What a Matrix Represent Part 2: Matrix Transformation Tutorial

Table of Contents

1. Introduction
2. Math Prerequisites

1. Importance of Correct Transformations
2. What a Matrix Represents
3. Matrix Multiplication
4. Transforming a Vector by a Matrix
5. Object Space Transformations
6. Camera Transformations
7. Inverse Transformations
8. Hierarchical Transformations

9. Precision
10. Conclusion

What a Matrix Represents

Before we continue, it will help you greatly to understand what the values in a matrix represent. A 4x4 matrix contains 4 vectors, which represent the world space coordinates of the x, y and z unit axis vectors, and the world space coordinate which is the origin of these axis vectors.

                X     Y     Z     C
            +=                       =+
            | += =+ += =+ += =+ += += |
            | |   | |   | |   | |   | |
            | | X | | X | | X | | 0 | |
            | |   | |   | |   | |   | |
            | |   | |   | |   | |   | |
            | | Y | | Y | | Y | | 0 | |
            | |   | |   | |   | |   | |
            | |   | |   | |   | |   | |
            | | Z | | Z | | Z | | 0 | |
            | += =+ += =+ += =+ |   | |
            | +===============+ |   | |
        O   |  X     Y     Z    | 1 | |
            | +===============+ += =+ |
            +=                       =+

The X column contains the world space coordinates of the local X axis
The Y column contains the world space coordinates of the local Y axis
The Z column contains the world space coordinates of the local Z axis

These vectors are unit vectors. A unit vector is a vector whose magnitude is 1. Basically, unit vectors are used to define directions, when magnitude is not really important.

The C column always contains the specified values
The O row contains the world space coordinates of the object's origin

You can make life easy for yourself by storing matrices which contain axis information in each object. I keep two matrices for every object; omatrix, which stores the object space matrix, and ematrix, which stores the eyespace matrix for the object.

A special matrix is the identity matrix:

        +=                       =+
        | += =+ += =+ += =+ += += |
        | |   | |   | |   | |   | |
        | | 1 | | 0 | | 0 | | 0 | |
        | |   | |   | |   | |   | |
        | |   | |   | |   | |   | |
        | | 0 | | 1 | | 0 | | 0 | |
        | |   | |   | |   | |   | |
        | |   | |   | |   | |   | |
        | | 0 | | 0 | | 1 | | 0 | |
        | += =+ += =+ += =+ |   | |
        | +===============+ |   | |
        |  0     0     0    | 1 | |
        | +===============+ += =+ |
        +=                       =+

Notice why the identity matrix is special? The identity matrix represents a set of object axes that are aligned with the world axes. Remember that the vectors stored in the matrix are unit vectors. Now, because the world x coordinate of the local x axis is 1, the world y and z coordinates of the local x axis are 0, and the origin vector is [0, 0, 0], the local x axis lies directly on the world x axis. The same is true for the local y and z axes.

The other special property of the identity matrix is given away in its name. If you are familiar with math, you know that there are identity elements in the set of any arithmetic operation. When an binary operation is performed on some operand and the identity element of the set, the operand is the result of the operation. For example, identity elements for multiplication and division are 1, and identity elements for addition and subtraction are 0. x + 0 = x; x - 0 = x; x * 1 = x; x / 1 = x. Similarly, [x] * [identity] = [x] (I will denote matrices in brackets [] throughout this doc, for example [x] is "matrix x").