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WebGL 3D - Directional Lighting

There are many ways to implement lighting. Probably the simplest is directional lighting.

Directional lighting assumes the light is coming uniformly from one direction. The sun on a clear day is often considered a directional light. It's so far way that its rays can be considered to be hitting the surface of an object all in parallel.

Computing directional lighting is actually pretty simple. If we know what direction the light is traveling and we know what direction the surface of the object is facing we can take the dot product of the 2 directions and it will give us the cosine of the angle between the 2 directions.

Here's an example

drag the points

Drag the points around, if you get them exactly opposite of each other you'll see the dot product is -1. If they are at the same spot exactly the dot product is 1.

How is that useful? Well if we know what direction the surface of our 3d object is facing and we know the direction the light is shining then we can just take the dot product of them and it will give us a number 1 if the light is pointing directly at the surface and -1 if they are pointing directly opposite.

rotate the direction

We can multiply our color by that dot product value and boom! Light!

One problem, how do we know which direction the surfaces of our 3d object are facing.

Introducing Normals

Normals are called that from the Latin word norma, a carpenter's square. Just as a carpenter's square has a right angle, normals are at a right angle to a line or surface. In 3D graphics a normal is the word for a unit vector that describes the direction a surface is facing.

Here are some normals for a cube and a sphere.

The lines sticking out of the objects represent normals for each vertex.

Notice the cube has 3 normals at each corner. That's because you need 3 different normals to represent the way each face of the cube is um, .. facing.

The normals are also colored based on their direction with positive x being red, up being green and positive z being blue.

So, let's go add normals to our F from our previous examples so we can light it. Since the F is very boxy and its faces are aligned to the x, y, or z axis it will be pretty easy. Things that are facing forward have the normal 0, 0, 1. Things that are facing away are 0, 0, -1. Facing left is -1, 0, 0, Facing right is 1, 0, 0. Up is 0, 1, 0 and down is 0, -1, 0.

function setNormals(gl) {
var normals = new Float32Array([
// left column front
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,

// top rung front
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,

// middle rung front
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,
0, 0, 1,

// left column back
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,

// top rung back
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,

// middle rung back
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,
0, 0, -1,

// top
0, 1, 0,
0, 1, 0,
0, 1, 0,
0, 1, 0,
0, 1, 0,
0, 1, 0,

// top rung right
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,

// under top rung
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,

// between top rung and middle
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,

// top of middle rung
0, 1, 0,
0, 1, 0,
0, 1, 0,
0, 1, 0,
0, 1, 0,
0, 1, 0,

// right of middle rung
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,

// bottom of middle rung.
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,

// right of bottom
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,
1, 0, 0,

// bottom
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,
0, -1, 0,

// left side
-1, 0, 0,
-1, 0, 0,
-1, 0, 0,
-1, 0, 0,
-1, 0, 0,
-1, 0, 0]);
gl.bufferData(gl.ARRAY_BUFFER, normals, gl.STATIC_DRAW);
}

and set them up. While we're at it let's remove the vertex colors so it's easier to see the lighting.

// look up where the vertex data needs to go.
var positionLocation = gl.getAttribLocation(program, "a_position");
-var colorLocation = gl.getAttribLocation(program, "a_color");
+var normalLocation = gl.getAttribLocation(program, "a_normal");

...

-// Create a buffer to put colors in
-var colorBuffer = gl.createBuffer();
-// Bind it to ARRAY_BUFFER (think of it as ARRAY_BUFFER = colorBuffer)
-gl.bindBuffer(gl.ARRAY_BUFFER, colorBuffer);
-// Put geometry data into buffer
-setColors(gl);

+// Create a buffer to put normals in
+var normalBuffer = gl.createBuffer();
+// Bind it to ARRAY_BUFFER (think of it as ARRAY_BUFFER = normalBuffer)
+gl.bindBuffer(gl.ARRAY_BUFFER, normalBuffer);
+// Put normals data into buffer
+setNormals(gl);

