was included in the book Generative Design (with full credit to and permission by me).

As I continued to play around with the idea, I wanted to make them even more organic. Rather than fixed a and b multipliers, I started letting these parameters wander a little bit, randomly. These wound up creating some really amazing images, which I called “Random Lissajous Webs”.

This concept became one of my more popular pieces. I was even commissioned some years later to do a series of designs using this technique that got used on wine bottles.

You can see more pictures of those here: https://www.anarchistwineco.com/ (and buy the wine!)

I’m not going to go through the code for all of these. This is more just for inspiration – how the concept of Lissajous curves led to something entirely different and very interesting.

The next chapter will look at a couple of related systems that are also very interesting ways of creating curves.

Chapter 3 in the Coding Curves Series

In this installment we’ll look at how to draw arcs, circles and ellipses. (And wander off on some tangents before we get done.)

It’s likely that your platform’s drawing api has at least some of this built in. For example, the HTML Canvas api does not have a circle method, but it does have an arc method as well as an ellipse method, either of which can be used to draw circles.

But it’s good to know how to do all of these manually. You’ll wind up using it someday, somewhere on some platform.

First let’s just look at arcs and circles. You can say that an arc is just part of a circle, or you could say that a circle is an arc that extends 360 degrees. So we could go at this either direction, but it makes sense to me to start with circles and specialize into arcs.

Note on Measurements

The drawing api I am using has the y-axis reversed from standard Cartesian coordinates. Negative values go up, positive down. This is very common for drawing apis which aren’t specifically made for math or science use cases. It’s the same with Processing, HTML canvas, Cairographics, .net graphics, and many others. Some apis do use Cartesian coordinates and have positive angles going counter-clockwise. Yet others, such as pygame, mix the systems – y values go down positively, but positive angles go couter-clockwise.

This affects the measurements of angles. Zero degrees is due east. In a Cartesian system, positive angles will move in a counter-clockwise direction and negative angles in a clockwise direction. In reversed y-axis systems, the opposite is true. In this chapter, I don’t make any attempt to “correct” this. The functions we’ll be creating here will mirror the built-in functions of many drawing apis.

But know that if you do want to create circle, arc and ellipse functions that operate per the Cartesian coordinates, that’s easy enough to do. And when you finish the section on arc, you’ll know how to do that.

Circles

Definition

A circle is usually defined something like “the set of points that are equidistant from a given center point”. But when you are trying to draw a specific circle, that’s not very useful. I don’t need a set of infinite points. I just need enough points to draw short line segments through that will form a circle.

You’ll also see the “equation of a circle” as x2 + y2 = r2. Also pretty useless from the perspective of trying to draw one.

But then you get to the parametric form:

x = a + r * cos(t)
y = b + r * sin(t)

Here, a, b is the center of the circle, r is the radius, and t is a parametric variable that ranges from 0 to 2 * PI.

This starts to get useful. We can define the center point and the radius and then run a for loop from 0 to 2 * PI to get a set of points that we can draw lines through.

Eternal reminder. All the code here is pseudocode. See the first post in this series for more info.

width = 600
height = 600
canvas(width, height)

cx = width / 2
cy = height / 2
radius = 250
for (t = 0; t < PI * 2; t += 0.01) {
  lineTo(cx + cos(t) * radius, cy + sin(t) * radius)
}
closePath()
stroke()

Depending on your drawing api, you might need to start off with a moveTo before the lineTos. If so, I trust you’ll figure that out. Otherwise, this is straightforward. cx, cy and radius are a, b and r from the above formula. And t is the angle we loop through.

Note the final closePath call there. That’s a feature of most drawing apis. It will draw a final line segment from where the last operation left off to the place where the current path started, closing the circle. It might be different on your platform but there should be something there.

This gives us…

One question is the 0.01 value in the for loop. At this point, this was just a rough guess. If you make it too big, like 0.2, then you’re going to be jumping around the circle in large jumps and it’s not going to look quite as good:

But if you make the increment too small, then you’re doing a lot of extra work for nothing. The larger the radius, the larger the circumference, and the more segments you need to use to make it smooth. The smaller the radius, the fewer the segments you need.

