This post is a continuation of a series of posts about WebGL. The first started with fundamentals and the previous was about 2D matrices. If you haven't read those please view them first.

In the last post we went over how 2D matrices worked. We talked about how translation, rotation, scaling, and even projecting from pixels into clip space can all be done by 1 matrix and some magic matrix math. To do 3D is only a small step from there.

In our previous 2D examples we had 2D points (x, y) that we multiplied by a 3x3 matrix. To do 3D we need 3D points (x, y, z) and a 4x4 matrix.

Let's take our last example and change it to 3D. We'll use an F again but this time a 3D 'F'.

The first thing we need to do is change the vertex shader to handle 3D. Here's the old shader.

<script id="2d-vertex-shader" type="x-shader/x-vertex"> attribute vec2 a_position; uniform mat3 u_matrix; void main() { // Multiply the position by the matrix. gl_Position = vec4((u_matrix * vec3(a_position, 1)).xy, 0, 1); } </script>

And here's the new one

<script id="3d-vertex-shader" type="x-shader/x-vertex"> attribute vec4 a_position; uniform mat4 u_matrix; void main() { // Multiply the position by the matrix. gl_Position = u_matrix * a_position; } </script>

It got even simpler!

Then we need to provide 3D data.

... gl.vertexAttribPointer(positionLocation, 3, gl.FLOAT, false, 0, 0); ... // Fill the buffer with the values that define a letter 'F'. function setGeometry(gl) { gl.bufferData( gl.ARRAY_BUFFER, new Float32Array([ // left column 0, 0, 0, 30, 0, 0, 0, 150, 0, 0, 150, 0, 30, 0, 0, 30, 150, 0, // top rung 30, 0, 0, 100, 0, 0, 30, 30, 0, 30, 30, 0, 100, 0, 0, 100, 30, 0, // middle rung 30, 60, 0, 67, 60, 0, 30, 90, 0, 30, 90, 0, 67, 60, 0, 67, 90, 0]), gl.STATIC_DRAW); }

Next we need to change all the matrix functions from 2D to 3D

Here are the 2D (before) versions of makeTranslation, makeRotation and makeScale

function makeTranslation(tx, ty) { return [ 1, 0, 0, 0, 1, 0, tx, ty, 1 ]; } function makeRotation(angleInRadians) { var c = Math.cos(angleInRadians); var s = Math.sin(angleInRadians); return [ c,-s, 0, s, c, 0, 0, 0, 1 ]; } function makeScale(sx, sy) { return [ sx, 0, 0, 0, sy, 0, 0, 0, 1 ]; }

And here are the updated 3D versions.

function makeTranslation(tx, ty, tz) { return [ 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, tx, ty, tz, 1 ]; } function makeXRotation(angleInRadians) { var c = Math.cos(angleInRadians); var s = Math.sin(angleInRadians); return [ 1, 0, 0, 0, 0, c, s, 0, 0, -s, c, 0, 0, 0, 0, 1 ]; }; function makeYRotation(angleInRadians) { var c = Math.cos(angleInRadians); var s = Math.sin(angleInRadians); return [ c, 0, -s, 0, 0, 1, 0, 0, s, 0, c, 0, 0, 0, 0, 1 ]; }; function makeZRotation(angleInRadians) { var c = Math.cos(angleInRadians); var s = Math.sin(angleInRadians); return [ c, s, 0, 0, -s, c, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, ]; } function makeScale(sx, sy, sz) { return [ sx, 0, 0, 0, 0, sy, 0, 0, 0, 0, sz, 0, 0, 0, 0, 1, ]; }

Notice we now have 3 rotation functions. We only needed one in 2D as we were effectively only rotating around the Z axis. Now though to do 3D we also want to be able to rotate around the X axis and Y axis as well. You can see from looking at them they are all very similar. If we were to work them out you'd see them simplify just like before

Z rotation

newX = x * c + y * s;

newY = x * -s + y * c;

Y rotation

newX = x * c + z * s;

newZ = x * -s + z * c;

X rotation

newY = y * c + z * s;

newZ = y * -s + z * c;

which gives you these rotations.