And at render time

-// Turn on the color attribute
-gl.enableVertexAttribArray(colorLocation);
-
-// Bind the color buffer.
-gl.bindBuffer(gl.ARRAY_BUFFER, colorBuffer);
-
-// Tell the attribute how to get data out of colorBuffer (ARRAY_BUFFER)
-var size = 3;                 // 3 components per iteration
-var type = gl.UNSIGNED_BYTE;  // the data is 8bit unsigned values
-var normalize = true;         // normalize the data (convert from 0-255 to 0-1)
-var stride = 0;               // 0 = move forward size * sizeof(type) each iteration to get the next position
-var offset = 0;               // start at the beginning of the buffer
-gl.vertexAttribPointer(
-    colorLocation, size, type, normalize, stride, offset)

+// Turn on the normal attribute
+gl.enableVertexAttribArray(normalLocation);
+
+// Bind the normal buffer.
+gl.bindBuffer(gl.ARRAY_BUFFER, normalBuffer);
+
+// Tell the attribute how to get data out of normalBuffer (ARRAY_BUFFER)
+var size = 3;          // 3 components per iteration
+var type = gl.FLOAT;   // the data is 32bit floating point values
+var normalize = false; // normalize the data (convert from 0-255 to 0-1)
+var stride = 0;        // 0 = move forward size * sizeof(type) each iteration to get the next position
+var offset = 0;        // start at the beginning of the buffer
+gl.vertexAttribPointer(
+    normalLocation, size, type, normalize, stride, offset)

Now we need to make our shaders use them

First the vertex shader we just pass the normals through to the fragment shader

attribute vec4 a_position;
-attribute vec4 a_color;
+attribute vec3 a_normal;

uniform mat4 u_matrix;

-varying vec4 v_color;
+varying vec3 v_normal;

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;

+  // Pass the normal to the fragment shader
+  v_normal = a_normal;
}

And the fragment shader we'll do the math using the dot product of the direction of the light and the normal

precision mediump float;

// Passed in from the vertex shader.
-varying vec4 v_color;
+varying vec3 v_normal;

+uniform vec3 u_reverseLightDirection;
+uniform vec4 u_color;

void main() {
+   // because v_normal is a varying it's interpolated
+   // so it will not be a unit vector. Normalizing it
+   // will make it a unit vector again
+   vec3 normal = normalize(v_normal);
+
+   float light = dot(normal, u_reverseLightDirection);

*   gl_FragColor = u_color;

+   // Lets multiply just the color portion (not the alpha)
+   // by the light
+   gl_FragColor.rgb *= light;
}

Then we need to lookup the locations of u_color and u_reverseLightDirection.

// lookup uniforms
var matrixLocation = gl.getUniformLocation(program, "u_matrix");
+  var colorLocation = gl.getUniformLocation(program, "u_color");
+  var reverseLightDirectionLocation =
+      gl.getUniformLocation(program, "u_reverseLightDirection");

and we need to set them

// Set the matrix.
gl.uniformMatrix4fv(matrixLocation, false, worldViewProjectionMatrix);

+  // Set the color to use
+  gl.uniform4fv(colorLocation, [0.2, 1, 0.2, 1]); // green
+
+  // set the light direction.
+  gl.uniform3fv(reverseLightDirectionLocation, m4.normalize([0.5, 0.7, 1]));

normalize, which we went over before, will make whatever values we put in there into a unit vector. The specific values in the sample are x = 0.5 which is positive x means the light is on the right pointing left. y = 0.7 which is positive y means the light is above pointing down. z = 1 which is positive z means the light is in front pointing into the scene. the relative values means the direction is mostly pointing into the scene and pointing more down then right.

And here it is

If you rotate the F you might notice something. The F is rotating but the lighting isn't changing. As the F rotates we want whatever part is facing the direction of the light to be the brightest.

To fix this we need to re-orient the normals as the object is re-oriented. Like we did for positions we can multiply the normals by some matrix. The most obvious matrix would be the world matrix. As it is right now we're only passing in 1 matrix called u_matrix. Let's change it to pass in 2 matrices. One called u_world which will be the world matrix. Another called u_worldViewProjection which will be what we're currently passing in as u_matrix

attribute vec4 a_position;
attribute vec3 a_normal;

*uniform mat4 u_worldViewProjection;
+uniform mat4 u_world;

varying vec3 v_normal;

void main() {
// Multiply the position by the matrix.
*  gl_Position = u_worldViewProjection * a_position;

*  // orient the normals and pass to the fragment shader
*  v_normal = mat3(u_world) * a_normal;
}

Notice we are multiplying a_normal by mat3(u_world). That's because normals are a direction so we don't care about translation. The orientation portion of the matrix is only in the top 3x3 area of the matrix.

