Matrices from applications
Matrices from applications in 2D and 3D graphics
Review: 2D linear transformations
We learned earlier that linear transformations of the plane $\mathbb{R}^2$ can be presented by $2 \times 2$ matrices.
Scaling: \[ \begin{bmatrix} a & 0 \\ 0 & a \end{bmatrix} \] scales vectors uniformly by some positive factor $a$
Scaling: \[ \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \] scales vectors by factor $a > 0$ in the $x$ direction and factor $b > 0$ in the $y$ direction.
Rotation \[ \begin{bmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \] rotates vectors counterclockwise about the origin by the angle $\theta$.
Reflection \[ \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \] or \[ \begin{bmatrix} 1 & 0 \\ 0 &-1 \end{bmatrix} \]
These are all nonsingular matrices. They can be combined via matrix-matrix product.
Scaling of $\mathbb{R}^3$
The matrices shown in the previous slide can be generalized to the 3-dimensional space $\mathbb{R}^3$ easily.
E.g., for $a,b,c > 0$, the matrix \[ \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \] scales the $x,y,z$ components of a vector by factors $a,b,c$, respectively, in the sense that \[ \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} ax \\ by \\ cz \end{bmatrix} \]
Reflection
Similarly, reflections along each axis (over a coordinate plane) can be represented by
\[ \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] long the $x$-axis (i.e. over the $yz$-plane)
\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] long the $y$-axis (i.e. over the $xz$-plane)
\[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \] long the $z$-axis (i.e. over the $xy$-plane)
For example, \[ \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x \\ -y \\ z \end{bmatrix} \]
Rotations
Rotations about the $x,y,z$ axis can be represented by matrices \begin{align*} R_x(\theta) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \\ \end{bmatrix} \\ R_y(\theta) &= \begin{bmatrix} \cos\theta &0 & \sin\theta \\ 0 &1 & 0 \\ -\sin\theta &0 & \cos\theta \\ \end{bmatrix} \\ R_z(\theta) &= \begin{bmatrix} \cos\theta & -\sin\theta & 0\\ \sin\theta & \cos\theta & 0\\ 0 & 0 & 1\\ \end{bmatrix} \end{align*}
Here, we follow a "right-handed" coordinate system convention. The signs will be different if a "left-handed" coordinate system is used.
These matrices are the basic building block of computer graphs, robotics, and many other applications. To learn more about this, read about "Euler angles".
There is a complete different way for representing rotations by using "quaternions".
Exercise
Recall that the composition of multiple transformations described above simply corresponds to multiplication of matrices.
E.g., the composition of one transformation represented by $R_1$ followed by a transformation represented by $R_2$ (that is, $R_1$ first, then $R_2$) is represented by the matrix product \[ R_2 R_1 \] (Notice the counterintuitive ordering)
A word about the orthogonal group
Among the transformations we saw so far, the subset of transformations that are "length-preserving" is of particular importance.
As expected, this group consists of exactly the orthogonal matrices, which are matrices satisfying the condition that \[ Q^\top Q = QQ^\top = I \]
It can be shown that this group is "generated" by rotations and reflections. That is, each matrix in this group can be expressed as a product of rotations and reflections.
Homogeneous coordinates
In computer graphs, we don't just want to transform vectors, we also want to apply affine transformations to points.
The homogeneous coordinates unifies both vectors (directions) and points (positions), and it is a useful coordinate system for 3D graphics.
Homogeneous coordinates
A vector (direction) $ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \in \mathbb{R}^3 $ is represented as $ \begin{bmatrix} a \\ b \\ c \\ 0 \end{bmatrix} $
A point in the 3D space $ \begin{bmatrix} x \\ y \\ z \end{bmatrix} $ is represented as $ \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} $
Then both linear and affine transformations can be represented as \[ \begin{bmatrix} t_{11} & t_{12} & t_{13} & \Delta x \\ t_{21} & t_{22} & t_{23} & \Delta y \\ t_{31} & t_{32} & t_{33} & \Delta z \\ 0 & 0 & 0 & 1 \end{bmatrix} \]
Pinhole camera matrix
The homogenous coordinates can be defined for any dimension. It is particularly convenient to write down matrix representations for cameras using homogeneous coordinates.
For example, an ideal pinhole camera can be represented by the matrix \[ \begin{bmatrix} f & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{bmatrix} \]
This is one of the simplest models. You can read more about "camera matrix models" to a wide variety of matrix models for digital cameras used in computer vision.