## Understanding OpenGL’s Matrices

I often get questions related to OpenGL’s matrices: how do they work, how do they get built, and so forth. This is a topic that I have been frequently confused by, myself, and I feel that it warrants further explanation.

To better understand OpenGL’s matrices, and how and why we use them, we first need to understand the OpenGL coordinate space.

### Normalized device coordinates

At the heart of things, OpenGL 2.0 doesn’t really know anything about your coordinate space or about the matrices that you’re using. OpenGL only requires that when all of your transformations are done, things should be in normalized device coordinates.

These coordinates range from -1 to +1 on each axis, regardless of the shape or size of the actual screen. The bottom left corner will be at (-1, -1), and the top right corner will be at (1, 1). OpenGL will then map these coordinates onto the viewport that was configured with glViewport. The underlying operating system’s window manager will then map that viewport to the appropriate place on the screen.

### Adjusting to the screen’s aspect ratio

While OpenGL wants things to be in normalized device coordinates, it’s hard to work with these directly. The first problem is that they always range from -1 to +1, so if you use these coordinates directly, your image might be stretched when switching from portrait mode to landscape mode.

The first thing you can do to get around this problem is to define an orthographic projection. Android has the orthoM method; other platforms will have something similar. Let’s take a closer look at Android’s method:

`orthoM(float[] m, int mOffset, float left, float right, float bottom, float top, float near, float far)`

To define a simple matrix that adjusts things for the screen’s aspect ratio, we might call orthoM as follows:

```float aspectRatio = (float) width / (float) height; orthoM(projectionMatrix, 0, -aspectRatio, aspectRatio, -1, 1, -1, 1);```

Let’s say that the screen dimensions are 800×600. The call would proceed as follows:

`orthoM(projectionMatrix, 0, -1.333, 1.333, -1, 1, -1, 1);`

Although the screen is wider than it’s tall, we automatically adjust the coordinate space to match by mapping -(800/600) to the left side and (800/600) to the right side.

This also works when we switch to portrait mode:

`orthoM(projectionMatrix, 0, -0.75, 0.75, -1, 1, -1, 1);`

We shrink the width in order to compensate for the smaller screen.

At the heart of things, the orthographic projection matrix will still convert things to the [-1, 1] range, since that’s what OpenGL expects. It just provides a way to adjust our coordinate space, so that we can see more of our scene if the screen is wider, and less if the screen is narrower.

### 3D projections

What about 3D projections? For those, we can use frustumM:

`frustumM(float[] m, int offset, float left, float right, float bottom, float top, float near, float far)`

We could define a simple 3D projection as follows:

`frustumM(projectionMatrix, 0, -aspectRatio, aspectRatio, -1, 1, 1, 100);`

The near & far range are handled differently: both have to be positive, and far has to be greater than near. We also have to watch out for the Z axis: frustumM will actually invert it, so that the negative Z points into the distance!

This has to do with convention: normalized device coordinates are in a left-handed coordinate system, while by convention, when we use a projection matrix, we work in a right-handed coordinate system.

Below is a good image illustrating the situation:

### The perspective divide

The perspective projection doesn’t actually create the 3D effect; for that, we need to do something called the perspective divide. Each coordinate in OpenGL actually has four components, X, Y, Z, and W. The projection matrix sets things up so that after multiplying with the projection matrix, each coordinate’s W will increase the further away the object is. OpenGL will then divide by w: X, Y, Z will be divided by W. The further away something is, the more it will be pulled towards the center of the screen.

This PDF goes into more detail: http://www.terathon.com/gdc07_lengyel.pdf

This image shows how the same coordinate gets closer to the center of the screen as the W value increases:

Say you have three XYZ source positions of the following:

```(3, 3,    -3)
(3, 3,    -6)
(3, 3, -1000)```

The second point is a little bit further, or more “into the screen” than the first, and the third point is much further away than the second point. An infinite projection matrix would convert the coordinates as follows:

```(3, 3,   1,    3)
(3, 3,   4,    6)
(3, 3, 998, 1000)```

That last component is W. Now, OpenGL will divide everything by W, so you get something like this:

```(1,     1,     0.33...)
(0.5,   0.5,   0.66...)
(0.003, 0.003, 0.998  )```

There are two side effects of this division by W:

1. The depth buffer becomes non-linear. There is a lot of Z precision up close, but less further away.
2. If you try to do translations, etc… on your vertices after the perspective projection, then you won’t get the results you expect. This is because many of these operations depend on the W being 1, while after perspective projection it can be something else.

Here is an example of the perspective divide: imagine that we have a perspective projection matrix that looks as follows:

``` 1.5, 0,  0,    0,
0,   1,  0,    0,
0,   0, -1.2, -2.2,
0,   0, -1,    0```

This will transform coordinates as follows:

```(1, 1, -1, 1) --> (1.5, 1, -1,   1)
(1, 1, -2, 1) --> (1.5, 1,  0.2, 2)
(2, 2, -2, 1) --> (3,   2,  0.2, 2)```

After division by W, we get this:

```(1.5, 1, -1,   1) --> (1.5,  1,   -1  )
(1.5, 1,  0.2, 2) --> (0.75, 0.5,  0.1)
(3,   2,  0.2, 2) --> (1.5,  1,    0.1)```

Notice that the projection matrix just sets up the W, and it’s the actual divide by OpenGL that does the perspective effect. To verify this, try it out with a matrix calculator:

http://www.math.ubc.ca/~israel/applet/mcalc/matcalc.html

### The view and model matrices

In OpenGL, we commonly use two additional matrices: the view and model matrices:

#### The model matrix

This matrix is used to move a model somewhere in the world. For example, let’s say we have a car model, and it’s defined such that it is centered around (0, 0, 0). We can place one car at (5, 5, 5) by setting up a model matrix that translates everything by (5, 5, 5), and drawing the car model with this matrix. We can then easily add a second car to the scene by just adjusting the translation. The model matrix helps us to push stuff out into the world.

#### The view matrix

The view matrix is functionally equivalent to a camera. It does the same thing as a model matrix, but it applies the same transformations equally to every object in the scene. Moving the whole world 5 units towards us is the same as if we had walked 5 units forwards.

### Order of operations

These matrices all have to be multiplied in a specific way, if we want our results to be correct. Let’s start with some basic definitions:

vertexmodel
This is an original vertex, as defined inside one of our object models.
vertexworld
This is a vertex in world coordinates. We get to here by using a model matrix to push the model out into the world.
vertexeye
This is a vertex in eye coordinates. We get here by using a view matrix to move the entire scene around.
vertexclip
This is a vertex in clip coordinates (also known as homogeneous coordinates): this is the coordinate space after projection, but before the perspective divide.
vertexndc
This is a vertex in normalized device coordinates, and this is what we end up with after the perspective divide.

As we can see, getting to vertexndc is just a matter of applying each transformation in order. Let’s try to formulate this as an expression:

vertexndc = PerspectiveDivide(ProjectionMatrix * vertexeye)
vertexndc = PerspectiveDivide(ProjectionMatrix * ViewMatrix * vertexworld)
vertexndc = PerspectiveDivide(ProjectionMatrix * ViewMatrix * ModelMatrix * vertexmodel)

OpenGL takes care of the perspective division for us, so we don’t actually need to worry about that. All we need to worry about is the order of operations; since matrix multiplication is non commutative, we’ll get a different result depending on the order.

### Column-major versus row-major order

A final point of confusion is often the layout of matrices in memory. OpenGL follows column-major order, meaning that the array offsets are specified like this:

```0  4   8  12
1  5   9  13
2  6  10  14
3  7  11  15```

m[0] … m[3] refer to the first column of the matrix.

## WebGL Lesson One: Getting Started

This is the first tutorial for learning OpenGL ES 2 on the web, using WebGL. In this lesson, we’ll look at how to create a basic WebGL instance and display stuff to the screen, as well as what you need in order to view WebGL in your browser. There will also be an introduction to shaders and matrices.

#### What is WebGL?

Previously, if you wanted to do real-time 3D graphics on the web, your only real option was to use a plugin such as Java or Flash. However, there is currently a push to bring hardware-accelerated graphics to the web called WebGL. WebGL is based on OpenGL ES 2, which means that we’ll need to use shaders. Since WebGL runs inside a web browser, we’ll also need to use JavaScript to control it.

#### Prerequisites

You’ll need a browser that supports WebGL, and you should also have the most recent drivers installed for your video card. You can visit Get WebGL to see if your browser supports WebGL and if not, it will tell you where you can get a browser that supports it.

The latest stable releases of Chrome and Firefox support WebGL, so you can always start there.

This lesson uses the following third-party libraries:

• webgl-utils.js — for basic initialization of an OpenGL context and rendering on browser request.
• glMatrix.js — for matrix operations.

#### Assumptions

The reader should be familiar with programming and 3D concepts on a basic level. The Khronos WebGL Public Wiki is a good place to start out.

#### Getting started

As I write this, I am also learning WebGL, so we’ll be learning together! We’ll look at how to get a context and start drawing stuff to the screen, and we’ll more or less follow lesson one for Android as this lesson is based on it. For those of you who followed the Android lesson, you may remember that getting an OpenGL context consisted of creating an activity and setting the content view to a GLSurfaceView object. We also provided a class which overrode GLSurfaceView.Renderer and provided methods which were called by the system.

With WebGL, it is just as easy to get things setup and running. The webgl-utils.js script provides us with two functions to get things going:

```function setupWebGL(canvas, opt_attribs);

function window.requestAnimFrame(callback, element);```

The setupWebGL() function takes care of initializing WebGL for us, as well as pointing the user to a browser that supports WebGL or further troubleshooting if there were errors initializing WebGL. More info on the optional parameters can be found at the WebGL Specification page, section 5.2.1.

The second function provides a cross-browser way of setting up a render callback. The browser will call the function provided in the callback parameter at a regular interval. The element parameter lets the browser know for which element the callback is firing.

In our script we have a function main() which is our main entry point, and is called once at the end of the script. In this function, we initialize WebGL with the following calls:

```    // Try to get a WebGL context
canvas = document.getElementById("canvas");

// We don't need a depth buffer.
// See https://www.khronos.org/registry/webgl/specs/1.0/ Section 5.2
gl = WebGLUtils.setupWebGL(canvas, { depth: false });```

If the calls were successful, then we go on to initialize our model data and set up our rendering callback.

#### Visualizing a 3D world

Like in lesson one for Android, we need to define our model data as an array of floating point numbers. These numbers can represent vertex positions, colors, or anything else that we need. Unlike OpenGL ES 2 on Android, WebGL does not support client-side buffers. This means that we need to load all of the data into WebGL using vertex buffer objects (VBOs). Thankfully, this is a pretty trivial step and it will be explained in more detail further below.

Before we transfer the data into WebGL, we’ll define it in client memory first using the Float32Array datatype. These typed arrays are an attempt to increase the performance of Javascript by adding typing information.

```		// Define points for equilateral triangles.
trianglePositions = new Float32Array([
// X, Y, Z,
-0.5, -0.25, 0.0,
0.5, -0.25, 0.0,
0.0, 0.559016994, 0.0]);

// This triangle is red, green, and blue.
triangle1Colors = new Float32Array([
// R, G, B, A
1.0, 0.0, 0.0, 1.0,
0.0, 0.0, 1.0, 1.0,
0.0, 1.0, 0.0, 1.0]);

...

```

All of the triangles can share the same position data, but we’ll define a different set of colors for each triangle.

##### Setting up initial parameters

After defining basic model data, our main function calls startRendering(), which takes care of setting up the viewport, building the shaders, and starting the rendering loop.

###### Setting up the viewport and projection matrix

First, we configure the viewport to be the same size as the canvas viewport. Note that this assumes a canvas that doesn’t change size, since we’re only doing this once.

```	// Set the OpenGL viewport to the same size as the canvas.
gl.viewport(0, 0, canvas.clientWidth, canvas.clientHeight);```

```	// Create a new perspective projection matrix. The height will stay the same
// while the width will vary as per aspect ratio.
var ratio = canvas.clientWidth / canvas.clientHeight;
var left = -ratio;
var right = ratio;
var bottom = -1.0;
var top = 1.0;
var near = 1.0;
var far = 10.0;

mat4.frustum(left, right, bottom, top, near, far, projectionMatrix);```
###### Configuring the view matrix and default parameters

Setting up the viewport and configuring the projection matrix is something we should do whenever the canvas has changed size. The next step is to set the default clear color as well as the view matrix.

```	// Set the background clear color to gray.
gl.clearColor(0.5, 0.5, 0.5, 1.0);

/* Configure camera */
// Position the eye behind the origin.
var eyeX = 0.0;
var eyeY = 0.0;
var eyeZ = 1.5;

// We are looking toward the distance
var lookX = 0.0;
var lookY = 0.0;
var lookZ = -5.0;

// Set our up vector. This is where our head would be pointing were we holding the camera.
var upX = 0.0;
var upY = 1.0;
var upZ = 0.0;

// Set the view matrix. This matrix can be said to represent the camera position.
var eye = vec3.create();
eye[0] = eyeX; eye[1] = eyeY; eye[2] = eyeZ;

var center = vec3.create();
center[0] = lookX; center[1] = lookY; center[2] = lookZ;

var up = vec3.create();
up[0] = upX; up[1] = upY; up[2] = upZ;

mat4.lookAt(eye, center, up, viewMatrix);```

In WebGL we can embed shaders in a few ways: we can embed them as a JavaScript string, we can embed them into the HTML of the page that contains the script, or we can put them in a separate file and link to that file from our script. In this lesson, we take the second approach:

```<script id="vertex_shader" type="x-shader/x-vertex"> uniform mat4 u_MVPMatrix; ```

``` ```

```... </script>```

```<script id="vertex_shader" type="x-shader/x-vertex"> precision mediump float; ```

``` ```

```... </script>```

We can then read in these scripts using the following code snippit:

```		// Read the embedded shader from the document.

