We will note some of the new features supported by WebGL 2.0 at the end of the course

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We will use WebGL 1.0. WebGL 2.0 is now being supported by most browsers but requires a better GPU so may not run on older computers or on most cell phones and tablets. See http://webglstats.com/. We will note some of the new features supported by WebGL 2.0 at the end of the course. three.js is starting to support WebGL 2.0.



OpenGL is a library of function calls for doing computer graphics. With it, you can create interactive applications that render high-quality color images composed of 2D and 3D geometric objects and images.

Additionally, the OpenGL API is independent of all operating systems, and their associated windowing systems. That means that the part of your application that draws can be platform independent. However, for OpenGL to be able to render, it needs a window to draw into. Generally, this is controlled by the windowing system on whatever platform you are working on. Likewise, interaction is not part of the API. Although the graphics part of the application is independent of the OS, applications must be recompiled on each architecture.


OpenGL ES is a smaller version of OpenGL that was designed for embedded systems which did have the hardware capability to run desktop OpenGL. OpenGL ES 2.0 is based on desktop OpenGL2.0 and thus is shader based. Each application must provide both a vertex shader and a fragment shader. Functions from the fixed function pipeline that were in ES 1.0 have been deprecated. OpenGL ES has become the standard API for developing 3D cell phone applications.

WebGL runs within the browser so is independent of the operating and window systems. It is an implementation of ES 2.0 and with a few exceptions the functions are identical. Because WebGL uses the HTML canvas element, it does not require system dependent packages for opening windows and interaction.


Basic graphics concepts are those in fundamentals course, including the basics of image formation.

For this course, experience with C/C++, Java, Python should suffice.

We assume you are comfortable running remote applications in a browser.


Cube is one of the geometries built into three.js. Because WebGL has a limited set of primitives, with WebGL we must model the cube with triangles. three.js includes libraries with interactive controls. With WebGL we can use libraries available on the Web such as jQuery or code devices through HTML. The standard lighting models are included with three.js. With WebGL we can build our lighting models in a variety of ways including within the shaders. Both three.js and WebGL support texture mapping using either images available in standard formats (gif, jpeg) or images defined in the code.


The initial version of OpenGL implemented a fixed-function pipeline, in which all the operations that OpenGL supported were fully-defined, and an application could only modify their operation by changing a set of input values (like colors or positions). The other point of a fixed-function pipeline is that the order of operations was always the same – that is, you can’t reorder the sequence operations occur. Modern GPUs and their features have diverged from this pipeline, and support of these previous versions of OpenGL are for supporting current applications. If you’re developing a new application, we strongly recommend using the techniques that we’ll discuss. Those techniques can be more flexible, and will likely perform better than using one of these early versions of OpenGL since they can take advantage of the capabilities of recent Graphics Processing Units (GPUs). To allow applications to gain access to these new GPU features, OpenGL version 2.0 officially added

programmable shaders into the graphics pipeline. This version of the pipeline allowed an application to create small programs, called shaders, that were responsible for implementing the features required by the application. In the 2.0 version of the pipeline, two programmable stages were made available:

•  vertex shading enabled the application full control over manipulation of the 3D geometry provided by the application

•  fragment shading provided the application capabilities for shading pixels (the terms classically used for determining a pixel’s color).

Until OpenGL 3.0, features have only been added (but never removed) from OpenGL, providing a lot of application backwards compatibility (up to the use of extensions). OpenGL version 3.0 introduced the

mechanisms for removing features from OpenGL, called the deprecation model.



WebGL is becoming increasingly more important because it is

supported by all browsers. Besides the advantage of being able to run without recompilation across platforms, it can easily be integrated with other Web applications and make use of a variety of portable packages available over the Web.

On Windows systems, Chrome and Firefox use an intermediate layer called ANGLE, which takes OpenGL calls and turns them into DirectX calls. This is done because the DirectX drivers are generally more efficient for Windows, since they've undergone more development.

Command-line options can disable the use of ANGLE.