If you’re going with a constant 0.01 increment, you’re drawing 628 segments for each circle. This is way too many for a small circle.

By shear trial and error, I’ve found that a workable resolution seems to be about 4.0/radius. This looks good on circles down to a radius of 5. And on a radius of 200 draws half as many line segments as an increment of 0.01 and looks just as good. You can do some experiments on your own and see what looks good to you as it may vary by system.

A Function

With this, we can make a circle drawing function like so:

function circle(x, y, r) {
  res = 4 / r
  for (t = 0; t < PI * 2; t += res) {
    lineTo(x + cos(t) * r, y + sin(t) *r)
  }
  closePath()
}

Note that I left off the stroke in the function. This way you can create the circle with this function and then choose to stroke it, fill it or do both. If you want, you can make a strokeCircle function and a fillCircle function. Here’s how you’d use this:

width = 600
height = 600
canvas(width, height)

circle(width / 2, height / 2, 200)
stroke()

Arcs

Now that we have that down, we can build on this to create an arc function. Again, your api may already have this, but lets’ do it anyway.

This’ll be pretty easy. It’s the same as the circle function, but instead of starting at 0 and ending at 2 * PI, we let the user say what angles they want to start and end at.

This is all pretty straightforward, I’ll just throw the code out here without any pre-explanation:

function arc(x, y, r, start, end) {
  res = 4 / r
  for (t = start; t < end; t += res) {
    lineTo(x + cos(t) * r, y + sin(t) *r)
  }
  lineTo(x + cos(end) * r, y + sin(end) *r)
}

Told you it was simple. Instead of hard-coding the start and end angles to 0 and 2 * PI, we make them parameters. Also, I removed the call to closePath and replaced it with a final lineTo that draws a last line to the end angle, just to be precise.

To use it:

width = 600
height = 600
canvas(width, height)

arc(width / 2, height / 2, 250, 0.5, 3.5)
stroke()

And that gives you:

But there are a couple of problems. What if I entered the start and end angles in the opposite order?

arc(width / 2, height / 2, 250, 3.5, 0.5)

This will jump out of the for loop right away because 3.5 is already greater than 0.5. Nothing gets drawn. What I probably wanted was to start at the angle of 3.5 and go around until I crossed the start of the circle and hit 0.5 again, like this:

One way to handle this is just to make sure that the end angle is greater than the start angle. We can do that by checking if it’s smaller and then adding 2 * PI to it until it is bigger.

function arc(x, y, r, start, end) {
  while (end < start) {
    end += 2 * PI
  }
  res = 4 / r
  for (t = start; t < end; t += res) {
    lineTo(x + cos(t) * r, y + sin(t) *r)
  }
  lineTo(x + cos(e) * r, y + sin(e) *r)
}

Now this should work as expected and produce the image shown above. One more thing though. We’re always making the assumption that we’re drawing the arc clockwise. We should allow the user to make that decision. Luckily this is pretty simple. We’ll just add another parameter, anticlockwise. If this is true, we just need to swap start and end and we should be good.

function arc(x, y, r, start, end, anticlockwise) {
  if (anticlockwise) {
    start, end = end, start
  }
  while (end < start) {
    end += 2 * PI
  }
  res = 4 / r
  for (t = start; t < end; t += res) {
    lineTo(x + cos(t) * r, y + sin(t) *r)
  }
  lineTo(x + cos(e) * r, y + sin(e) *r)
}

If you’re lucky, your language will let you do the swap like this:

start, end = end, start

If not, you’ll have to go the old fashioned route:

temp = start
start = end
end = temp

Now this code:

arc(width / 2, height / 2, 250, 3.5, 0.5, false)
stroke()

will give you this arc:

And this code:

arc(width / 2, height / 2, 250, 3.5, 0.5, true)
stroke()

will give you this arc:

Both start at an angle of 3.5 and draw an arc to 0.5. One goes one way, the other goes the opposite way.

As mentioned at the beginning of the article, positives angles going clockwise is the default I chose here, unlike Cartesian coordinates. Now that you know how to draw arcs in either direction, you are free to make your default whichever way you want.