We also need to update the projection function. Here's the old one

function make2DProjection(width, height) { // Note: This matrix flips the Y axis so 0 is at the top. return [ 2 / width, 0, 0, 0, -2 / height, 0, -1, 1, 1 ]; }

which converted from pixels to clip space. For our first attempt at expanding it to 3D let's try

function make2DProjection(width, height, depth) { // Note: This matrix flips the Y axis so 0 is at the top. return [ 2 / width, 0, 0, 0, 0, -2 / height, 0, 0, 0, 0, 2 / depth, 0, -1, 1, 0, 1, ]; }

Just like we needed to convert from pixels to clip space for X and Y, for
Z we need to do the same thing. In this case I'm making the Z axis pixel
units as well. I'll pass in some value similar to `width`

for the `depth`

so our space will be 0 to `width`

pixels wide, 0 to `height`

pixels tall, but
for `depth`

it will be `-depth / 2`

to `+depth / 2`

.

Finally we need to to update the code that computes the matrix.

// Compute the matrices var projectionMatrix = make2DProjection(canvas.clientWidth, canvas.clientHeight, 400); var translationMatrix = makeTranslation(translation[0], translation[1], translation[2]); var rotationXMatrix = makeXRotation(rotation[0]); var rotationYMatrix = makeYRotation(rotation[1]); var rotationZMatrix = makeZRotation(rotation[2]); var scaleMatrix = makeScale(scale[0], scale[1], scale[2]); // Multiply the matrices. var matrix = matrixMultiply(scaleMatrix, rotationZMatrix); matrix = matrixMultiply(matrix, rotationYMatrix); matrix = matrixMultiply(matrix, rotationXMatrix); matrix = matrixMultiply(matrix, translationMatrix); matrix = matrixMultiply(matrix, projectionMatrix); // Set the matrix. gl.uniformMatrix4fv(matrixLocation, false, matrix);

And here's that sample.

The first problem we have is that our geometry is a flat F which makes it hard to see any 3D. To fix that let's expand the geometry to 3D. Our current F is made of 3 rectangles, 2 triangles each. To make it 3D will require a total of 16 rectangles. That's quite a few to list out here. 16 rectangles with 2 triangles per rectangle and 3 vertices per triangle is 96 vertices. If you want to see all of them view the source of the sample.

We have to draw more vertices so

// Draw the geometry. gl.drawArrays(gl.TRIANGLES, 0, 16 * 6);

And here's that version

Moving the sliders it's pretty hard to tell that it's 3D. Let's try coloring each rectangle a different color. To do this we will add another attribute to our vertex shader and a varying to pass it from the vertex shader to the fragment shader.

Here's the new vertex shader

<script id="3d-vertex-shader" type="x-shader/x-vertex"> attribute vec4 a_position; attribute vec4 a_color; uniform mat4 u_matrix; varying vec4 v_color; void main() { // Multiply the position by the matrix. gl_Position = u_matrix * a_position; // Pass the color to the fragment shader. v_color = a_color; } </script>

And we need to use that color in the fragment shader

<script id="3d-vertex-shader" type="x-shader/x-fragment"> precision mediump float; // Passed in from the vertex shader. varying vec4 v_color; void main() { gl_FragColor = v_color; } </script>

We need to lookup the location to supply the colors, then setup another buffer and attribute to give it the colors.