{
}

// Pass in the shader source.

I mentioned a bit earlier that WebGL doesn’t support client-side buffers, so we need to upload our data into WebGL itself using buffer objects. This is actually pretty straightforward:

```    // Create buffers in OpenGL's working memory.
trianglePositionBufferObject = gl.createBuffer();
gl.bindBuffer(gl.ARRAY_BUFFER, trianglePositionBufferObject);
gl.bufferData(gl.ARRAY_BUFFER, trianglePositions, gl.STATIC_DRAW);

triangleColorBufferObject1 = gl.createBuffer();
gl.bindBuffer(gl.ARRAY_BUFFER, triangleColorBufferObject1);
gl.bufferData(gl.ARRAY_BUFFER, triangle1Colors, gl.STATIC_DRAW);

...

```

First we create a buffer object using createBuffer(), then we bind the buffer. Then we pass in the data using gl.bufferData() and tell OpenGL that this buffer will be used for static drawing; this hints to OpenGL that we will not be updating this buffer often.

###### WebGL versus OpenGL ES 2

You may have noticed that the WebGL API is a bit different than the base OpenGL ES 2 API: functions and variable names have had their “gl” or “GL_” prefixes removed. This actually makes the API a bit cleaner to use and read. At the same time, some functions have been modified a bit to mesh better with the JavaScript environment.

###### Setting up a rendering callback

We finally kick off the rendering loop by calling window.requestAnimFrame():

```	// Tell the browser we want render() to be called whenever it's time to draw another frame.
window.requestAnimFrame(render, canvas);```
##### Rendering to the screen

The code to render to the screen is pretty much a transpose of the lesson one code for Android. One main difference is that we call window.requestAnimFrame() at the end to request another animation frame.

```	// Request another frame
window.requestAnimFrame(render, canvas);```

#### Recap

If everything went well, you should end up with an animated canvas like the one just below:

Your browser does not support the canvas tag. This is a static example of what would be seen.

If you would like more explanations behind the shaders or other aspects of the program, please be sure to check out lesson one for Android.

##### Debugging

Debugging in JavaScript in the browser is a little more difficult than within an integrated environment such as Eclipse, but it can be done using tools such as Chrome’s inspector. You can also use the WebGL Inspector, which is a plugin that lets you delve into WebGL’s internals and get a better idea of what’s going on.

##### Embedding into WordPress

WebGL can easily be embedded into your posts and pages! You need a canvas, script includes for any third-party libraries, and a script body for your main script (this can also be an include).

Example of a canvas:
`<pre><canvas id="canvas" width="550" height="375">Your browser does not support the canvas tag. This is a static example of what would be seen.</canvas></pre>`

Example of a script include:
`<pre><script type="text/javascript" src="http://www.learnopengles.com/wordpress/wp-content/uploads/2011/06/webgl-utils.js"></script></pre>`

Example of an embedded script:
```<pre><script type="text/javascript"> /** * Lesson_one.js */```

``` ... ```

`</script></pre>`

The <pre> tag is important; otherwise WordPress will mangle your scripts and insert random paragraph tags and other stuff inside. Also, once you’ve inserted this code, you have to stick to using the HTML editor, as the visual editor will also mangle or delete your scripts.

#### Exploring further

Try changing the animation speed, vertex points, or colors, and see what happens!

The full source code for this lesson can be downloaded from the project site on GitHub.