Now that we have a solid arc function, we can actually go back and remove some duplication from our circle function, changing it to this:

function circle(x, y, r) {
  arc(x, y, r, 0, 2 * PI, true)
}

This draws an arc from 0 to 2 * PI, which is a circle.

Segments and Sectors

There are a couple of other functions you can create if you find them useful. A segment is an arc that is joined by a line segment between its beginning and end (a chord). We can do this by drawing an arc and then just calling closePath or whatever does that on your system.

function segment(x, y, r, start, end, anticlockwise) {
  arc(x, y, r, start, end, anticlockwise)
  closePath()
}

Here’s a segment that goes from an angle of 2.5 to 4.5:

And a sector is an arc that is joined by line segments that go to the center of the circle. We can do that by executing a lineTo to the center point and then calling closePath

function sector(x, y, r, start, end, anticlockwise) {
  arc(x, y, r, start, end, anticlockwise)
  lineTo(x, y)
  closePath()
}

Here is a sector drawn with the same angles as the segment example:

Now you’re well on you way to making pie charts!

Polygons

Before we move on to ellipses, I want to give you one bonus function: regular polygons. This isn’t what I would normally think of as a curve, but mathematically, it might be. Anyway, it’s low hanging fruit, right there for the picking, so let’s do it.

When we were talking about resolution, we saw how a low resolution circle starts to look chunky. You can see the individual line segments that make it up. Well, we can push that bug even further and turn it into a feature.If we push the resolution so low that we only wind up drawing six segments in our circle (exactly six), we have a hexagon. Five create a pentagon, four a square and three a triangle. We just have to specify how many sides we want, and divide 2 * PI by that number to get the resolution that will make that shape.

Here’s one take:

function polygon(x, y, radius, sides) {
  res = PI * 2 / sides
  for (i = 0; i < PI * 2; i+= res) {
    lineTo(x + cos(i) * radius, y + sin(i) * radius)
  }
  closePath()
}

Now you can call it like:

polygon(300, 300, 250, 5)
stroke()

and get a pentagon like this:

You might want to specify an initial rotation, you can do that like so:

function polygon(x, y, radius, sides, rotation) {
  res = PI * 2 / sides
  for (i = 0; i < PI * 2; i+= res) {
    lineTo(x + cos(i + rotation) * radius, y + sin(i + rotation) * radius)
  }
  closePath()
}

Now you can say

polygon(300, 300, 250, 5, 0.5)
stroke()

and have the polygon rotated a bit.

Try it with different numbers of sides.

A fun effect is to create a series of polygons of different sizes, each slightly rotated.

angle = 0
for (r = 5; r <= 255; r += 10) {
  polygon(300, 300, r, 5, angle)
  stroke()
  angle += 0.05
}

This creates a nice pattern like so:

Might be a bit off-topic, but hey, there are five new emergent curves there! I’ll accept it.

Ellipses

Final bit of this installment, ellipses.

Well, let’s look at the definition of an ellipse, from Wikipedia…

a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant.

https://en.wikipedia.org/wiki/Ellipse

hmm… I get it, but doesn’t really help us to draw it. How about…

Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane

https://en.wikipedia.org/wiki/Ellipse

Nope. Let’s keep reading…

An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant.

https://en.wikipedia.org/wiki/Ellipse

OK, this is going nowhere. But as before, we can eventually find the parametric formula, which I’ve tweaked a bit to be similar to the one we had for the circle.

x = a + rx * cos(t)
y = b + ry * sin(t)

Here, in addition to the a and b that form the center position of the circle, we have rx and ry which I find easiest to think about as “radius x” and “radius y”, though these names will probably make mathematicians cringe. But for an un-rotated ellipse, rx will wind up being equal to half the ellipse’s width, and ry half its height.