Once our JS and HTML code is interpreted and executes with a basic OpenGL pipeline. Generally speaking, data flows from your application through the GPU to generate an image in the frame buffer. Your application will provide vertices, which are collections of data that are composed to form geometric objects, to the OpenGL pipeline. The vertex processing stage uses a vertex shader to process each vertex, doing any computations necessary to determine where in the frame buffer each piece of geometry should go.

After all the vertices for a piece of geometry are processed, the rasterizer determines which pixels in the frame buffer are affected by the geometry, and for each pixel, the fragment processing stage is employed, where the fragment shader runs to determine the final color of the pixel.

In your OpenGL/WebGL applications, you’ll usually need to do the following tasks:

•  specify the vertices for your geometry

•  load vertex and fragment shaders (and other shaders, if you’re using them as well)

•  issue your geometry to engage the pipeline for processing

Of course, OpenGL and WebGL are capable of many other operations as well, many of which are outside of the scope of this introductory course. We have included references at the end of the notes for your further research and development.



Although OpenGL source code for rendering should be the same across multiple platforms, the code must be recompiled for each

architecture, In addition, the non-rendering parts of an application such as opening windows and input are not part of of OpenGL and can be very different on different systems. Almost all OpenGL applications are designed to run locally on the computer on which they live.


A typical WebGL application consists of a mixture of HTML5, JavaScript and GLSL (shader) code. The application can be located almost

anywhere and is accessed through its URL . All browsers can run JavaScript and all modern browsers support HTML. The rendering part of the application is in JavaScript and renders into the HTML5 Canvas element. Thus, the WebGL code is obtained from a server (either locally or remote) and is compiled by the browser’s JavaScript engine into code that run on the local CPU and GPU.



OpenGL is not object oriented. three.js is a scene graph which supports a variety of objects including geometric objects (boxes, spheres,

planes), cameras, materials, lights and controls and methods for creating and manipulating them. three.js also supports high level functionality such as shadows and kinematics. three.js supports rendering by WebGL, SVG and the HTML Canvas although most applications use WebGL.


WebGL can be looked at in two ways> First as an API that we can use to build graphics applications. Second as a rendering library that that can be accessed by other APIs such three.js.


You can also download the demo programs from http://bit.ly/





You’ll find that a few techniques for programming with modern WebGL goes a long way. In fact, most programs – in terms of WebGL activity – are very repetitive. Differences usually occur in how objects are

rendered, and that’s mostly handled in your shaders.

There four steps you’ll use for rendering a geometric object are as follows:

1.  First, you’ll load and create WebGL shader programs from shader source programs you create

2.  Next, you will need to load the data for your objects into WebGL’s memory. You do this by creating buffer objects and loading data into them.

3.  Continuing, WebGL needs to be told how to interpret the data in your buffer objects and associate that data with variables that you’ll use in your shaders. We call this shader plumbing.

4.  Finally, with your data initialized and shaders set up, you’ll render your objects



HTML (hypertext markup language) is the standard for describing Web pages. A page consists of a several elements which are described by tags, HTML5 introduced the canvas element which provides a window that WebGL can render into. Note other applications can also render into the canvas or on the same page.

Generally, we use HTML to set up the canvas, bring in the necessary files and set up other page elements such as buttons and sliders. We can embed our JavaScript WebGL code in the same file or have the HTML file load the JavaScript from a file or URL. Likewise with the shaders.


In OpenGL, as in other graphics libraries, objects in the scene are composed of geometric primitives, which themselves are described by vertices. A vertex in modern OpenGL is a collection of data values associated with a location in space. Those data values might include colors, reflection information for lighting, or additional coordinates for use in texture mapping. Locations can be specified on 2, 3 or 4

dimensions but are stored in 4 dimensional homogeneous coordinates.

The homogenous coordinate representation of a point has w = 1 and for a vector w = 0. Perspective cameras can change the value of w. We return to normal 3D coordinates by perspective division which replaces p = [x, y, z, w] by p’= [x/w, y/w, z/w].

Vertices must be organized in OpenGL server-side objects called vertex buffer objects (also known as VBOs), which need to contain all of the


To form 3D geometric objects, you need to decompose them into geometric primitives that WebGL can draw. WebGL (and modern desktop OpenGL) only knows how to draw three things: points, lines, and triangles, but can use collections of the same type of primitive to optimize rendering.


