Give Me a Sine

Tim Hammerquist November 05, 2019 #javascript #math

One quiet night I was remembering how much fun I had plotting graphics on my TRS-80 Color Computer and lamenting the lack of any decent GUI platforms on modern computers. Specifically, there was one trig function plotter from my CoCo's manual that I thought would be fun to port.

Polar Dares

The book and code are long gone, but the output looked a little like Fig. 1. But how should we reproduce it now?

I still have echoes of PTSD from wrangling cross-platform JavaScript and CSS back when it was still called 'DHTML,' but I thought the Canvas HTML element's 2D API would be a good, erm, canvas for my polar masterpiece! The 2D API provides exactly the Cartesian plotting interface we need.

First we pick an overall size for the square canvas. Halve it on each axis to give us an origin. Then we fetch the canvas element itself and create a 2D context for it, setting its dimensions while we're at it.

Inside the loop, we plot each value of t (θ) over the range [0, 2π) by fractional increments. We compute r using the trig function cos. We then translate the polar coordinates (r, t) to a Cartesian point (x, y) and plot it relative to our origin.

The canvas routines like fillRect will automatically perform subpixel antialiasing ("dithering") for fractional coordinates, which can consume unnecessary CPU. To minimize this, we use a JavaScript bitwise trick to round to the nearest integer. This isn't crucial in this routine, but may be helpful soon.

const size = 150, origin = size / 2;
const canvas = document.getElementById("graph-js1");
canvas.height = canvas.width = size;
const context = canvas.getContext('2d');

for (let t = 0; t < 2 * Math.PI; t += 0.007) {
  const r = 0.9 * origin * Math.cos(2 * t);
  const x = origin + r * Math.cos(t);
  const y = origin - r * Math.sin(t);
  context.fillRect(~~(0.5 + x), ~~(0.5 + y), 1, 1);
}

So here is a minimal implementation. Feel free to play around with the code. In the next section, we'll look at how to make it feel more authentically TRS-80.

Recursive Nostalgia

BASIC's FOR loop translates nicely to JavaScript's for loop. But graphics plotting on the CoCo was not only imperative and synchronous, but a lot slower! How could I reproduce the satisfyingly tortoise-like plot speed of the Motorola 6809E (approximated in Fig. 2) with modern JavaScript?

In most other languages this would be a simple sleep() or wait() call, but JavaScript is not only inherently asynchronous, but single-threaded. JavaScript won't let us tie it up in a sleep call because that one thread has to keep the entire browser page running!

In JavaScript, inserting delays in code is handled using time-outs and callbacks. But to use that we have to rethink how our program is structured. Larger tasks need to be broken down into discrete steps. Each invocation of our function performs one step. Then before we exit the function, we set a "timeout" — a delay after which we perform the next step. And in that intervening time, the browser can handle other things, like network I/O, or user interaction.

So we need to break our larger task ("Plot values from 0 to 2π") into discrete steps ("Plot one point at regular intervals"). The body of the function is essentially the same as the body of the for loop above:

calculate the (x, y) coordinates for t
plot the new coordinates
set a timeout to plot the next t

Steps (1) and (2) — the body of our for loop above — will form the body of the new graphLoop callback function, along with a setTimeout delay just long enough to create that authentic sub-MHz CPU feel.

Fig. 2: r = cos(3⋅θ)

The next invocation of the loop won't know what value of t to use, so we'll have to pass that along with. And since the setTimeout function doesn't send parameters, we have to create a 0-arity lambda that does this.

There are also a few quirks of the Klipse widget that I had to work around. Every code change causes the whole block to be re-evaluated, so I took advantage of JavaScript's atrocious variable scoping to allow the outdated graphLoop calls to exit cleanly. Here are the highlights:

f is defined as a var to make it visible across re-evaluations of the code.
f is an arrow function, which creates a unique object every evaluation.
On entry to graphLoop, we check if f has been recreated the code. If so, return without drawing or setting a new time-out.
When the full graph period ([0, 2π)) is complete, clear the graph and start over at 0.

Here is the final code currently driving Fig. 2.

var f = (t) => Math.cos(3 * t);
var paused = false;

const delay = 15;
const size = 300;
const origin = size / 2;

const canvas = document.getElementById("graph-js2");
canvas.height = canvas.width = size;
canvas.onclick = () => (paused = !paused);

const context = canvas.getContext('2d');
context.clearRect(0, 0, size, size);

function graphLoop(_f, t) {
  if (_f !== f) return;   // exit loop if code modified

  if (!paused) {
    const r = 0.9 * origin * _f(t);
    const x = origin + r * Math.cos(t);
    const y = origin + r * Math.sin(t);
    context.fillRect(~~(0.5 + x), ~~(0.5 + y), 1, 1);

    if (t >= Math.PI) {
      context.clearRect(0, 0, size, size);
      t = 0;
    } else {
      t += 0.003;
    }

    setTimeout(() => graphLoop(_f, t), delay);
  } else {
    setTimeout(() => graphLoop(_f, t), 200);
  }
}

graphLoop(f, 0);

The Challenge

We spent the previous section refactoring our for loop so we could slow the plotting down. In this exercise, we're not going to plot trig functions one point at a time. We're going to plot about a hundred functions per second, end-to-end. Let's play with Canvas for a while.

Fig. 3: r = sin(n⋅θ)
Click box to start plot.

const caption = document.getElementById("fig3-caption");
const canvas = document.getElementById('graph-js3');
const graph = canvas.getContext('2d');
const size = 300;
const origin = size / 2;
canvas.height = canvas.width = size;

// let the user stop/start the animation
let paused = true;
canvas.onclick = () => { paused = !paused; };

const graphLoop = (n) => {
    caption.innerHTML = `Click box to ${paused ? "start" : "pause"} plot`;

    if (paused) {
        setTimeout(() => graphLoop(n), 200);
    } else {
        graph.clearRect(0, 0, size, size);
        for (let t = 0; t < 2 * Math.PI; t += 0.004) {
            const r = 0.96 * origin * Math.sin(n * t);
            const x = origin + r * Math.cos(t);
            const y = origin - r * Math.sin(t);
            graph.fillRect(~~(0.5 + x), ~~(0.5 + y), 1, 1);
        }
        setTimeout(() => graphLoop(n + 0.002), 16);
    }
};
graphLoop(0);

You'll notice lots of consts in the code, because almost none of these variables will change. E.g., there should be only one canvas element and only one caption.

The most of the logic is only required to start/stop the animation by clicking on the canvas. The rest of the code is a loop that does nothing but calculate the polar coordinates (r, θ), convert them to Cartesian (x, y), and plot them on the canvas.

Note: A previous revision of this post used ClojureScript to render Fig. 3.