So we can make a function:

function ellipse(x, y, rx, ry) {
  res = 4.0 / max(rx, ry)
  for (t = 0; t < 2 * PI; t += res) {
    lineTo(x + cos(t) * rx, y + sin(t) * ry)
  }
  closePath()
}

About the only thing worth mentioning here is that to get the resolution value, I divide 4.0 by the largest of the two “radii”. you might think of a better way, but this is good enough for me. Now you can call it like:

ellipse(300, 300, 250, 150)
stroke()

And get:

Bonus

Sometimes I can’t stop writing. This next part isn’t really so much about creating curves… or maybe it is. You decide. But rather than draw line segments between each point on a circle (or arc, or polygon, or ellipse), we could just draw some other shape there. We’ll have to increase the interval that we use to draw the curve so all the shapes don’t mash together. In fact, the polygon method is perfect for this. This lets us draw a circle with a set number of circles. I’m not even going to explain this code. You should get it.

width = 600
height = 600
canvas(width, height)

cx = width / 2
cy = height / 2

res = PI * 2 / 20 // to draw 20 circles
for (t = 0; t < PI * 2; t += res) {
  x = cx + cos(t) * 200
  y = cy + sin(t) * 200
  circle(x, y, 20)
  stroke()
}

Which gives us:

Summary

I’m already elaborating on this in my head, but we’re off-topic enough, and this installment is long enough.

So far things have been pretty basic, but hopefully still interesting. From here on, they will get a bit more complex and hopefully even more interesting.

Chapter 2 in the Coding Curves Series

This is going to be a relatively simple one, but we’ll get into a few different applications. I’m not going to take a deep dive into what trigonometry is, there will be no images of triangles with the little square in the corner telling you which angle is 90°. No definitions of adjacent, opposite, hypotenuse. And NO soh-cah-toa!!!

I’m going to assume you know all that stuff. And if you don’t, a google search for basics of trigonometry will net you about 18,800,000 results in under one second. Proof:

So if you have no idea what I’m talking about in that first paragraph, stop here and do some learning on that stuff first.

Wait, what am I doing? Totally ignoring an opportunity for shameless self-promotion on my own blog. OK, here’s a playlist of trigonometry videos by myself. The best material on the subject out of all of those 18 million results, or your money back.

Coding Math – Trigonometry Playlist

But we have to start somewhere…

OK, there are a few basic trig functions that your language should have somewhere in it. Maybe they are global, maybe part of a math library. For this post, we’ll be dealing with what you’ll probably find named sin, cos, and tan. These stand for sine, cosine and tangent, which you should know about or have just learned more about above.

Let’s start with sin. You pass it a number and it gives you a number back. If you give it 0.0, it returns 0.0. As you increase that argument incrementally, the result you get will slowly increase up to 1.0 – or almost 1.0 anyway. Then as you continue to increase it it will go back down to 0.0, then down to -1.0 and back up to 0.0. As you continue to increase the argument, the result will continue to oscillate between -1.0 and 1.0.

Try it out:

for (i = 0.0; i < 6.28; i += 0.1) {
  print(sin(i))
}

In your language, you might need println or some kind of console or log function to display values. But you should get some output that looks like:

As I predicted, it starts at 0, goes up to 0.99957… and then starts going down again. Your output should continue going down to something like -0.999923… and then back up to almost 0.0.

Note that it did exactly one cycle of 0 to 1 to -1 to 0. Not a coincidence. It’s because the for loop ends at 6.28, which is approximately 2 * PI. Or 2 * 3.14… In most programming languages, the trig functions work on radians rather than degrees. This is another point I’ll assume you have some understanding of. But a radian is approximately 57 degrees. And 3.14… (PI) radians is 180 degrees. 6.28… (2*PI) radians is 360 degrees.

So the result of sin will go from 0 to 1 to -1 to 0 exactly one time from 0 to 360 degrees, or 0 to 6.28… radians.

If you change the for loop to end at PI (most languages have that as a constant somewhere) and the numbers should go from 0.0 up to 1.0, back to 0.0 and stop there. That’s half a cycle.

Bring on the Curves!

OK, let’s draw a sine wave. Size your drawing area to 800×500. It’ll also be helpful to set variables (or constants) to the width and height values. Then have a for loop with a variable x going from 0 to 800, using that width variable. Then we’ll take the sine of x and that will be our y value. We’ll add that to half the height so our sine wave will run along the center of the image. Draw a line segment to x, y and we’re all set.