The next few slides will introduce our example program, one which simply displays a cube with different colors at each vertex. We aim for simplicity in this example, focusing on the WebGL techniques, and not on optimal performance. This example is animated with rotation about the three coordinate axes and interactive buttons that allow the user to change the axis of rotation and start or stop the rotation.


To simplify our application development, we define a few types and constants to make our code more readable and organized.

Our cube, like any other cube, has six square faces, each of which we’ll draw as two triangles. In order to size memory arrays to hold the

necessary vertex data, we define the constant numVertices.

As we shall see, GLSL has vec2, vec3 and vec4 types. All are stored as four element arrays: [x, y, z, w]. The default for vec2’s is to set z = 0 and w =1. For vec3’s the default is to set w = 1.

MV.js also contains many matrix and viewing functions. The package is available on the course website or at www.cs.unm.edu/~angel/WebGL.

MV.js is not necessary for writing WebGL applications but its functions simplify development of 3D applications.



To provide data for WebGL to use, we need to stage it so that we can load it into the VBOs that our application will use. In your applications, you might load these data from a file, or generate them on the fly. For each vertex, we want to use two bits of data – vertex attributes in OpenGL speak – to help process each vertex to draw the cube. In our case, each vertex has a position in space, and an associated color. To store those values for later use in our VBOs, we create two arrays to hold the per vertex data. Note that we can organize our data in other ways such as with a single array with interleaved positions and colors.

We note that JavaSript arrays are objects and are not equivalent to simple C/C++/Java arrays. JS arrays are objects with attributes and methods.


In our example we’ll copy the coordinates of our cube model into a VBO for WebGLto use. Here we set up an array of eight coordinates for the corners of a unit cube centered at the origin.

You may be asking yourself: “Why do we have four coordinates for 3D data?” The answer is that in computer graphics, it’s often useful to include a fourth coordinate to represent three-dimensional coordinates, as it allows numerous mathematical techniques that are common operations in graphics to be done in the same way. In fact, this four- dimensional coordinate has a proper name, a homogenous coordinate.

We could also use a vec3 type, i.e.

vec3(-0.5, -0.5, 0.5)

which will be stored in 4 dimensions on the GPU.

In this example, we will again use the default camera so our vertices all fit within the default view volume.



Just like our positional data, we’ll set up a matching set of colors for each of the model’s vertices, which we’ll later copy into our VBO. Here we set up eight RGBA colors. In WebGL, colors are processed in the pipeline as floating-point values in the range [0.0, 1.0]. Your input data can take any for; for example, image data from a digital photograph usually has values between [0, 255]. WebGL will (if you request it), automatically convert those values into [0.0, 1.0], a process called normalizing values.


flatten() is in MV.js.

Alternately, we could use typed arrays as we did for the triangle example and avoid the use of flatten for one-dimensional arrays.

However. we will still need to convert matrices from two-dimensional to one-dimensional arrays to send them to the shaders. In addition, there are potential efficiency differences between using JS arrays vs typed arrays. It’s a very small change to use typed Arrays in MV.js. See the website.



As our cube is constructed from square cube faces, we create a small function, quad(), which takes the indices into the original vertex color and position arrays, and copies the data into the VBO staging arrays. If you were to use this method (and we’ll see better ways in a moment), you would need to remember to reset the Index value between setting up your VBO arrays.

Note the use of the array method push() so we do not have to use indices for the point and color array elements


Here we complete the generation of our cube’s VBO data by specifying the six faces using index values into our original positions and

colors arrays. It’s worth noting that the order that we choose our vertex indices is important, as it will affect something called backface culling later.

We’ll see later that instead of creating the cube by copying lots of data, we can use our original vertex data along with just the indices we passed into quad() here to accomplish the same effect. That

technique is very common, and something you’ll use a lot. We chose this to introduce the technique in this manner to simplify the OpenGL concepts for loading VBO data.