Don’t hesitate to ask any questions or offer feedback, and thanks for stopping by!

```

uniform mat4 u_MVPMatrix;   // A constant representing the combined model/view/projection matrix.

attribute vec4 a_Position;  // Per-vertex position information we will pass in.
attribute vec4 a_Color;     // Per-vertex color information we will pass in.

varying vec4 v_Color;       // This will be passed into the fragment shader.

void main()                 // The entry point for our vertex shader.
{
v_Color = a_Color;      // Pass the color through to the fragment shader.
// It will be interpolated across the triangle.

// gl_Position is a special variable used to store the final position.
// Multiply the vertex by the matrix to get the final point in normalized screen coordinates.
gl_Position = u_MVPMatrix * a_Position;
}

precision mediump float;       // Set the default precision to medium. We don't need as high of a
// precision in the fragment shader.
varying vec4 v_Color;          // This is the color from the vertex shader interpolated across the
// triangle per fragment.
void main()                    // The entry point for our fragment shader.
{
gl_FragColor = v_Color;    // Pass the color directly through the pipeline.
}

/**
* Lesson_one.js
*/

// We make use of the WebGL utility library, which was downloaded from here:
// https://cvs.khronos.org/svn/repos/registry/trunk/public/webgl/sdk/demos/common/webgl-utils.js
//
// It defines two functions which we use here:
//
// // Creates a WebGL context.
// WebGLUtils.setupWebGL(canvas);
//
// Requests an animation callback. See: https://developer.mozilla.org/en/DOM/window.requestAnimationFrame
// window.requestAnimFrame(callback, node);
//
// We also make use of the glMatrix file which can be downloaded from here:
//

/** Hold a reference to the WebGLContext */
var gl = null;

/** Hold a reference to the canvas DOM object. */
var canvas = null;

/**
* Store the model matrix. This matrix is used to move models from object space (where each model can be thought
* of being located at the center of the universe) to world space.
*/
var modelMatrix = mat4.create();

/**
* Store the view matrix. This can be thought of as our camera. This matrix transforms world space to eye space;
* it positions things relative to our eye.
*/
var viewMatrix = mat4.create();

/** Store the projection matrix. This is used to project the scene onto a 2D viewport. */
var projectionMatrix = mat4.create();

/** Allocate storage for the final combined matrix. This will be passed into the shader program. */
var mvpMatrix = mat4.create();

/** Store our model data in a Float32Array buffer. */
var trianglePositions;
var triangle1Colors;
var triangle2Colors;
var triangle3Colors;

/** Store references to the vertex buffer objects (VBOs) that will be created. */
var trianglePositionBufferObject;
var triangleColorBufferObject1;
var triangleColorBufferObject2;
var triangleColorBufferObject3;

/** This will be used to pass in the transformation matrix. */
var mvpMatrixHandle;

/** This will be used to pass in model position information. */
var positionHandle;

/** This will be used to pass in model color information. */
var colorHandle;

/** Size of the position data in elements. */
var positionDataSize = 3;

/** Size of the color data in elements. */
var colorDataSize = 4;

{
var error;

{

{
}

// Pass in the shader source.

// Get the compilation status.

// If the compilation failed, delete the shader.
if (!compiled)
{
}
}

{
throw("Error creating shader " + sourceScriptId + ": " + error);
}

}

// Helper function to link a program
{
// Create a program object and store the handle to it.
var programHandle = gl.createProgram();

if (programHandle != 0)
{
// Bind the vertex shader to the program.

// Bind the fragment shader to the program.

// Bind attributes
gl.bindAttribLocation(programHandle, 0, "a_Position");
gl.bindAttribLocation(programHandle, 1, "a_Color");

// If the link failed, delete the program.
{
gl.deleteProgram(programHandle);
programHandle = 0;
}
}

if (programHandle == 0)
{
throw("Error creating program.");
}

return programHandle;
}

//Called when we have the context
function startRendering()
{
/* Configure viewport */
// Set the OpenGL viewport to the same size as the canvas.
gl.viewport(0, 0, canvas.clientWidth, canvas.clientHeight);

// Create a new perspective projection matrix. The height will stay the same
// while the width will vary as per aspect ratio.
var ratio = canvas.clientWidth / canvas.clientHeight;
var left = -ratio;
var right = ratio;
var bottom = -1.0;
var top = 1.0;
var near = 1.0;
var far = 10.0;

mat4.frustum(left, right, bottom, top, near, far, projectionMatrix);

/* Configure general parameters */

// Set the background clear color to gray.
gl.clearColor(0.5, 0.5, 0.5, 1.0);

/* Configure camera */
// Position the eye behind the origin.
var eyeX = 0.0;
var eyeY = 0.0;
var eyeZ = 1.5;

// We are looking toward the distance
var lookX = 0.0;
var lookY = 0.0;
var lookZ = -5.0;

// Set our up vector. This is where our head would be pointing were we holding the camera.
var upX = 0.0;
var upY = 1.0;
var upZ = 0.0;

// Set the view matrix. This matrix can be said to represent the camera position.
var eye = vec3.create();
eye[0] = eyeX; eye[1] = eyeY; eye[2] = eyeZ;
var center = vec3.create();
center[0] = lookX; center[1] = lookY; center[2] = lookZ;
var up = vec3.create();
up[0] = upX; up[1] = upY; up[2] = upZ;
mat4.lookAt(eye, center, up, viewMatrix);

// Create a program object and store the handle to it.

// Set program handles. These will later be used to pass in values to the program.
mvpMatrixHandle = gl.getUniformLocation(programHandle, "u_MVPMatrix");
positionHandle = gl.getAttribLocation(programHandle, "a_Position");
colorHandle = gl.getAttribLocation(programHandle, "a_Color");

// Tell OpenGL to use this program when rendering.
gl.useProgram(programHandle);

// Create buffers in OpenGL's working memory.
trianglePositionBufferObject = gl.createBuffer();
//    checkError();
gl.bindBuffer(gl.ARRAY_BUFFER, trianglePositionBufferObject);
//    checkError();
gl.bufferData(gl.ARRAY_BUFFER, trianglePositions, gl.STATIC_DRAW);
//    checkError();

triangleColorBufferObject1 = gl.createBuffer();
//    checkError();
gl.bindBuffer(gl.ARRAY_BUFFER, triangleColorBufferObject1);
//    checkError();
gl.bufferData(gl.ARRAY_BUFFER, triangle1Colors, gl.STATIC_DRAW);
//    checkError();

triangleColorBufferObject2 = gl.createBuffer();
//    checkError();
gl.bindBuffer(gl.ARRAY_BUFFER, triangleColorBufferObject2);
//    checkError();
gl.bufferData(gl.ARRAY_BUFFER, triangle2Colors, gl.STATIC_DRAW);
//    checkError();

triangleColorBufferObject3 = gl.createBuffer();
//    checkError();
gl.bindBuffer(gl.ARRAY_BUFFER, triangleColorBufferObject3);
//    checkError();
gl.bufferData(gl.ARRAY_BUFFER, triangle3Colors, gl.STATIC_DRAW);
//    checkError();

// Tell the browser we want render() to be called whenever it's time to draw another frame.
window.requestAnimFrame(render, canvas);
}

// Callback called each time the browser wants us to draw another frame
function render(time)
{
// Clear the canvas
gl.clear(gl.COLOR_BUFFER_BIT);

// Do a complete rotation every 10 seconds.
var time = Date.now() % 10000;
var angleInDegrees = (360.0 / 10000.0) * time;
var angleInRadians = angleInDegrees / 57.2957795;

var xyz = vec3.create();

// Draw the triangle facing straight on.
mat4.identity(modelMatrix);
drawTriangle(triangleColorBufferObject1);

// Draw one translated a bit down and rotated to be flat on the ground.
mat4.identity(modelMatrix);
xyz[0] = 0; xyz[1] = -1; xyz[2] = 0;
mat4.translate(modelMatrix, xyz);
mat4.rotateX(modelMatrix, 90 / 57.2957795);
xyz[0] = 0; xyz[1] = 0; xyz[2] = 1;
drawTriangle(triangleColorBufferObject2);

// Draw one translated a bit to the right and rotated to be facing to the left.
mat4.identity(modelMatrix);
xyz[0] = 1; xyz[1] = 0; xyz[2] = 0;
mat4.translate(modelMatrix, xyz);
mat4.rotateY(modelMatrix, 90 / 57.2957795);
xyz[0] = 0; xyz[1] = 0; xyz[2] = 1;
drawTriangle(triangleColorBufferObject3);

// Send the commands to WebGL
gl.flush();

// Request another frame
window.requestAnimFrame(render, canvas);
}

function checkError()
{
var error = gl.getError();

if (error)
{
throw("error: " + error);
}
}

// Draws a triangle from the given vertex data.
function drawTriangle(triangleColorBufferObject)
{
// Pass in the position information
//	console.log("positionHandle=" +  positionHandle);
//	console.log("colorHandle=" +  colorHandle);
gl.enableVertexAttribArray(positionHandle);
//    checkError();

gl.bindBuffer(gl.ARRAY_BUFFER, trianglePositionBufferObject);
gl.vertexAttribPointer(positionHandle, positionDataSize, gl.FLOAT, false,
0, 0);
//    checkError();

// Pass in the color information
gl.enableVertexAttribArray(colorHandle);
//    checkError();

gl.bindBuffer(gl.ARRAY_BUFFER, triangleColorBufferObject);
gl.vertexAttribPointer(colorHandle, colorDataSize, gl.FLOAT, false,
0, 0);
//    checkError();

// This multiplies the view matrix by the model matrix, and stores the result in the modelview matrix
// (which currently contains model * view).
mat4.multiply(viewMatrix, modelMatrix, mvpMatrix);
// Matrix.multiplyMM(mMVPMatrix, 0, mViewMatrix, 0, mModelMatrix, 0);

// This multiplies the modelview matrix by the projection matrix, and stores the result in the MVP matrix
// (which now contains model * view * projection).
mat4.multiply(projectionMatrix, mvpMatrix, mvpMatrix);

// Matrix.multiplyMM(mMVPMatrix, 0, mProjectionMatrix, 0, mMVPMatrix, 0);

gl.uniformMatrix4fv(mvpMatrixHandle, false, mvpMatrix);
//    checkError();
gl.drawArrays(gl.TRIANGLES, 0, 3);
//    checkError();
//    console.log("Made it past one frame");
}

// Main entry point
function main()
{
// Try to get a WebGL context
canvas = document.getElementById("canvas");

// We don't need a depth buffer. See https://www.khronos.org/registry/webgl/specs/1.0/ Section 5.2 for more info.
gl = WebGLUtils.setupWebGL(canvas, { depth: false });

if (gl != null)
{
// Init model data.

// Define points for equilateral triangles.
trianglePositions = new Float32Array([
// X, Y, Z,
-0.5, -0.25, 0.0,
0.5, -0.25, 0.0,
0.0, 0.559016994, 0.0]);

// This triangle is red, green, and blue.
triangle1Colors = new Float32Array([
// R, G, B, A
1.0, 0.0, 0.0, 1.0,
0.0, 0.0, 1.0, 1.0,
0.0, 1.0, 0.0, 1.0]);

// This triangle is yellow, cyan, and magenta.
triangle2Colors = new Float32Array([
// R, G, B, A
1.0, 1.0, 0.0, 1.0,
0.0, 1.0, 1.0, 1.0,
1.0, 0.0, 1.0, 1.0]);

// This triangle is white, gray, and black.
triangle3Colors = new Float32Array([
// R, G, B, A
1.0, 1.0, 1.0, 1.0,
0.5, 0.5, 0.5, 1.0,
0.0, 0.0, 0.0, 1.0]);

startRendering();
}
}

// Execute the main entry point
main();
```