Remember, this is all psuedocode. You’ll have to convert this to the language and drawing api of your choice. More details in the series intro.

width = 800
height = 500
canvas(width, height)

for (x = 0; x < width; x++) {
  y = height / 2 + sin(x)
  lineTo(x, y)
}
stroke()

Depending on your drawing api, you might have to start with a moveTo(0, height/2) before you do the lineTos. When you’re done, you should have something like this image:

That’s a sine wave, but maybe not quite what you would expect. There are two problems, which is good, because they both lead into the next two things we need to talk about: wavelength/frequency and amplitude. Here, there are too many waves – the wavelength is too short (or the frequency is too high), and the waves themselves are mere blips – the amplitude is too low.

Amplitude

The amplitude is easy enough to handle. The sin function returns from -1.0 to 1.0, so we just need to multiply that number by an amplitude value and we’ll be good. The height times something like 0.45 will make the wave just smaller than the size of the canvas.

width = 800
height = 500
amplitude = height * 0.45
canvas(width, height)

for (x = 0; x < width; x++) {
  y = height / 2 + sin(x) * amplitude
  lineTo(x, y)
}
stroke()

That gives us height, but there are still too many waves so it’s hard to even comprehend this as a sine wave. We’ll fix that next.

Wavelength and Frequency

Wavelength is… wait for it… the length of a wave. Or the length between the same point on two consecutive waves. In the real world, these are actually physical distance measurements, with units anywhere from meters to nanometers, depending on what kind of wave it is you are measuring.

Another way of measuring waves is frequency – how many waves there are in a given interval (of space or time).

Wavelength and frequency are inversely related. A low wavelength (small distance between waves) equals a high frequency (many more waves in a given space). A high wavelength equals a low frequency.

You can write code to use either method of measurement. Let’s start with frequency.

Frequency

This is often specified as “cycles per second” (cps) rather than cycles in a given distance. Radio waves, for example, hit a receiver of some kind and we can just count how many hit it each second. The term “Hertz”, abbreviated “Hz” is the same as cycles per second. 100 Hz is 100 cycles per second. One kilohertz is 1000 cps, one megahertz is a million cps, etc.

Since we’re not dealing with moving waves though, it’ll be easier for us to measure frequency in terms of how many cycles occur in a given space. What you come up with here is up to your given application, but since we’re currently drawing a sine wave across an 800-pixel wide canvas, we can say that a frequency of one means that we should see one cycle occur as the wave goes across that canvas.

There’s a bit of math involved, so I’ll throw the pseudocode out there, then we’ll go through it.

width = 800
height = 500
amplitude = height * 0.45
freq = 1.0
canvas(width, height)

for (x = 0; x < width; x++) {
  y = height / 2 + sin(x / width * PI * 2 * freq) * amplitude
  lineTo(x, y)
}
stroke()

First we create a new variable, freq and set it to 1.0.

Then instead of just taking the sine of x, we divide x by width. This results in values from 0.0 to 1.0 as we move across the canvas.

Then we’ll multiply that by PI * 2. This will give us values from 0.0 to 6.28…, which should be familiar from the very first code example. This alone would make the results go from 0 to 1 to -1 and back to 0 exactly one time, as we saw. One cycle.

Finally, we multiply that by freq. That’s set to 1.0 now, so we’ll still get one cycle. With all this in place, you should be seeing this:

OK, that’s more like it. But you might be more used to the sine wave going up first, then down. If you’re seeing it go down and then up, like in my example, it’s because your drawing surface has the y-axis oriented such that positive values go down, the reverse of Cartesian coordinates. If you know how to use the 2d transform methods of your drawing api, you can fix it that way. I’ll go for a simpler fix of just subtracting y from height in the lineTo call.

width = 800
height = 500
amplitude = height * 0.45
freq = 1.0
canvas(width, height)

for (x = 0; x < width; x++) {
  y = height / 2 + sin(x / width * PI * 2 * freq) * amplitude
  lineTo(x, height - y)
}
stroke()

And now we have:

Now you can mess with those two variables, amplitude and freq to create different waves. Here I set amplitude to 50 and freq to 5:

Wavelength

Now let’s encode this to use wavelength instead of frequency. In this case we’ll be defining wavelength in terms of how many pixels it will take the wave to complete a full cycle. Let’s say we want one cycle to be 100 pixels long. Again, here’s the code, explanation to follow:

width = 800
height = 500
amplitude = height * 0.45
wavelength = 100
canvas(width, height)

for (x = 0; x < width; x++) {
  y = height / 2 + sin(x / wavelength * PI * 2) * amplitude
  lineTo(x, height - y)
}
stroke()

This time, inside the sin function call we divide x by wavelength. So for the first 100 pixels, we’ll get 0.0 to 1.0, then for the next 100 pixels 1.0 to 2.0, etc.

This gets multiplied by PI * 2, so for every 100 pixels we’ll get some multiple of 6.28… and will execute a complete wave cycle.

The result:

Since our canvas is 800px and the wavelength is 100px, we get eight full cycles, as expected.

Resolution

A quick word about resolution. Here, we are moving through the canvas in intervals of a single pixel per iteration, so our sine waves look pretty smooth. But it might be too much. If you are doing a lot of this and you want to limit the lines drawn, you can do so, with the possible loss of some resolution. We’ll just create a res variable and add that to x in the for loop and see what that does. We’ll start with a resolution of 10.

width = 800
height = 500
amplitude = height * 0.45
freq = 3
res = 10
canvas(width, height)

for (x = 0; x < width; x += res) {
  y = height / 2 + sin(x / width * PI * 2 * freq) * amplitude
  lineTo(x, height - y)
}
stroke()

On the plus side, we are drawing only 10% of the lines we were drawing before. But you can already see that things have gotten a bit chunky. Going up to 20 for res reduces the lines drawn by half again. But now things are looking rough.

But the important thing is knowing the options, their benefits and tradeoffs.

Cosine

I’m going to keep this one really quick. Because everything I’ve said about sine holds true for cosine, except that the whole cycle is shifted over a bit. Going back to a single cycle and simply swapping out cos for sin …

width = 800
height = 500
amplitude = height * 0.45
freq = 1.0
canvas(width, height)

for (x = 0; x < width; x++) {
  y = height / 2 + cos(x / width * PI * 2 * freq) * amplitude
  lineTo(x, height - y)
}
stroke()

Here, an input of 0.0 gives us 1.0. And then we drop to 0.0, -1.0, back through 0.0 and end on 1.0 as we move from inputs of 0.0 to 2 * PI. It’s the same wave shifted over 90 degrees (or PI / 2 radians). It’s easier to see if we move the frequency up a bit.

I don’t really have a whole lot to say about cosine waves for this post. But we’ll be revisiting them again later in the series.

A Function

We can use everything we’ve covered so far to make a reusable function that draws a sine wave between two points. Again, I’ll do the pseudocode drop and then the explanation.

function sinewave(x0, y0, x1, y1, freq, amp) {
	dx = x1 - x0
	dy = y1 - y0
	dist = sqrt(dx*dx + dy*dy)
	angle = atan2(dy, dx)
	push()
	translate(x0, y0)
	rotate(angle)
	for (x = 0.0; x < dist; x++) {
		y = sin(x / dist * freq * PI * 2) * amp
		lineTo(x, -y)
	}
	stroke()
	pop()
}

This function has six parameters, the x and y coords of the start and end points, frequency and amplitude.

We get the distance between the two points on each axis as dx and dy.

Then, using these, we calculate the distance between the two points using the Pythagorean theorem and the sqrt function which should be somewhere in your language.

And we get the angle between the two points using the atan2 function that should also be available. If you don’t understand what’s happening so far, I suggest you go look at the references at the beginning of the post.

Now this function is assuming that your drawing api has some 2d transformation methods. If so, it probably also has a way to push and pop transforms from a stack. In HTML’s Canvas api for example, these would be context.save() and context.restore().

We want to push the current transform to the stack so we can transform the canvas, do our drawing, and then restore the earlier state when we are done. So we call push.