... var colorLocation = gl.getAttribLocation(program, "a_color"); ... // Create a buffer for colors. var buffer = gl.createBuffer(); gl.bindBuffer(gl.ARRAY_BUFFER, buffer); gl.enableVertexAttribArray(colorLocation); // We'll supply RGB as bytes. gl.vertexAttribPointer(colorLocation, 3, gl.UNSIGNED_BYTE, true, 0, 0); // Set Colors. setColors(gl); ... // Fill the buffer with colors for the 'F'. function setColors(gl) { gl.bufferData( gl.ARRAY_BUFFER, new Uint8Array([ // left column front 200, 70, 120, 200, 70, 120, 200, 70, 120, 200, 70, 120, 200, 70, 120, 200, 70, 120, // top rung front 200, 70, 120, 200, 70, 120, ... ... gl.STATIC_DRAW); }

Now we get this.

Uh oh, what's that mess? Well, it turns out all the various parts of that 3D 'F', front, back, sides, etc get drawn in the order they appear in in our geometry. That doesn't give us quite the desired results as sometimes the ones in the back get drawn after the ones in the front.

Triangles in WebGL have the concept of front facing and back facing. A front facing triangle has its vertices go in a clockwise direction. A back facing triangle has its vertices go in a counter clockwise direction

WebGL has the ability to draw only forward facing or back facing triangles. We can turn that feature on with

gl.enable(gl.CULL_FACE);

which we do just once, right at the start of our program. With that feature turned on, WebGL defaults to "culling" back facing triangles. "Culling" in this case is a fancy word for "not drawing".

Note that as far as WebGL is concerned, whether or not a triangle is considered to be going clockwise or counter clockwise depends on the vertices of that triangle in clip space. In other words, WebGL figures out whether a triangle is front or back AFTER you've applied math to the vertices in the vertex shader. That means for example a clockwise triangle that is scaled in X by -1 becomes a counter clockwise triangle or a clockwise triangle rotated 180 degrees becomes a couter clockwise triangle. Because we had CULL_FACE disabled we can see both clockwise(front) and counter clockwise(back) triangles. Now that we've turned it on, any time a front facing triangle flips around either because of scaling or rotation or for whatever reason, WebGL won't draw it. That's a good thing since as your turn something around in 3D you generally want whichever triangles are facing you to be considered front facing.

With CULL_FACE turned on this is what we get

Hey! Where did all the triangles go? It turns out, many of them are facing the wrong way. Rotate it and you'll see them appear when you look at the other side. Fortunately it's easy to fix. We just look at which ones are backward and exchange 2 of their vertices. For example if one backward triangle is

1, 2, 3, 40, 50, 60, 700, 800, 900,

we just flip the last 2 vertices to make it forward.

1, 2, 3, 700, 800, 900, 40, 50, 60,

Going through and fixing all the backward triangles gets us to this

That's closer but there's still one more problem. Even with all the triangles facing in the correct direction and with the back facing ones being culled we still have places where triangles that should be in the back are being drawn over triangles that should be in front.

Enter the DEPTH BUFFER.

A depth buffer, sometimes called a Z-Buffer, is a rectangle of *depth*
pixels, one depth pixel for each color pixel used to make the image. As
WebGL draws each color pixel it can also draw a depth pixel. It does this
based on the values we return from the vertex shader for Z. Just like we
had to convert to clip space for X and Y, Z is also in clip space or (-1
to +1). That value is then converted into a depth space value (0 to +1).
Before WebGL draws a color pixel it will check the corresponding depth
pixel. If the depth value for the pixel it's about to draw is greater
than the value of the corresponding depth pixel then WebGL does not draw
the new color pixel. Otherwise it draws both the new color pixel with the
color from your fragment shader AND it draws the depth pixel with the new
depth value. This means, pixels that are behind other pixels won't get
drawn.

We can turn on this feature nearly as simply as we turned on culling with

gl.enable(gl.DEPTH_TEST);

We also need to clear the depth buffer back to 1.0 before we start drawing.

// Draw the scene. function drawScene() { // Clear the canvas AND the depth buffer. gl.clear(gl.COLOR_BUFFER_BIT | gl.DEPTH_BUFFER_BIT); ...

And now we get

which is 3D!

In the next post I'll go over how to make it have perspective.