## Android Lesson One: Getting Started

This is the first tutorial on using OpenGL ES 2 on Android. In this lesson, we’re going to go over the code step-by-step, and look at how to create an OpenGL ES 2 context and draw to the screen. We’ll also take a look at what shaders are and how they work, as well as how matrices are used to transform the scene into the image you see on the screen. Finally, we’ll look at what you need to add to the manifest to let the Android market know that you’re using OpenGL ES 2.

#### Prerequisites

Before we begin, you’ll want to make sure you have the following tools installed on your machine:

Unfortunately, the Android emulator does not support OpenGL ES 2, so you’ll need access to an actual Android device in order to run the tutorial. The Android Emulator now supports OpenGL ES 2 in recent versions of the Android SDK. Most recent devices should also support OpenGL ES 2 these days, so all you need to do is enable developer mode and hook the phone up to your machine.

#### Assumptions

The reader should be familiar with Android and Java on a basic level. The Android Tutorials are a good place to start.

#### Getting started

We’ll go over all of the code below and explain what each part does. You can copy the code on a segment by segment basis by creating your own project, or you can download the completed project at the end of the lesson. Once you have the tools installed, go ahead and create a new Android project in Eclipse. The names don’t matter, but for the lesson I will be referring to the main activity as LessonOneActivity.

Let’s take a look at the code:

```	/** Hold a reference to our GLSurfaceView */
private GLSurfaceView mGLSurfaceView;```

The GLSurfaceView is a special view which manages OpenGL surfaces for us and draws it into the Android view system. It also adds a lot of features which make it easier to use OpenGL, including but not limited to:

• It provides a dedicated render thread for OpenGL so that the main thread is not stalled.
• It supports continuous or on-demand rendering.
• It takes care of the screen setup for you using EGL, the interface between OpenGL and the underlying window system.

The GLSurfaceView makes setting up and using OpenGL from Android relatively painless.

```	@Override
public void onCreate(Bundle savedInstanceState)
{
super.onCreate(savedInstanceState);

mGLSurfaceView = new GLSurfaceView(this);

// Check if the system supports OpenGL ES 2.0.
final ActivityManager activityManager = (ActivityManager) getSystemService(Context.ACTIVITY_SERVICE);
final ConfigurationInfo configurationInfo = activityManager.getDeviceConfigurationInfo();
final boolean supportsEs2 = configurationInfo.reqGlEsVersion >= 0x20000;

if (supportsEs2)
{
// Request an OpenGL ES 2.0 compatible context.
mGLSurfaceView.setEGLContextClientVersion(2);

// Set the renderer to our demo renderer, defined below.
mGLSurfaceView.setRenderer(new LessonOneRenderer());
}
else
{
// This is where you could create an OpenGL ES 1.x compatible
// renderer if you wanted to support both ES 1 and ES 2.
return;
}

setContentView(mGLSurfaceView);
}```

The onCreate() of our activity is the important part where our OpenGL context gets created and where everything starts happening. In our onCreate(), the first thing we do after calling the superclass is creating our GLSurfaceView. We then need to figure out if the system supports OpenGL ES 2. To do this, we get an ActivityManager instance which lets us interact with the global system state. We can then use this to get the device configuration info, which will tell us if the device supports OpenGL ES 2.

Once we know if the device supports OpenGL ES 2 or not, we tell the surface view we want an OpenGL ES 2 compatible surface and then we pass in a custom renderer. This renderer will be called by the system whenever it’s time to adjust the surface or draw a new frame. We can also support OpenGL Es 1.x by passing in a different renderer, though we would need to write different code as the APIs are different. For this lesson we’ll only look at supporting OpenGL ES 2.