While we’ve talked a lot about VBOs, we haven’t detailed how one goes about creating them. Vertex buffer objects, like all (memory) objects in WebGL (as compared to geometric objects) are created in the same way, using the same set of functions. In fact, you’ll see that the pattern of calls we make here are like other sequences of calls for doing other WebGL operations.

In the case of vertex buffer objects, you’ll do the following sequence of function calls:

1.  Generate a buffer’s by calling gl.createBuffer()

2.  Next, you’ll make that buffer the “current” buffer, which means it’s the selected buffer for reading or writing data values by calling gl.bindBuffer(), with a type of GL_ARRAY_BUFFER. There are different types of buffer objects, with an array buffer being the one


To complete the “plumbing” of associating our vertex data with variables in our shader programs, you need to tell WebGL where in our buffer object to find the vertex data, and which shader variable to pass the data to when we draw. The above code snippet shows that process for our two data sources. In our shaders (which we’ll discuss in a

moment), we have two variables: vPosition, and vColor, which we will associate with the data values in our VBOs that we copied form our vertex positions and colors arrays.

The calls to gl.getAttribLocation() will return a compiler-generated index which we need to use to complete the connection from our data to the shader inputs. We also need to “turn the valve” on our data by enabling its attribute array by calling gl.enableVertexAttribArray() with the selected attribute location.

Here we use the flatten function to extract the data from the JS arrays and put them into the simple form expected by the WebGL functions, basically one dimensional C-style arrays of floats.



To initiate the rendering of primitives, you need to issue a drawing routine. While there are many routines for this in OpenGL, we’ll discuss the most fundamental ones. The simplest routine is glDrawArrays(), to which you specify what type of graphics primitive you want to draw (e.g., here we’re rending triangles), which vertex in the enabled vertex attribute arrays to start with, and how many vertices to send. If we use triangle strips or triangle fans, we only need to store four vertices for each face of the cube rather than six.

This is the simplest way of rendering geometry in WebGL. You merely need to store you vertex data in sequence, and then gl.drawArrays() takes care of the rest. However, in some cases, this won’t be the most memory efficient method of doing things. Many geometric objects




The vertex shader the stage between the application and the raster. It operates in four dimensions and is used primarily for geometric

operations such as changes in representations from the object space to the camera space and lighting computations. A vertex shader must output a position in clip coordinates or discard the vertex. It can also output other attributes such as colors and texture coordinates to the rasterizer.


The final shading stage that OpenGL supports is fragment shading which allows an application per-pixel-location control over the color that may be written to that location. Fragments, which are on their way to the framebuffer, but still need to do some pass some additional processing to become pixels. However, the computational power available in shading fragments is a great asset to generating images.

In a fragment shader, you can compute lighting values – similar to what we just discussed in vertex shading – per fragment, which gives much better results, or add bump mapping, which provides the illusion of greater surface detail. Likewise, we’ll apply texture maps, which allow us to increase the detail for our models without increasing the

geometric complexity.



Generally, GLS Lcode is compiled by WebGL as opposed to the HML and JS code which is interpreted. After successful compilation the shaders are put into a program object which is linked with the

application code. WebGL allows for multiple program objects and thus multiple shaders within an application.



As with any programming language, GLSL has types for variables.

However, it includes vector-, and matrix-based types to simplify the operations that occur often in computer graphics.

In addition to numerical types, other types like texture samplers are used to enable texture operations. We’ll discuss texture samplers in the texture mapping section.


The vector and matrix classes of GLSL are first-class types, with arithmetic and logical operations well defined. This helps simplify your code, and prevent errors.

Note in the above example, overloading ensures that both a*m and m*a are defined although they will not in general produce the same result.


In addition to types, GLSL has numerous qualifiers to describe a variable usage. The most common of those are:

•  attribute qualifiers indicate the shader variable will receive data flowing into the shader, either from the application,

•  varying qualifier which tag a variable as data output where data will flow to the next shader stage

•  uniform qualifiers for accessing data that doesn’t change across a draw operation

Recent versions of GLSL replace attribute and varying qualifiers by in and out qualifiers



GLSL also provides a rich library of functions supporting common operations. While pretty much every vector- and matrix-related function available you can think of, along with the most common mathematical functions are built into GLSL, there’s no support for operations like reading files or printing values. Shaders are data-flow engines with data coming in, being processed, and sent on for further processing.


