CS315 Lab 3: 3D Transformations

Highlights of this lab:

This lab is an introduction to Matrix Transformation


After the lab lecture, you have one week to modify the files in Lab3.zip:

Lecture Notes

A. The Classic OpenGL Transformation Pipeline

The classic OpenGL pipeline had two main stages of vertex transformation, each with its own transformation matrix. These were built into the graphics hardware. These days, other transformation pipelines have become possible since transformations are done in the vertex shader. However, in this lab, as in the textbook, we will try to implement the classic pipeline.

3D to 2D flow diagram.

Each vertex in the scene passes through two main stages of transformations:

There is one global matrix internally for each of the two stage above:

Given a 3D vertex of a polygon, P = [x, y, z, 1]T, in homogeneous coordinates, applying the model view transformation matrix to it will yield a vertex in eye relative coordinates:

P’ = [x’, y’, z’, 1]T = Mmodelview*P.

By applying projection to P’, a 2D coordinate in homogeneous form is produced:

P” = [x”, y”, 1]T = Mprojection*P’.

The final coordinate [x”, y”] is in a normalized coordinate form and can be easily mapped to a location on the screen to be drawn.

Setting Up The Modelview and Projection Matrices in your shader

Since OpenGL Core Profile always uses shaders, neither the modelview nor the projection matrix is available. You have to set them up yourself. The matrices will be allocated and given their values in the main program, and they will be applied to vertices in the shader program.

To help us create and manipulate matrices in our main program we will use the matrix classes and helper functions in mat.h . Each matrix will be initialized to identity if you use the default constructor. So to create our initial modelview and projection matrices we would declare two mat4 objects like so:

    var mv = new mat4();   // create a modelview matrix and set it to the identity matrix.
    var p = new mat4();    // create a projection matrix and set it to the identity matrix.

These two matrices can be modified either by assigning or post-multiplying transformation matrices on to them like this:


p  = perspective(45.0f, aspect, 0.1f, 10.0f); // Set the projection matrix to 
                                              // a perspective transformation 

mv = mult( mv, rotateY(45) ); // Rotate the modelview matrix by 45 degrees 
                              // around the Y axis.

As in this example, we will usually set the projection matrix p by assignment, and accumulate transformations in the modelview matrix mv by post multiplying.

You will use uniforms to send your transformations to the vertex shader and apply them to incoming vertices. Last lab you did this for colours by making vector type uniforms and for point sizes by making a float uniform. Uniforms can also be matrices.

//other declarations

//Uniform declarations
uniform mat4 mv; //declare modelview matrix in shader
uniform mat4 p;  //declare projection matrix in shader

void main()
  //other shader code

  //apply transformations to incoming points (vPosition)
  gl_Position = p * mv * vPosition; 

  //other shader code

To set the value of uniform shader variables you must first request their location like this:

//Global matrix variables
var projLoc;
var mvLoc;

//In your init code
// Get location of projection matrix in shader
projLoc = gl.getUniformLocation(program, "p");

// Get location of modelview matrix in shader
mvLoc = gl.getUniformLocation(program, "mv");

Then, you use a uniformMatrix* function with the uniform location and a local variable to set their value. Do this whenever you need to update a matrix - usually when the window is resized or right before you draw something. To set the value of our 4x4 float type matrices we will use the form uniformMatrix4fv:

//in display routine, after applying transformations to mv
//and before drawing a new object:
gl.uniformMatrix4fv(mvLoc, gl.FALSE, flatten(transpose(mv))); // copy mv to uniform value in shader

//after calculating a new projection matrix
//or as needed to achieve special effects
gl.uniformMatrix4fv(projLoc, gl.FALSE, flatten(transpose(p))); // copy p to uniform value in shader

Important: Notice that we use the flatten() and transpose() functions from MVnew.js. You are probably used to flatten() by now. The transpose() function is necessary because most CPU oriented languages expect matrices to be in Row Major order, but your GPU and GLSL expect matrices to be in Column Major. Some Javascript math matrix libraries are written to match GLSL, but MVnew.js is not. In a full featured OpenGL, the second argument can be set to true to ask for a transpose, but the OpenGL ES and WebGL standards require that argument to be false.

B. Elementary Transformations

This section covers the 3D transformation matrix functions provided with your textbook. There are alternative libraries that come with their own in-depth discussion of similar transformations, such as Mozilla's Matrix math for the web article.

Defining a Coordinate System – Which Way Does Z-point?


translate(dx, dy, dz);

Where [dx, dy, dz] is the translation vector.