Finally, we set the content view to our GLSurfaceView, which tells Android that the activity’s contents should be filled by our OpenGL surface. To get into OpenGL, it’s as easy as that!

```	@Override
protected void onResume()
{
// The activity must call the GL surface view's onResume() on activity onResume().
super.onResume();
mGLSurfaceView.onResume();
}

@Override
protected void onPause()
{
// The activity must call the GL surface view's onPause() on activity onPause().
super.onPause();
mGLSurfaceView.onPause();
}```

GLSurfaceView requires that we call its onResume() and onPause() methods whenever the parent Activity has its own onResume() and onPaused() called. We add in the calls here to round out our activity.

#### Visualizing a 3D world

In this section, we’ll start looking at how OpenGL ES 2 works and how we can start drawing stuff onto the screen. In the activity we passed in a custom GLSurfaceView.Renderer to the GLSurfaceView, which will be defined here. The renderer has three important methods which will be automatically called by the system whenever events happen:

public void onSurfaceCreated(GL10 glUnused, EGLConfig config)
This method is called when the surface is first created. It will also be called if we lose our surface context and it is later recreated by the system.
public void onSurfaceChanged(GL10 glUnused, int width, int height)
This is called whenever the surface changes; for example, when switching from portrait to landscape. It is also called after the surface has been created.
public void onDrawFrame(GL10 glUnused)
This is called whenever it’s time to draw a new frame.

You may have noticed that the GL10 instance passed in is referred to as glUnused. We don’t use this when drawing using OpenGL ES 2; instead, we use the static methods of the class GLES20. The GL10 parameter is only there because the same interface is used for OpenGL ES 1.x.

Before our renderer can display anything, we’ll need to have something to display. In OpenGL ES 2, we pass in stuff to display by specifying arrays of numbers. These numbers can represent positions, colors, or anything else we need them to. In this demo, we’ll display three triangles.

```	// New class members
/** Store our model data in a float buffer. */
private final FloatBuffer mTriangle1Vertices;
private final FloatBuffer mTriangle2Vertices;
private final FloatBuffer mTriangle3Vertices;

/** How many bytes per float. */
private final int mBytesPerFloat = 4;

/**
* Initialize the model data.
*/
public LessonOneRenderer()
{
// This triangle is red, green, and blue.
final float[] triangle1VerticesData = {
// X, Y, Z,
// R, G, B, A
-0.5f, -0.25f, 0.0f,
1.0f, 0.0f, 0.0f, 1.0f,

0.5f, -0.25f, 0.0f,
0.0f, 0.0f, 1.0f, 1.0f,

0.0f, 0.559016994f, 0.0f,
0.0f, 1.0f, 0.0f, 1.0f};

...

// Initialize the buffers.
mTriangle1Vertices = ByteBuffer.allocateDirect(triangle1VerticesData.length * mBytesPerFloat)
.order(ByteOrder.nativeOrder()).asFloatBuffer();

...

mTriangle1Vertices.put(triangle1VerticesData).position(0);

...
}```

So, what’s all this about? If you’ve ever used OpenGL 1, you might be used to doing things this way:

```glBegin(GL_TRIANGLES);
glVertex3f(-0.5f, -0.25f, 0.0f);
glColor3f(1.0f, 0.0f, 0.0f);
...
glEnd();```

Things don’t work that way in OpenGL ES 2. Instead of defining points via a bunch of method calls, we define an array instead. Let’s take a look at our array again:

```final float[] triangle1VerticesData = {
// X, Y, Z,
// R, G, B, A
-0.5f, -0.25f, 0.0f,
1.0f, 0.0f, 0.0f, 1.0f,
...```

This represents one point of the triangle. We’ve set things up so that the first three numbers represent the position (X, Y, and Z), and the last four numbers represent the color (red, green, blue, and alpha (transparency)). You don’t need to worry too much about how this array is defined; just remember that when we want to draw stuff in OpenGL ES 2, we need to pass it data in chunks instead of passing it in one at a time.

##### Understanding buffers
```// Initialize the buffers.
mTriangle1Vertices = ByteBuffer.allocateDirect(triangle1VerticesData.length * mBytesPerFloat)
.order(ByteOrder.nativeOrder()).asFloatBuffer();
...
mTriangle1Vertices.put(triangle1VerticesData).position(0);```

We do our coding in Java on Android, but the underlying implementation of OpenGL ES 2 is actually written in C. Before we pass our data to OpenGL, we need to convert it into a form that it’s going to understand. Java and the native system might not store their bytes in the same order, so we use a special set of buffer classes and create a ByteBuffer large enough to hold our data, and tell it to store its data using the native byte order. We then convert it into a FloatBuffer so that we can use it to hold floating-point data. Finally, we copy our array into the buffer.

This buffer stuff might seem confusing (it was to me when I first came across it!), but just remember that it’s an extra step we need to do before passing our data to OpenGL. Our buffers are now ready to be used to pass data into OpenGL.

As a side note, float buffers are slow on Froyo and moderately faster on Gingerbread, so you probably don’t want to be changing them too often.

##### Understanding matrices
```	 // New class definitions
/**
* Store the view matrix. This can be thought of as our camera. This matrix transforms world space to eye space;
* it positions things relative to our eye.
*/
private float[] mViewMatrix = new float[16];

@Override
public void onSurfaceCreated(GL10 glUnused, EGLConfig config)
{
// Set the background clear color to gray.
GLES20.glClearColor(0.5f, 0.5f, 0.5f, 0.5f);

// Position the eye behind the origin.
final float eyeX = 0.0f;
final float eyeY = 0.0f;
final float eyeZ = 1.5f;

// We are looking toward the distance
final float lookX = 0.0f;
final float lookY = 0.0f;
final float lookZ = -5.0f;

// Set our up vector. This is where our head would be pointing were we holding the camera.
final float upX = 0.0f;
final float upY = 1.0f;
final float upZ = 0.0f;

// Set the view matrix. This matrix can be said to represent the camera position.
// NOTE: In OpenGL 1, a ModelView matrix is used, which is a combination of a model and
// view matrix. In OpenGL 2, we can keep track of these matrices separately if we choose.
Matrix.setLookAtM(mViewMatrix, 0, eyeX, eyeY, eyeZ, lookX, lookY, lookZ, upX, upY, upZ);

...```

Another ‘fun’ topic is matrices! These will become your best friends whenever you do 3D programming, so you’ll want to get to know them well.

When our surface is created, the first thing we do is set our clear color to gray. The alpha part has also been set to gray, but we’re not doing alpha blending in this lesson so this value is unused. We only need to set the clear color once since we will not be changing it later.