Fundamental to shader processing are a couple of built-in GLSL variable which are the terminus for operations. Vertex data, which can be processed by up to four shader stages in desktop OpenGL, are all ended by setting a positional value into the built-in variable,


Additionally, fragment shaders provide seversal of built-in variables.

For example, gl_FragCoord is a read-only variable, while

gl_FragDepth is a read-write variable. Recent versions of OpenGL allow fragment shaders to output to other variables of the user’s designation as well.



Here’s the simple vertex shader we use in our cube rendering example.

It accepts two vertex attributes as input: the vertex’s position and color, and does very little processing on them; in fact, it merely copies the input into some output variables (with gl_Position being implicitly declared). The results of each vertex shader execution are passed further down the pipeline, and ultimately end their processing in the fragment shader.



Here’s the associated fragment shader that we use in our cube

example. While this shader is as simple as they come – merely setting the fragment’s color to the input color passed in, there’s been a lot of processing to this point. Every fragment that’s shaded was generated by the rasterizer, which is a built-in, non-programmable (i.e., you don’t write a shader to control its operation). What’s magical about this process is that if the colors across the geometric primitive (for multi- vertex primitives: lines and triangles) is not the same, the rasterizer will interpolate those colors across the primitive, passing each iterated value into our color variable.

The precision for floats must be specified. All WebGL implementations must support medium precision.


Shaders need to be compiled before they can be used in your program.

As compared to C programs, the compiler and linker are implemented within WebGL , and accessible through function calls from within your program. The diagram illustrates the steps required to compile and link each type of shader into your shader program. A program must contain a vertex shader (which replaces the fixed-function vertex processing), a fragment shader (which replaces the fragment coloring stages).

Just a with regular programs, a syntax error from the compilation stage, or a missing symbol from the linker stage could prevent the successful generation of an executable program. There are routines for verifying the results of the compilation and link stages of the compilation

process, but are not shown here. Instead, we’ve provided a routine that


To simplify our lives, we created a routine that simplifies loading, compiling, and linking shaders: InitShaders(). It implements the shader compilation and linking process shown on the previous slide. It also does full error checking, and will terminate your program if there’s an error at some stage in the process (production applications might choose a less terminal solution to the problem, but it’s useful in the classroom).

InitShaders() accepts two parameters, each a filename to be loaded as source for the vertex and fragment shader stages, respectively.

The value returned from InitShaders() will be a valid GLSL program id that you can pass into glUseProgram().



OpenGL shaders, depending on which stage their associated with, process different types of data. Some data for a shader changes for each shader invocation. For example, each time a vertex shader executes, it’s presented with new data for a single vertex; likewise for fragment, and the other shader stages in the pipeline. The number of executions of a particular shader rely on how much data was

associated with the draw call that started the pipeline – if you call glDrawArrays() specifying 100 vertices, your vertex shader will be called 100 times, each time with a different vertex.

Other data that a shader may use in processing may be constant

across a draw call, or even all the drawing calls for a frame. GLSL calls those uniform varialbes, since their value is uniform across the



Once you know the names of variables in a shader – whether they’re attributes or uniforms – you can determine their location using one of the glGet*Location() calls.

If you don’t know the variables in a shader (if, for instance, you’re writing a library that accepts shaders), you can find out all the shader variables by using the glGetActiveAttrib() function.


You’ve already seen how one associates values with attributes by calling glVertexAttribPointer(). To specify a uniform’s value, we use one of the glUniform*() functions. For setting a vector type, you’ll use one of the glUniform*() variants, and for matrices you’ll use a glUniformMatrix

*() form.




Uniform qualified variables are constant for an execution of

gl.drawArrays(), i.e. are constant for each instantiation of the vertex shader.


