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
GLint projLoc;
GLint 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 uniform* (GLES 2.0 man page) (WebGL Spec) 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 routing, after applying transformations to mv
//and before drawing a new object:
gl.uniformMatrix4fv(mvLoc, gl.FALSE, 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, p); // copy p to uniform value in shader

B. Elementary Transformations


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.



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

rotate(angle, x, y, z);

Classic OpenGL only defined a single rotation function capable of rotating about an arbitrary vector. A similar function is in MV.js. However, typically we rotate about only one of the major axes. These simple rotations are then concatenated to produce the arbitrary rotation desired.

The effect of 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) =

Rotation matrices for the y-axis or z-axis can be achieved respectively by these functions 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.TRUE, mv);

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

C. The Order of Transformations

D. Modeling Transformation vs. Viewing Transformation


The lookAt() Function: define a viewing transformation

mat4 lookAt (vec3 eye, vec3 at, vec3 up)
  eye: specifies the position of the eye point
  at:  specifies the position of the reference point
  up:  specifies the direction of the up vector

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.

E. Saving and Restoring the Matrix

F. Viewport and Projection Transformations

Viewport Transformation

The gl.viewport() function takes four parameters, which are used to specify the lower-left corner coordinates and the width and height of the viewport, or the drawable area in your OpenGL view. It is best to call it only once you know how big the window is. That means it should be in your rehape function.

Projection Transformation

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.


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

     left, right: 
         Specify the coordinates for the left and right vertical
         clipping planes;
     bottom, top: 
         Specify the coordinates for the bottom and top horizontal
         clipping planes;
     near, far: 
         Specify the distances to the near and far depth
         clipping planes.  Both distances must be positive.  

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( GLfloat fovy, GLfloat aspect,
                    GLfloat zNear, GLfloat zFar )


          Specifies the field of view angle, in degrees, in the y

          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);


          Specifies the distance from the viewer to the near clipping
          plane (always positive);


          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 original aspect ratio is 1.0.
    1. What happens when the aspect ratio is 1.0 and you change the canvas dimensions to width="512" height="256" ?
    2. How about width="256" height="512" ?
    3. Modify the aspect ratio in the 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 constants. 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 top, bottom, left, right, 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.
  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. I have also added r/R to rotate the arm on the X axis so you can see it from above, and t/T button to switch between solid and wire cubes.

/5 marks


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