The second thing we do is setup our view matrix. There are several different kinds of matrices we use and they all do something important:

1. The model matrix. This matrix is used to place a model somewhere in the “world”. For example, if you have a model of a car and you want it located 1000 meters to the east, you will use the model matrix to do this.
2. The view matrix. This matrix represents the camera. If we want to view our car which is 1000 meters to the east, we’ll have to move ourselves 1000 meters to the east as well (another way of thinking about it is that we remain stationary, and the rest of the world moves 1000 meters to the west). We use the view matrix to do this.
3. The projection matrix. Since our screens are flat, we need to do a final transformation to “project” our view onto our screen and get that nice 3D perspective. This is what the projection matrix is used for.

A good explanation of this can be found over at SongHo’s OpenGL Tutorials. I recommend reading it a few times until you grasp the idea well; don’t worry, it took me a few reads as well!

In OpenGL 1, the model and view matrices are combined and the camera is assumed to be at (0, 0, 0) and facing the -Z direction.

We don’t need to construct these matrices by hand. Android has a Matrix helper class which can do the heavy lifting for us. Here, I create a view matrix for a camera which is positioned behind the origin and looking toward the distance.

##### Defining vertex and fragment shaders
```		final String vertexShader =
"uniform mat4 u_MVPMatrix;      \n"		// A constant representing the combined model/view/projection matrix.

+ "attribute vec4 a_Position;     \n"		// Per-vertex position information we will pass in.
+ "attribute vec4 a_Color;        \n"		// Per-vertex color information we will pass in.

+ "varying vec4 v_Color;          \n"		// This will be passed into the fragment shader.

+ "void main()                    \n"		// The entry point for our vertex shader.
+ "{                              \n"
+ "   v_Color = a_Color;          \n"		// Pass the color through to the fragment shader.
// It will be interpolated across the triangle.
+ "   gl_Position = u_MVPMatrix   \n" 	// gl_Position is a special variable used to store the final position.
+ "               * a_Position;   \n"     // Multiply the vertex by the matrix to get the final point in
+ "}                              \n";    // normalized screen coordinates.```

In OpenGL ES 2, anything we want to display on the screen is first going to have to go through a vertex and fragment shader. The good thing is that these shaders are really not as complicated as they appear. Vertex shaders perform operations on each vertex, and the results of these operations are used in the fragment shaders which do additional calculations per pixel.

Each shader basically consists of input, output, and a program. First we define a uniform, which is a combined matrix containing all of our transformations. This is a constant across all vertices and is used to project them onto the screen. Then we define two attributes for position and color. These attributes will be read in from the buffer we defined earlier on, and specify the position and color of each vertex. We then define a varying, which interpolates values across the triangle and passes it on to the fragment shader. When it gets to the fragment shader, it will hold an interpolated value for each pixel.

Let’s say we defined a triangle with each point being red, green, and blue, and we sized it so that it will take up 10 pixels on the screen. When the fragment shader runs, it will contain a different varying color for each pixel. At one point, that varying will be red, but halfway between red and blue it may be a more purplish color.

Aside from setting the color, we also tell OpenGL what the final position of the vertex should be on the screen. Then we define the fragment shader:

```		final String fragmentShader =
"precision mediump float;       \n"		// Set the default precision to medium. We don't need as high of a
// precision in the fragment shader.
+ "varying vec4 v_Color;          \n"		// This is the color from the vertex shader interpolated across the
// triangle per fragment.
+ "void main()                    \n"		// The entry point for our fragment shader.
+ "{                              \n"
+ "   gl_FragColor = v_Color;     \n"		// Pass the color directly through the pipeline.
+ "}                              \n";```

This is the fragment shader which will actually put stuff on the screen. In this shader, we grab the varying color from the vertex shader, and just pass it straight through to OpenGL. The point is already interpolated per pixel since the fragment shader runs for each pixel that will be drawn.

More information can be found on the OpenGL ES 2 quick reference card.

```		// Load in the vertex shader.

{
// Pass in the shader source.

// Get the compilation status.
final int[] compileStatus = new int[1];

// If the compilation failed, delete the shader.
if (compileStatus[0] == 0)
{
}
}

{
throw new RuntimeException("Error creating vertex shader.");
}```

First, we create the shader object. If this succeeded, we’ll get a reference to the object. Then we use this reference to pass in the shader source code, and then we compile it. We can obtain the status from OpenGL and see if it compiled successfully. If there were errors, we can use GLES20.glGetShaderInfoLog(shader) to find out why. We follow the same steps to load the fragment shader.

```		// Create a program object and store the handle to it.
int programHandle = GLES20.glCreateProgram();

if (programHandle != 0)
{
// Bind the vertex shader to the program.

// Bind the fragment shader to the program.

// Bind attributes
GLES20.glBindAttribLocation(programHandle, 0, "a_Position");
GLES20.glBindAttribLocation(programHandle, 1, "a_Color");

final int[] linkStatus = new int[1];

// If the link failed, delete the program.
{
GLES20.glDeleteProgram(programHandle);
programHandle = 0;
}
}

if (programHandle == 0)
{
throw new RuntimeException("Error creating program.");
}```

Before we can use our vertex and fragment shader, we need to bind them together into a program. This is what connects the output of the vertex shader with the input of the fragment shader. It’s also what lets us pass in input from our program and use the shader to draw our shapes.

We create a new program object, and if that succeeded, we then attach our shaders. We want to pass in the position and color as attributes, so we need to bind these attributes. We then link the shaders together.

```	//New class members
/** This will be used to pass in the transformation matrix. */
private int mMVPMatrixHandle;

/** This will be used to pass in model position information. */
private int mPositionHandle;

/** This will be used to pass in model color information. */
private int mColorHandle;

@Override
public void onSurfaceCreated(GL10 glUnused, EGLConfig config)
{
...

// Set program handles. These will later be used to pass in values to the program.
mMVPMatrixHandle = GLES20.glGetUniformLocation(programHandle, "u_MVPMatrix");
mPositionHandle = GLES20.glGetAttribLocation(programHandle, "a_Position");
mColorHandle = GLES20.glGetAttribLocation(programHandle, "a_Color");

// Tell OpenGL to use this program when rendering.
GLES20.glUseProgram(programHandle);
}```

After we successfully linked our program, we finish up with a couple more tasks so we can actually use it. The first task is obtaining references so we can pass data into the program. Then we tell OpenGL to use this program when drawing. Since we only use one program in this lesson, we can put this in the onSurfaceCreated() instead of the onDrawFrame().

##### Setting the perspective projection
```	// New class members
/** Store the projection matrix. This is used to project the scene onto a 2D viewport. */
private float[] mProjectionMatrix = new float[16];

@Override
public void onSurfaceChanged(GL10 glUnused, int width, int height)
{
// Set the OpenGL viewport to the same size as the surface.
GLES20.glViewport(0, 0, width, height);

// Create a new perspective projection matrix. The height will stay the same
// while the width will vary as per aspect ratio.
final float ratio = (float) width / height;
final float left = -ratio;
final float right = ratio;
final float bottom = -1.0f;
final float top = 1.0f;
final float near = 1.0f;
final float far = 10.0f;

Matrix.frustumM(mProjectionMatrix, 0, left, right, bottom, top, near, far);
}```

Our onSurfaceChanged() is called at least once and also whenever our surface is changed. Since we only need to reset our projection matrix whenever the screen we’re projecting onto has changed, onSurfaceChanged() is an ideal place to do it.