The effect of calling this function is to create the translation matrix defined by the parameters [dx, dy, dz] which you should concatenate to the global model view matrix:

Mmodelview = Mmodelview * T(dx, dy, dz);

Where T(dx, dy, dz) =

In general, a new transformation matrix is always concatenated to the global matrix from the right. This is often called post-multiplication.

    mv = mult( mv, translate(0,0,-6) ); //Translate by -6 units on z-axis


There are two forms of rotation in MVnew.js.

rotate(angle, vec3(x, y, z));

The first is similar to the only one available in Classic OpenGL. It is capable of rotating by angle degrees about an arbitrary vector. However, it is often easier to rotate about only one of the major axes:

These simple rotations are then concatenated to produce the arbitrary rotation desired. For example:

    mv = mult( mv, rotate(20,vec3(0,1,0)) ); // Rotate 20 degrees CCW around Y axis

Rotating around only one axis at a time is so common that many matrix libraries provide special functions dedicated to each axis.


In the second form, angle is the angle of counterclockwise rotation in degrees, and * is one of X, Y or Z.

The method for calling a rotation matrix is similar to translation. For example, this:

    mv = mult( mv, rotateX(a) );

will have the following effect:

Mmodelview = Mmodelview * Rx(a);

Where Rx(a) denotes the rotation matrix about the x-axis for degree a: Rx(a) =

Applying rotation around the y-axis or z-axis can be achieved respectively by these function calls:

    mv = mult( mv, rotateY(a) ); // rotation about the y-axis
mv = mult( mv, rotateZ(a) ); // rotation about the z-axis


scale(sx, sy, sz);

where sx, sy and sz are the scaling factors along each axis with respect to the local coordinate system of the model. The scaling transformation allows a transformation matrix to change the dimensions of an object by shrinking or stretching along the major axes centered on the origin.

Example: to make the wire cube in this week's sample code three times as high, we can stretch it along the y-axis by a factor of 3 by using the following commands.

     // make the y dimension 3 times larger
     mv = mult( mv, scale(1, 3, 1));

     //Send mv to the shader
     gl.uniformMatrix4fv(mvLoc, gl.FALSE, mv);

     // draw the cube
     gl.drawArrays(gl.LINE_STRIP, wireCubeStart, wireCubeVertices); 

C. The Order of Transformations

D. Modeling Transformation vs. Viewing Transformation


viewing with lookAt

mat4 lookAt (vec3 eye, vec3 at, vec3 up)

The lookAt() function defines a viewing transformation


The lookAt() function makes it easy to move both the "from" and the "to" points in a linear manner. For example, if you need to pan along the wall of a building located away from the origin and aligned along no axes in particular, you could simply take the "to" point to be one corner of the building and calculate the "from" as a constant distance from the "to" point. To pan along the building, just vary the "to" point.

The ModelView Matrix

You may be confused about the exact distinction between modelling and viewing. This confusion stems from the fact that we are imitating classic OpenGL which uses one matrix to represent both the modeling and viewing steps for everything that is drawn - the Modelview matrix. The transformation used to describe the model and the transformation used to describe the viewpoint's location and orientation coexist in that one matrix. This approach results in a simpler set of calculations in the graphics pipeline - it is much faster to calculate the modelview matrix once on the CPU than to potentially do it thousands of times in the shader program - once for every single vertex to be drawn. Matrix multiplication is not commutative but rather associative, which means that the product of ((AB)C) is the same as (A(BC)). Thus OpenGL's Modelview matrix is logically the product of a viewing matrix and a modeling matrix.

Mmodelview = Mviewing * Mmodeling

What this means is that your viewing transformations must be entered into the Modelview matrix before modeling transformations.

E. Saving and Restoring the Matrix

Whichever method you use, you will almost always need to either reset the matrix to the identity matrix, or save and restore a previous matrix state. To reset to the identity matrix use code like this:

mv = mat4(); //restore mv to the identity matrix

To save and restore a matrix you can use a matrix stack. Classic OpenGL had one built in, but, like the rest of the matrix functions, it is missing in modern OpenGL varieties and must be provided by an external library. You can use any stack-like data structure that can handle your math library's matrix class. Javascript's arrays, which provide .push() and .pop() functions, are perfect.

To make a matrix stack in Javascript, write code like this:

Add to your global variables
//global modelview matrix stack
var matStack = [];

In display, use .push() and .pop() around transforms that should only affect one or a limited set of objects.

//Apply transforms to modelview matrix
//Draw objects
//... etc ...

//restore old modelview matrix
mv = matStack.pop();

You can store any mat4 matrix on the matrix stack so long as you remember to pop back to the correct matrices in the correct sequence.

F. Viewport and Projection Transformations

Once you have learned Modelview transformations, the next step is to understand projection modes and viewport mapping.

Viewport Transformation

The gl.viewport() function is used to specify a viewport, or the drawable area in your WebGL canvas. It can be cause your draws to use all or only a portion of the canvas. It is best to call it at least as often as the canvas changes size and only after you know the size of the canvas. In WebGL, that means it could be in your init() function, but if you are using a variable sized canvas it would be best to call gl.viewport() as part of rendering.

gl.viewport(x, y, width, height)


        // Correctly uses the size of the canvas as stored in the gl context
        // to set a full size viewport
        gl.viewport(0, 0, gl.canvas.width, gl.canvas.height);

Projection Transformations

There are two basic methods of converting 3D images into 2D ones.