The multiplication operation is overloaded so we can use vec4 = float*vec4 in shader but we must set the w component back to zero or the effect of the scaling will be cancelled by the perspective division when we go from homogeneous coordinates back to three dimensions in the pipeline



id allows us to refer to button in JS file. Text between <button> and </

button> tags is placed on button. Clicking on button generates an event which we will handle with a listener in JS file.

Note buttons are part of HTML not WebGL.


The event of clicking a button (onclick) occurs on the display

(document). Thus the name of the event is document.button_id.onclick.

For a button, the event returns no information other than it has been clicked. The first three buttons allow us to change the axis about which we increment the angle in the render function. The variable flag is toggled between true and force. When it is true, we increment the angle in the render function.



This completes the rotating cube example.

Other interactive elements such as menus, sliders and text boxes are only slightly more complex to add since they return extra information to the listener. We can obtain position information from a mouse click in a similar manner.




Matrix operations are supported directly in GLSL where matrices and vectors are atomic types. In the application code, we either carry out the operations in our code or use a library such as MV.js or glMatrix.


Recall that WebGL uses four dimensional homogeneous coordinates (x, y, z, w). If we use 3D in our application, w defaults to 1.

Clip coordinates and screen coordinates are the only ones required by WebGL. However, applications prefer to use their own coordinates and convert to clip coordinates in the vertex shader.



Note that human vision and a camera lens have cone-shaped viewing volumes. OpenGL (and almost all computer graphics APIs) describe a pyramid-shaped viewing volume. Therefore, the computer will “see”

differently from the natural viewpoints, especially along the edges of viewing volumes. This is particularly pronounced for wide-angle “fish- eye” camera lenses.

These transformations were built into the original fixed-function OpenGL, Although the functions that used these coordinate systems have been deprecated (other than the viewport transformation), most applications prefer to build in all these transformations.



The processing required for converting a vertex from 3D or 4D space into a 2D window coordinate is done by the transform stage of the graphics pipeline.

The operations in that stage are illustrated above. Each box represent a matrix multiplication operation. In graphics, all our matrices are 4×4 matrices (they’re homogenous, hence the reason for homogenous coordinates).

When we want to draw an geometric object, like a chair for instance, we first determine all the vertices that we want to associate with the chair. Next, we determine how those vertices should be grouped to form geometric primitives, and the order we’re going to send them to the graphics subsystem. This process is called modeling. Quite often, we’ll model an object in its own little 3D coordinate system. When we want to add that object into the scene we’re developing, we need to determine its world coordinates. We do this by

specifying a modeling transformation, which tells the system how to move from one coordinate system to another.

Modeling transformations, in combination with viewing transforms, which dictate where the viewing frustum is in world coordinates, are the first transformation that a vertex goes through. Next, the projection transform is applied which maps the vertex into another space called clip coordinates, which is where clipping occurs. After clipping, we divide by the w value of the vertex, which is modified by projection. This division operation is what allows the farther-objects-being-smaller activity. The transformed, clipped

coordinates are then mapped into the window.


By using 4×4 matrices, OpenGL can represent all geometric

transformations using one matrix format. Perspective projections and translations require the 4th row and column. Otherwise, these

operations would require an vector-addition operation, in addition to the matrix multiplication.

While OpenGL specifies matrices in column-major order, this is often confusing for “C” programmers who are used to row-major ordering for two-dimensional arrays. OpenGL provides routines for loading both column- and row-major matrices. However, for standard OpenGL transformations, there are functions that automatically generate the matrices for you, so you don’t generally need to be concerned about this until you start doing more advanced operations.



Another essential part of the graphics processing is setting up how much of the world we can see. We construct a viewing frustum, which defines the chunk of 3-space that we can see. There are two types of views: a

perspective view, which you’re familiar with as it’s how your eye works, is used to generate frames that match your view of reality–things farther from your appear smaller. This is the type of view used for video games, simulations, and most graphics applications in general.

The other view, orthographic, is used principally for engineering and design situations, where relative lengths and angles need to be preserved.

