Kristoffer Dyrkorn, November 4, 2022

8. Let’s fix this

(This article is part of a series. You can jump to the previous section or the next section if you would like to.)

In this section, we will convert the rasterizer to use fixed point coordinates. We have already implemented a FixedPointVector class to help us, so the walkthrough here only considers the changes to the application and to the rasterizer itself.

Updating the code

When calculating the determinant we now refer to input vectors as FixedPointVectors. Apart from that, there are no changes - the two vector classes we use have the same API.

getDeterminant(a, b, c) {
    const ab = new FixedPointVector(b);

    const ac = new FixedPointVector(c);

    return ab[1] * ac[0] - ab[0] * ac[1];

At the start of the triangle draw method, we convert the incoming floating point screen coordinates to fixed point coordinates by using the built-in constructor in FixedPointVectors:

draw(screenCoordinates, color) {
    // get screen coordinates for this triangle
    const va = new FixedPointVector(screenCoordinates[]);
    const vb = new FixedPointVector(screenCoordinates[this.vb]);
    const vc = new FixedPointVector(screenCoordinates[]);

    const determinant = this.getDeterminant(va, vb, vc);

    // backface culling: only draw if determinant is positive
    // in that case, the triangle is ccw oriented - ie front-facing
    if (determinant <= 0) {


When we create the bounding box, we no longer have the Math.floor() and Math.ceil() functions avaiable, so we round the values up and down manually when we convert from fixed point numbers to normal integers:

const xmin = Math.min(va[0], vb[0], vc[0]) >> FixedPointVector.SHIFT;
const ymin = Math.min(va[1], vb[1], vc[1]) >> FixedPointVector.SHIFT;

const xmax = (Math.max(va[0], vb[0], vc[0]) + FixedPointVector.DIVISION_CEILING) >> FixedPointVector.SHIFT;
const ymax = (Math.max(va[1], vb[1], vc[1]) + FixedPointVector.DIVISION_CEILING) >> FixedPointVector.SHIFT;

Also, our w vector and the candidate point vector p need to change type:

   // w = edge distances
    const w = new FixedPointVector();

    // p = screen coordinates
    const p = new FixedPointVector();

    for (let y = ymin; y <= ymax; y++) {
      for (let x = xmin; x <= xmax; x++) {
        // there is no need to round off the result.
        // the input is an integer, and although we add 0.5 to it,
        // we then multiply by 2^n (n>0), which means the result will always be an integer
        p[0] = (x + 0.5) * FixedPointVector.MULTIPLIER;
        p[1] = (y + 0.5) * FixedPointVector.MULTIPLIER;

The final part now is to update the fill rule to operate on fixed point numbers. Again, the APIs of the two vector classes are the same, so the change is minimal.

  isLeftOrTopEdge(start, end) {
    const edge = new FixedPointVector(end);

    const leftEdge = edge[1] > 0;
    const topEdge = edge[1] == 0 && edge[0] < 0;
    return leftEdge || topEdge;

Regarding the fill rule, there is an important detail here: When we used integer coordinates in the rasterizer, the adjustment value was 1 - since that was the numerical resolution (ULP). Independent of the choice of number representation, we still want the adjustment value to equal the available resolution. And that value is now 1 - in fixed point representation. So although the code does not seem to have changed, the meaning of the number 1 definately has changed.

if (this.isLeftOrTopEdge(vb, vc)) w[0]--;
if (this.isLeftOrTopEdge(vc, va)) w[1]--;
if (this.isLeftOrTopEdge(va, vb)) w[2]--;

And that is all that’s needed! Sweet! We now have a fully working and correct rasterizer that gives us smooth animation, due to subpixel resolution support.

If you want to test out various subpixel resolutions and see the effects yourself, you can adjust the value of the FixedPointVector.SHIFT constant (at the end of this file). Try out values like 0 (no subpixels - ie back to a pure-integer version), 1, 2, 4, and 8 for example.

FixedPointVector.SHIFT = 4;

Measuring performance

However, the code - as it is now - is not particularly fast. Let’s add some simple timing code around the triangle draw function in the application code:

let start =;
greenTriangle.draw(rotatedVertices, greenColor);
triangleDrawTime += - start;

if (frameCounter % 100 == 0) {
    console.log(`Triangle time: ${(triangleDrawTime / frameCounter).toFixed(2)} ms`);

When running the code on my machine (a MacBook Pro with an M1 processor), drawing the green triangle takes around 2.3 ms. That is actually quite a long time. Here, each triangle consists of only a few pixels (remember, the pixels we see are quite large) and does not require a lot of work to draw, but we still would not be able to draw and animate more than 7 triangles on screen before the movement would start stuttering. The browser draws 60 frames per second, so for everyting to run smoothly we must keep drawing at at least the same tempo. That means we have a time budget of 16.7 ms per frame to draw and animate everything. And 7 triangles times 2.3 ms per triangle equals 16.1 ms, so 7 triangles will be the max.

The profiler in my browser tells me that we spend a lot of time calculating determinants, evaluating the fill rule and instantiating FixedPointVectors. Could we speed up our code? Yes we can! In the next section we will do just that.

The code for this section and the utility classes is available. The demo app is available as well. Press space to show/hide the blue triangle, and p to turn the animation on/off.