Projection is handled by the MProjection matrix. You do not usually concatenate to the projection matrix as you do with the modelview matrix.

Orthographic Projection


  void ortho( left,   right, 
              bottom, top, 
              near,   far )


ortho() describes an orthographic projection matrix. (left, bottom, -near) and (right, top, -near) specify the points on the near clipping plane that are mapped to the lower left and upper right corners of the window, respectively, assuming that the eye is located at (0, 0, 0). -far specifies the location of the far clipping plane. Both near and far must be positive.

The following figure approximates an orthographic (actually it is for frustum() - see below) volume and the ortho() parameters


In old OpenGL systems, a function with the same parameters as ortho() could create perspective transformations. It was called frustum() and though it was powerful, it was not very intuitive. There is a much simpler perspective command, called perspective(). Like frustum() it generates a perspective viewing volume but only a simple one. It lacks the flexibility of frustum which can be manipulated to achieve special effects.

  void perspective( fovy,  aspect,
                    zNear, zFar )
Parameters: fovy: Specifies the field of view angle, in degrees, in the y direction; aspect: Specifies the aspect ratio that determines the field of view in the x direction. The aspect ratio is the ratio of x (width) to y (height); zNear: Specifies the distance from the viewer to the near clipping plane (always positive); zFar: Specifies the distance from the viewer to the far clipping plane (always positive).

perspective() specifies a viewing frustum into the world coordinate system. In general, the aspect ratio in perspective should match the aspect ratio of the associated viewport. For example, aspect=2.0 means the viewer's angle of view is twice as wide in x as it is in y. If the viewport is twice as wide as it is tall, it displays the image without distortion.

The following shows perspective viewing volume and the perspective() parameters



Goals of this assignment:

Master the use of the standard matrix transformations:

Part 1

Start with boxes.html and boxes.js from Lab3.zip.

As written, this program draws a basic coordinate system with a green x-axis, a red y-axis, and a blue z-axis. These will be referred to in the instructions as the axes

With the initial camera settings you are looking directly down the z-axis so you will not see it.

Make the following changes. Write your answers to the questions in steps 1, 2, 4 and 10.

  1. Comment out the lookAt() call and replace it with a translate() with parameters ( 0, 0, -10 )
    Is there any change in the display? Why? Why not?
  2. Comment out both the lookAt() and translate() lines. What happens? Why?
  3. Restore the lookAt() call.
  4. Take a look at the perspective() call. The aspect ratio you were originally given was 1.0.
    1. What happens when the aspect ratio is 1.0 and you change the canvas dimensions in boxes.html to width="512" height="256" ?
    2. How about width="256" height="512" ?
    3. The aspect ratio allows us to compensate for different shapes of canvas. Study the recommendations in point 2. of WebGL Anti-Patterns, then modify the aspect ratio in your perspective call so that it is an appropriate ratio of width to height based on the actual dimensions of the viewing area. Test the result with the two suggested canvas shapes to be sure you got it right.
  5. Draw a wireCube centered at (0, 0, 0) relative to the axes. You can use the provided buffers and related shape data. Do this in the render function.
  6. Move this cube so that it is centered at (1, 0, 0) relative to the axes.
  7. Draw a second cube after the first - in a new colour if you can - and rotate it 45 degrees around the y-axis.
  8. Place this rotated cube directly above the first cube. It will be centered at (1, 1, 0) relative to the axes. Be careful of the order of transformations.
  9. The perspective view makes the two cubes look a little awkward. Try using orthographic projection instead of the perspective call. The function for that is: ortho. Use left, right, bottom, top, near and far values that include the whole scene and not much more. See the picture for expected results (some deviation is OK):

    Please leave a commented perspective call in your program so I can see your answer to step 4c.
  10. Rotate everything (using modeling transformations NOT lookAt) so that you are looking down at the top of the boxes and seeing the blue z-axis (and no red y-axis). See the picture for expected results:

    If you wanted to leave your x and y axes unchanged, but still see the top of the boxes, like this:

    how would you change your code?
  11. Rotate everything so that you can see all three axes along with the two cubes. See the picture for expected results:

    You may use different angles of course.

/5 marks

Part 2

Start with robot_arm.html and robot_arm.js from Lab3.zip.

  1. First load the application and see how it works. Try pressing lower and uppercase 'e' to move the elbow. Try pressing lower and uppercase 's' to move the shoulder
  2. Now, add three fingers and a thumb to the robot.
    Use matStack.push() and matStack.pop() to separate the transformations for each digit. Do not attempt to "untransform" with an inverse rotate, translate or scale.
  3. Finally, add some code that will make the finger and thumb move apart when 'f' is pressed and and together when 'F' is pressed. The center of rotation should be at the wrist.
    Your completed robot hand might look something like the following.
  4. robot arm demo
    Your browser does not support WebGL.

    You can interact with this sample solution to see how your arm might work. Click on it and use the keys described above. I have also added some additional controls:

/5 marks