##### Drawing stuff to the screen!
```	// New class members
/**
* Store the model matrix. This matrix is used to move models from object space (where each model can be thought
* of being located at the center of the universe) to world space.
*/
private float[] mModelMatrix = new float[16];

@Override
public void onDrawFrame(GL10 glUnused)
{
GLES20.glClear(GLES20.GL_DEPTH_BUFFER_BIT | GLES20.GL_COLOR_BUFFER_BIT);

// Do a complete rotation every 10 seconds.
long time = SystemClock.uptimeMillis() % 10000L;
float angleInDegrees = (360.0f / 10000.0f) * ((int) time);

// Draw the triangle facing straight on.
Matrix.setIdentityM(mModelMatrix, 0);
Matrix.rotateM(mModelMatrix, 0, angleInDegrees, 0.0f, 0.0f, 1.0f);
drawTriangle(mTriangle1Vertices);

...
}```

This is where stuff actually get displayed on the screen. We clear the screen so we don’t get any weird hall of mirror effects, and we want our triangles to animate smoothly so we rotate them using time. Whenever you animate something on the screen, it’s usually better to use time instead of framerate.

The actual drawing is done in drawTriangle:

```	// New class members
/** Allocate storage for the final combined matrix. This will be passed into the shader program. */
private float[] mMVPMatrix = new float[16];

/** How many elements per vertex. */
private final int mStrideBytes = 7 * mBytesPerFloat;

/** Offset of the position data. */
private final int mPositionOffset = 0;

/** Size of the position data in elements. */
private final int mPositionDataSize = 3;

/** Offset of the color data. */
private final int mColorOffset = 3;

/** Size of the color data in elements. */
private final int mColorDataSize = 4;

/**
* Draws a triangle from the given vertex data.
*
* @param aTriangleBuffer The buffer containing the vertex data.
*/
private void drawTriangle(final FloatBuffer aTriangleBuffer)
{
// Pass in the position information
aTriangleBuffer.position(mPositionOffset);
GLES20.glVertexAttribPointer(mPositionHandle, mPositionDataSize, GLES20.GL_FLOAT, false,
mStrideBytes, aTriangleBuffer);

GLES20.glEnableVertexAttribArray(mPositionHandle);

// Pass in the color information
aTriangleBuffer.position(mColorOffset);
GLES20.glVertexAttribPointer(mColorHandle, mColorDataSize, GLES20.GL_FLOAT, false,
mStrideBytes, aTriangleBuffer);

GLES20.glEnableVertexAttribArray(mColorHandle);

// This multiplies the view matrix by the model matrix, and stores the result in the MVP matrix
// (which currently contains model * view).
Matrix.multiplyMM(mMVPMatrix, 0, mViewMatrix, 0, mModelMatrix, 0);

// This multiplies the modelview matrix by the projection matrix, and stores the result in the MVP matrix
// (which now contains model * view * projection).
Matrix.multiplyMM(mMVPMatrix, 0, mProjectionMatrix, 0, mMVPMatrix, 0);

GLES20.glUniformMatrix4fv(mMVPMatrixHandle, 1, false, mMVPMatrix, 0);
GLES20.glDrawArrays(GLES20.GL_TRIANGLES, 0, 3);
}```

Do you remember those buffers we defined when we originally created our renderer? We’re finally going to be able to use them. We need to tell OpenGL how to use this data using GLES20.glVertexAttribPointer(). Let’s look at the first call.

```		// Pass in the position information
aTriangleBuffer.position(mPositionOffset);
GLES20.glVertexAttribPointer(mPositionHandle, mPositionDataSize, GLES20.GL_FLOAT, false,
mStrideBytes, aTriangleBuffer);
GLES20.glEnableVertexAttribArray(mPositionHandle);```

We set our buffer position to the position offset, which is at the beginning of the buffer. We then tell OpenGL to use this data and feed it into the vertex shader and apply it to our position attribute. We also need to tell OpenGL how many elements there are between each vertex, or the stride.

Note: The stride needs to be defined in bytes. Although we have 7 elements (3 for the position, 4 for the color) between vertices, we actually have 28 bytes, since each floating point number takes up 4 bytes. Forgetting this step may not cause any errors, but you will be wondering why you don’t see anything on the screen.

Finally, we enable the vertex attribute and move on to the next attribute. A little bit further down we build a combined matrix to project points onto the screen. We could do this in the vertex shader, too, but since it only needs to be done once we may as well just cache the result. We pass in the final matrix to the vertex shader using GLES20.glUniformMatrix4fv() and GLES20.glDrawArrays() converts our points into a triangle and draws it on the screen.

#### Recap

Whew! This was a big lesson, and kudos to you if you made it all the way through. We learned how to create our OpenGL context, pass shape data, load in a vertex and pixel shader, set up our transformation matrices, and finally bring it all together. If everything went well, you should see something similar to the screenshot on the right.

This lesson was a lot to digest and you may need to go over the steps a few times to understand it well. OpenGL ES 2 takes more setup work to get going, but once you’ve been through the process a few times you’ll remember the flow by the back of your hand.

##### Publishing on Android Market

When developing apps, we wouldn’t want people unable to run those apps to see them in the market, otherwise we could end up with a lot of bad reviews and ratings when the app crashes on their device. To prevent an OpenGL ES 2 app from appearing on a device which doesn’t support it, you can add this to your manifest:

```<uses-feature android:glEsVersion="0x00020000" android:required="true" />```

This tells the Market that your app requires OpenGL ES 2, and it will hide your app from devices which don’t support it.

#### Exploring further

Try changing the animation speed, vertex points, or colors, and see what happens!

The full source code for this lesson can be downloaded from the project site on GitHub.

I also recommend looking at the code in ApiDemos, which can be found under the Samples folder in your Android SDK. The code in there helped me out a lot when I was preparing this lesson.

Please don’t hesitate to ask any questions or offer feedback, and thanks for stopping by!