Then we translate to the first point and rotate to the angle we just calculated. At this point we just need to draw a sine wave using dist, freq and amp exactly as we’ve been doing.

Since our canvas is translated exactly as we want it, we can just call lineTo(x, -y) which just corrects the sine wave like we did before.

When we’re done, we stroke that path and call pop (or restore or whatever your api uses) to leave things how we found them.

We can now use this function like so:

width = 800
height = 500
canvas(width, height)

sinewave(100, 100, 700, 400, 10, 40)

This draws a sine wave starting at 100, 100, going to 700, 400, with a frequency of 10 and amplitude of 40.

If your drawing api does not have transform methods, I pity you. You can still do this kind of thing, but it will be much more complex. And beyond the scope of this post.

As mentioned in the comments below, the “frequency” aspect of this method might be a bit confusing, because the way it’s coded “frequency” means “number of cycles between point A and B”. So a long wave drawn with this method would have a larger wavelength than another, shorter line drawn with the same frequency. To address this, you might want to change the function to use wavelength instead. This would ensure that all waves drawn with the same wavelength would look similar, no matter how long they were. Another option would be to adopt a frame of reference for frequency, like “so many cycles per 1000 pixels” or something. What you do here depends on your needs and application, so I’ll leave that up to you.

Tangent

Drawing tangent curves is not nearly as useful as sine and cosine waves, but we’ll cover it for completeness.

Unlike the two functions we’ve covered so far, tangent does not constrain itself to the range of -1 to 1. In fact, it’s range is infinite. Let’s do the same thing we did for sin and just trace out the values.

for (i = 0.0; i < 6.28; i += 0.1) {
  print(tan(i))
}

You’ll get something like this:

Again we start at 0, but quickly go up to 14, then jump to -34. But that’s deceiving. The values are truncated because we are moving in relatively large steps of 0.1. What is actually happening is that we are rapidly going up to positive infinity as the input angle approaches PI / 2 radians (or 90 degrees). Once it crosses that value, it jumps towards negative infinity, swiftly rises to 0.0 again at PI radians (180 degrees) and then repeats the cycle again once more before it gets to 2 * PI radians (360 degrees). This isn’t a linear progression though – it executes a curve as it approaches and leaves 0. Let’s code it up using frequency to see a single tangent wave cycle.

width = 800
height = 500
amplitude = 10
freq = 1.0
canvas(width, height)

for (x = 0; x < width; x++) {
  y = height / 2 + tan(x / width * PI * 2 * freq) * amplitude
  lineTo(x, height - y)
}
stroke()

Same thing here again, but swapped out tan for sin. I also reduced the amplitude just so we could see the curve better. In fact, it’s probably a misnomer to talk about the amplitude of a tangent wave since its amplitude is actually infinite. But this value does affect the shape of the curve.

One thing to note is those near-vertical lines that shoot down from positive to negative infinity. Those aren’t technically part of the plot of the tangent. Mathematically, it implies that the tangent could have a value somewhere along that line at some point, but it will not. It is actually undefined at an angle of 90 and 270 degrees. And near infinity/negative infinity just before and after those values.

Not sure if this one will be of much use to you, but there it is.

Summary

Well, we got through the first few curves in this series. Good ones to have under the belt as a lot of other curves will uses these functions in one way or another. See you soon!

Background

For a number of years, I’ve been wanting to writing a book called “Coding Curves”. There are all kinds of really fascinating curves that are fun to code and understand and create very interesting images. I’ve started this book at least three times. I’m feeling the urge to to it again, but I’m going to be more realistic about it this time. I don’t think I’m going to sit down and write a book cover to cover and get it formatted and edited and self publish it. I’ve accepted that that’s just not going to happen. But I want to write about this stuff.

The Plan

So I’m going to do it as a series of blog posts. I don’t know how often these posts will come out. Maybe once a week if I’m diligent. Maybe more often if I get on a roll. Maybe less if I get bored, distracted, or busy with other things. But even if there’s a gap, I can pick back up where I left off and continue.

Maybe, when it’s all done (will it ever be?) there will be enough there to actually compile into a book. Who knows.