For a perspective, we locate the eye at the apex of the frustum pyramid. We can see any objects which are between the two planes perpendicular to eye (they’re called the near and far clipping planes, respectively). Any vertices between near and far, and inside the four planes that connect them will be rendered. Otherwise, those vertices are clipped out and discarded. In some cases a primitive will be entirely outside of the view, and the system will

discard it for that frame. Other primitives might intersect the frustum, which we clip such that the part of them that’s outside is discarded and we create new vertices for the modified primitive.

While the system can easily determine which primitives are inside the frustum, it’s wasteful of system bandwidth to have lots of primitives discarded in this manner. We utilize a technique named culling to determine exactly which primitives need to be sent to the graphics processor, and send only those primitives to maximize its efficiency.


In OpenGL, the default viewing frusta are always configured in the same manner, which defines the orientation of our clip coordinates.

Specifically, clip coordinates are defined with the “eye” located at the origin, looking down the –z axis. From there, we define two distances:

our near and far clip distances, which specify the location of our near and far clipping planes. The viewing volume is then completely by specifying the positions of the enclosing planes that are parallel to the view direction.



lookAt() generates a viewing matrix based on several points.

LookAt() provides natural semantics for modeling flight application, but care must be taken to avoid degenerate numerical situations, where the generated viewing matrix is undefined.

An alternative is to specify a sequence of rotations and translations that are concatenated with an initial identity matrix.

lookAt() is in MV.js as are translation, scaling and rotation functions that we discuss in the next few slides.

Note: that the name modelview matrix is appropriate since moving objects in the model front of the camera is equivalent to moving the camera to view a set of objects.


Here’s an example vertex shader for rotating our cube. We generate the matrices in the shader (as compared to in the application), based on the input angle theta. It’s useful to note that we can vectorize

numerous computations. For example, we can generate a vector of sines and cosines for the input angle, which we’ll use in further computations.

This example is but one way to use the shaders to carry out

transformations. We could compute the transformation in the application each iteration and send it to the shader as a uniform. Which is best can depend on the speed of the GPU and how much other work we need to do in the CPU.


Completing our shader, we compose two of three rotation matrices (one around each axis). In generating our matrices, we use one of the many matrix constructor functions (in this case, specifying the 16 individual elements). It’s important to note in this case, that our matrices are column-major, so we need to take care in the placement of the values in the constructor.



We complete our shader here by generating the last rotation matrix and then using the composition of those matrices to transform the input vertex position. We also pass-thru the color values by assigning the input color to an output variable.


Finally, we merely need to supply the angle values into our shader through our uniform plumbing. In this case, we track each of the axes rotation angle, and store them in a vec3 that matches the angle

declaration in the shader. We also keep track of the uniform’s location so we can easily update its value.

This is not the only approach. We could also have generated the

transformation matrix in the application and send it to the vertex shader as a uniform, which may be more efficient. Either way, the

transformation approach should be better than transforming all the vertices in the application and resending them to the shaders.



Lighting is an important technique in computer graphics. Without lighting, objects tend to look like they are made from plastic.

The models used in most WebGL applications divide lighting into three parts: material properties, light properties and global lighting


While we’ll discuss the mathematics of lighting in terms of computing illumination in a vertex shader, the almost identical computations can be done in a fragment shader to compute the lighting effects per-pixel, which yields much better results.


The lighting normal determines how the object reflects light around a vertex. If you imagine that there is a small mirror at the vertex, the lighting normal describes how the mirror is oriented, and consequently how light is reflected.


WebGL can use the shade at one vertex to shade an entire polygon (constant shading) or interpolate the shades at the vertices across the polygon (smooth shading), the default.

The original lighting model that was supported in hardware and OpenGL was due to Phong and later modified by Blinn.


Here we declare numerous variables that we’ll use in computing a color using a simple lighting model. All the uniform values are passed in from the application and describe the material and light properties being rendered. We can send these values to either the vertex or fragment shader, depending on how we want to do lighting computation, either on per vertex basis or a per fragment basis.




Textures are images that can be thought of as continuous and be one, two, three, or four dimensional. By convention, the coordinates of the image are s, t, r and q. Thus for the two dimensional image above, a point in the image is given by its (s, t) values with (0, 0) in the lower-left corner and (1, 1) in the top-right corner.