Now we have to look those uniforms up

// lookup uniforms
*  var worldViewProjectionLocation =
*      gl.getUniformLocation(program, "u_worldViewProjection");
+  var worldLocation = gl.getUniformLocation(program, "u_world");

And we have to change the code that updates them

*// Set the matrices
*gl.uniformMatrix4fv(
*    worldViewProjectionLocation, false,
*    worldViewProjectionMatrix);
*gl.uniformMatrix4fv(worldLocation, false, worldMatrix);

and here's that

Rotate the F and notice which ever side is facing the light direction gets lit.

There is one problem which I don't know how to show directly so I'm going to show it in a diagram. We're multiplying the normal by the u_world matrix to re-orient the normals. What happens if we scale the world matrix? It turns out we get the wrong normals.

click to toggle normals

I've never bothered to understand the solution but it turns out you can get the inverse of the world matrix, transpose it, which means swap the columns for rows, and use that instead and you'll get the right answer.

In the diagram above the purple sphere is unscaled. The red sphere on the left is scaled and the normals are being multiplied by the world matrix. You can see something is wrong. The blue sphere on the right is using the world inverse transpose matrix.

Click the diagram to cycle through different representations. You should notice when the scale is extreme it's very easy to see the normals on the left (world) are not staying perpendicular to the surface of the sphere where as the ones on the right (worldInverseTranspose) are staying perpendicular to the sphere. The last mode makes them all shaded red. You should see the lighting on the 2 outer spheres is very different based on which matrix is used. It's hard to tell which is correct which is why this is a subtle issue but based on the other visualizations it's clear using the worldInverseTranspose is correct.

To implement this in our example let's change the code like this. First we'll update the shader. Technically we could just update the value of u_world but it's best if we rename things so they're named what they actually are otherwise it will get confusing.

attribute vec4 a_position;
attribute vec3 a_normal;

uniform mat4 u_worldViewProjection;
*uniform mat4 u_worldInverseTranspose;

varying vec3 v_normal;

void main() {
// Multiply the position by the matrix.
gl_Position = u_worldViewProjection * a_position;

// orient the normals and pass to the fragment shader
*  v_normal = mat3(u_worldInverseTranspose) * a_normal;
}

Then we need to look that up

-  var worldLocation = gl.getUniformLocation(program, "u_world");
+  var worldInverseTransposeLocation =
+      gl.getUniformLocation(program, "u_worldInverseTranspose");

And we need to compute and set it

var worldViewProjectionMatrix = m4.multiply(viewProjectionMatrix, worldMatrix);
var worldInverseMatrix = m4.inverse(worldMatrix);
var worldInverseTransposeMatrix = m4.transpose(worldInverseMatrix);

// Set the matrices
gl.uniformMatrix4fv(
worldViewProjectionLocation, false,
worldViewProjectionMatrix);
-gl.uniformMatrix4fv(worldLocation, false, worldMatrix);
+gl.uniformMatrix4fv(
+    worldInverseTransposeLocation, false,
+    worldInverseTransposeMatrix);

and here's the code to transpose a matrix

var m4 = {
transpose: function(m) {
return [
m, m, m, m,
m, m, m, m,
m, m, m, m,
m, m, m, m,
];
},

...

Because the effect is subtle and because we aren't scaling anything there's no noticeable difference but at least now we're prepared.

I hope this first step into lighting was clear. Next up point lighting.

Alternatives to mat3(u_worldInverseTranspose) * a_normal

In our shader above there's a line like this

v_normal = mat3(u_worldInverseTranspose) * a_normal;

We could have done this

v_normal = (u_worldInverseTranspose * vec4(a_normal, 0)).xyz;

Because we set w to 0 before multiplying that would end up multiplying the translation from the matrix by 0 effectively removing it. I think that's the more common way to do it. The mat3 way looked cleaner to me but I've often done it this way too.

Yet another solution would be to make u_worldInverseTranspose a mat3. There are 2 reasons not to do that. One is we might have other needs for the full u_worldInverseTranspose so passing the entire mat4 means we can use with for those other needs. Another is that all of our matrix functions in JavaScript make 4x4 matrices. Making a whole other set for 3x3 matrices or even converting from 4x4 to 3x3 is work we'd rather not do unless there was a more compelling reason.