Update (2/19/23)

It’s “done”. Which is not to say that I’ll never add to it, but I’ve covered what I set out to cover initially, and I’m pretty happy with it. I may eventually compile it all into an ebook that people can download as a single file and read like that. Probably under a pay-what-you-want type system. But for now, other projects await.

Table of Contents

This is the plan. It may differ, but we gotta start somewhere. I’ll link the posts as they come out here, so this post can continue to serve as an index.

1. Intro (you’re reading it)
2. Trig Curves
3. Arcs, Circles, Ellipses
4. Lissajous Curves
5. Harmonographs
6. Pintographs
7. Parabolas
8. Bezier curves
9. Roulette curves
10. Spirals
11. Roses
12. Guilloche Patterns
13. Superellipses and Superformulas
14. Miscellaneous Curve

In general we’ll be covering two-dimensional curves. Some of these subjects will take multiple posts to cover. I’ll update the TOC as it evolves. There may be more subjects too, and I’m open to suggestions.

The Audience and Purpose

The audience for this course is people who are interested in exploring different types of curves and want practical techniques for drawing them. It may be artists, designers, game developers, UI developers, or recreational mathematicians. This is NOT a mathematics course, though there will be plenty of math in it. A lot of the explanations I give will be very incomplete. There will hopefully be enough information in the materials to give you the understanding you need to draw the given curves and expand on them. But at best, each chapter will be a mere introduction to a topic. Hopefully nothing will be straight out incorrect, but I’m happy to hear about it if there is, and will make my best efforts to correct it.

Personally, I love finding some new mathematical formula that I can apply to create some new, interesting shape, then tweaking the parameters to see how it changes that shape, and eventually animating those parameters. And finally combining the formula for one curve with the formula for another one and possibly making some shape that nobody has every seen before. If any of that rings true for you too, you will probably enjoy reading through this.

The Format

Most of what I’ve written before has been tailored to a specific language. For example, I wrote Playing With Chaos using JavaScript and HTML 5 Canvas. The idea was that it should be generic enough to convert to any language or platform, but I felt I needed to base it in some concrete language.

After recently reading a couple of really fantastic books on ray tracing (covered here), which were completely language agnostic and only included pseudocode samples, I’ve changed my thinking about this. So all “code samples” in this series will actually be “pseudocode samples”.

So write in whatever language or platform you want. The main requirements will be:

A. The ability to set up and size a canvas or drawing surface of some kind.

B. The ability to draw lines of different colors and widths on said canvas.

C. Typical language features and control structures like functions, variables, for loops, conditionals, etc. We’ll probably need some kind of array or list as well as some structured object type, so we can have a point object that has x and y properties. Most of this is all table stakes for any modern language, though it may look different in each one.

Additionally, it will be nice if you can draw circles (or at least arcs, which can become circles) and rectangles, and be able to fill an area with a given color. Extra credit if your drawing api lets you write text to the canvas.

A simple pseudocode example might look like this:

canvas(800, 800)

setLineWidth(5)
setColor(1, 0, 0) // red
moveTo(100, 100)
lineTo(700, 700)
stroke()

This should output an image like this:

I’ll avoid going as deep as OOP or FP as those can look quite different on different platforms, but we’ll need some functions, so I’ll keep those pretty basic:

canvas(500, 100)
foo(100)

function foo(count) {
  for (i = 0; i < count; i++) {
    moveTo(i * 10, 10)
    lineTo(i * 10, 90)
  }
  stroke()
}

Resulting in:

A few things to note here.

First, I’m calling the function before it’s defined. That might work in your language, or it might not. If not, you’ll have to rearrange things.

Secondly, the drawing api functions are written like global functions. It’s likely that in your drawing api, these will need to be called as methods of some kind of canvas object, which may have to be passed into the function, or might be able to be defined globally. I’m going to ignore that and assume that you can figure stuff like that out. The focus here is on covering concepts, not best coding practices.

Also, I’m not bothering with types for vars, args or returns, unless in some instance it becomes non-obvious and important.

As I get deeper into the series, my pseudocode style might change somewhat. If so, I’ll come back here and update things.

Wait for it!

Look for the first installment soon!