A texture map for a two-dimensional geometric object in (x, y, z) world coordinates maps a point in (s, t) space to a corresponding point on the screen.


When you want to map a texture onto a geometric primitive, you need to provide texture coordinates. Valid texture coordinates are between 0 and 1, for each texture dimension, and usually manifest in shaders as vertex attributes. We’ll see how to deal with texture coordinates outside the range [0, 1] in a moment.


In the simplest approach, we must perform these four steps.

Textures reside in texture memory. When we assign an image to a texture it is copied from processor memory to texture memory where pixels are formatted differently.

Texture coordinates are actually part of the state as are other vertex attributes such as color and normals. As with colors, WebGL

interpolates texture inside geometric objects.

Because textures are discrete and of limited extent, texture mapping is subject to aliasing errors that can be controlled through filtering.

Texture memory is a limited resource and having only a single active texture can lead to inefficient code.


Specifying the texels for a texture is done using the

gl.texImage_2D() call. This will transfer the texels in CPU memory to OpenGL, where they will be processed and converted into an internal format.

The level parameter is used for defining how WebGL should use this image when mapping texels to pixels. Generally, you’ll set the level to 0, unless you are using a texturing technique called mipmapping.


Just like vertex attributes were associated with data in the application, so too with textures. You access a texture defined in your application using a texture sampler in your shader. The type of the sampler needs to match the type of the associated texture. For example, you would use a sampler2D to work with a two-dimensional texture created with gl.texImage2D( GL_TEXTURE_2D, … );

Within the shader, you use the texture() function to retrieve data values from the texture associated with your sampler. To the texture() function, you pass the sampler as well as the texture coordinates where you want to pull the data from.

Note: the overloaded texture() method was added into GLSL version


Some common uses:

desktop OpenGL: used for large scientific applications OpenGL ES: used to design smart phone apps

WebGL: Web applications, increasing interest in using it for games three.js: CADV

Vulkan: high-end games


There are a couple of major areas we haven’t discussed. One is image processing/texture. Support for texture and large amounts of texture memory make GPUs well suited for imaging applications. Support for off-screen rendering leads to a variety of applications such as shadow mapping.



Only atomic primitives in JS are numbers (64 bit floats), strings and booleans. Everything else is an object. Objects inherit from a prototype object and thus even the simplest objects have some members and functions that are defined in the prototype. JS uses function scope rather than block scope as in most other languages. There are global variables defined outside of your program, e.g. window.

Typing is dynamic so we can change the type of a variable anyplace in the program.

JavaScript is a large language that has matured over the past few years. However, there are multiple ways to accomplish a task with JS, some good and some bad. See for example Crockford, JavaScript, the


WebGL 2.0 is now supported in almost all browsers. Other types of shaders are widely available as WebGL extensions.



Buffer array objects let us put multiple vertex attributes together. When combined with transform feedback, simulation applications can be executed entirely in the GPU.


Many of these variants allow programmers more familiar with Java and Python to write JS code that is more familiar to them and avoids some of the “gotchas” in JS. Some variants also allow the programmer to write more concise code.

Some of the ES6 additions allow for more familiar object types and scoping.



Generally, OpenGL programs are fairly small and the driver large.

Consequently, it is straightforward to write an OpenGL application since the complexity is in the driver. But that limits the ability of the application programmer to take advantage of many options that have been set in the driver. Vulkan takes the opposite view and puts a tremendous amount of control in the application and requires a relatively small driver. Thus with Vulkan an application can adjust to the hardware, e.g.

an integrated processor vs separate CPU and GPU. For the most applications, we can get the performance we need with WebGL and OpenGL.



OpenGL may also continue to develop in parallel with Vulkan since OpenGL has a large user community that does not need to deal with the complexity of Vulkan.




All the above books except Angel and Shreiner, Interactive Computer Graphics (Addison-Wesley) and Learning three.js, are in the Addison- Wesley Professional series of OpenGL books.




Many example programs, a JS matrix-vector package and the

InitShader function are under the Book Support tab at www.cs.unm.edu/






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