Geometry

How Does Matrix Work: Part 2

Reading time: 6 mins.

Relation Between Matrices and Cartesian Coordinate Systems

If you imagine that you have a point $P_x$ with coordinates (1, 0, 0) and want to rotate this point around the z-axis by 10 degrees clockwise, what would be the new point's coordinates? Using what we have learned on rotation matrices, we know these new coordinates can be found using simple trigonometry. The x coordinates of the unique rotated point are given by cos(-10), and the y coordinate is given by sin(-10) (remember that the trigonometric functions in C++ expect the angles to be expressed in radians). If we do the same thing, but this time with a point $P_y$ which is (0, 1, 0), then the x coordinate of this point after the rotation will be equal to -sin(-10), and the y coordinate will be similar to cos(-10). You can observe that the first line (or row) of the rotation matrix that rotates points around the z-axis ($R_Z$) contains the same trigonometric functions as those we used to compute the new coordinates of the point $P_x$ after rotation. The same observation can be made for the second line of the matrix, which also contains the same trigonometric functions as those we used to compute the new coordinates for $P_y$:

$$ \begin{array}{ll} Px_x = \cos(\theta)&Px_y = \sin(\theta)\\ Py_x=-\sin(\theta)& Py_y=\cos(\theta)\end{array} $$

As you can see, as we rotate these axes around the z-vector, the new coordinates can be computed using for $P_x$, the first row of the matrix, and for $P_y$, the second row. If you repeat the exercise for $P_z$ and the rotation matrix $R_X$ or $R_Y$, you will see that the new coordinates of $P_z$ can be computed using the third row of any one of these matrices (depending on which axis your rotate $P_z$ around).

The key idea in understanding matrices is that each row of the matrix represents an axis (or the bases) of a coordinate system. This is important as, later, you will learn how to create matrices to transform points and vectors from one coordinate system to another (change of basis) by simply replacing the rows of the matrix with the coordinates of each axis of that coordinate system you want to transform your vectors or points into:

$$ \begin{bmatrix} \color{red}{c_{00}}& \color{red}{c_{01}}&\color{red}{c_{02}}\\ \color{green}{c_{10}}& \color{green}{c_{11}}&\color{green}{c_{12}}\\ \color{blue}{c_{20}}& \color{blue}{c_{21}}&\color{blue}{c_{22}}\\ \end{bmatrix} \begin{array}{l} \rightarrow \quad \color{red} {x-axis}\\ \rightarrow \quad \color{green} {y-axis}\\ \rightarrow \quad \color{blue} {z-axis}\\ \end{array} $$

This common technique in CG will be described in the following chapters. Matrices are less of a mystery when you understand that they are just a way of storing the coordinates of a coordinate system where the rows of the matrix are the axes of this coordinate system or orientation matrix as we call it sometimes.

Orthogonal Matrices

In linear algebra, the type of matrices we have just described in this chapter and the previous one (the rotation matrices) are called orthogonal matrices. An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors. We have mentioned previously that each row from the matrix represents an axis of a Cartesian coordinate system. Suppose the matrix is a rotation matrix or the result of several rotation matrices multiplied by each other. In that case, each row necessarily represents an axis of unit length (because the elements of the rows are constructed from the sine and cosine trigonometric functions, which are used to compute the coordinates of points lying on the unit circle). You can see them as a Cartesian coordinate system initially aligned with the world coordinate system (the identity matrix's rows represent the axes of the world coordinate system) and rotated around one particular axis or a random axis. Orthogonal matrices have a few interesting properties, but the most useful one in Computer Graphics is that the transpose of an orthogonal matrix is equal to its inverse. Assuming Q is an orthogonal matrix, we can write:

$Q^T=Q^{-1}$ which entails that $QQ^T=I$,

where I is the identity matrix (see the chapter on Matrix Operations to learn more about matrix inversion, the transpose of a matrix, and the matrix identity).

Affine Transformations

You will sometimes find the terms affine transformations used in place of matrix transformation. This technical term is more accurate to designate the transformations you get from using the type of matrices we have described so far. In short, an affine transformation is a transformation that preserves straight lines. For example, the translation, rotation, and shearing matrix are all affine transformations, as are their combinations. The other type of transformation we will be studying in Computer Graphics is called projective transformation (perspective projection is a projective transformation). As you may have guessed, such transformations do not necessarily preserve parallelism between lines (check the lessons on the perspective and orthographic projection matrix in the Foundation of 3D Rendering section).

Summary

**Figure 6:** as the point rotates, its coordinates with respect to the world coordinate system (red and green axes) change. But they stay the same with respect to the coordinate system defined by the rotation matrix.

Not only have you learned in this chapter (and the previous one) how to create rotation matrices, but we have also given you a way of visualizing what a matrix is: each row of the matrix represents one axis of a cartesian coordinate system. The orientation (rotation), size (scale), and position (translation) of this coordinate system represent the transformation that will be applied to the points when they are multiplied by this matrix. The key idea is that points are initially defined in a particular coordinate system (let's call it A). Suppose a point is attached to a local coordinate system B (the matrix) and we move, rotate, and translate that local coordinate system (i.e., the matrix). In that case, the point coordinates will not change with respect to the local coordinate system B. The point is constrained to the transformation applied to the local coordinate system B (it moves with it). However, the coordinates of that point will change in coordinate system A. Multiplying the point whose coordinates are expressed in regards to A by the matrix B will provide us with the point's new coordinates in the coordinate system A. This is illustrated in figure 6.

It would be best to remember how to find the formula for the basic rotation matrices from that chapter. That the order by which you multiply this basic matrix is important. And finally (and that's almost the most important), a matrix can be seen as a local cartesian system where each row of the matrix represents one axis of that local coordinate system. Such a matrix is also called an orientation matrix. We will explain why in the chapter Creating an Orientation Matrix or Local Coordinate System.