# Matrices In Action, Grafix 2.0

The 4-by-4 matrices in the panels on the following screenshots are at the heart of computer graphics. They describe objects moving in three-dimensional space and are essential to MATLAB's Handle Graphics, to CAD (Computer Added Design) packages, to CGI (Computer Graphics Imagery) in films, and to most popular video games.

### Contents

#### Grafix 2.0

Here is the opening screen from version 2.0 of `Grafix`, my tool for investigating the matrices involved in 3-D computer graphics. The MATLAB code for `Grafix` is availble here.

I am interested in the matrix in the panel, which I call `M`. Many matrices like this one describe the dyamic transformations to be made on a set of target objects in a complex three-dimensional scene. This particular `M` is the product of a scaling and a rotation that results in the size and orientation of the plane shown.

I also want to point out the coordinate axes being used. This is `view(3)`, MATLAB's default 3-D cordinate system. The positive $x$-axis goes up and to the right on the screen, the positive $y$-axis up and to the left, and the positive $z$-axis goes straight up.

#### Rotations

The *homogeneous coordinates* system used in modern computer graphics makes it possible to describe rotations, translations and many other operations with 4-by-4 matrices. These matrices operate on vectors with the position of an object in the first three components and, for now, a one as the fourth component, eg. [ $x$, $y$, $z$, 1 ]',

Rotations are described by products of these matrices, each of which operates on only two of the first three components of the vector. The first matrix, $R_x$, leaves $x$ unchanged while it rotates $y$ and $z$. The second matrix, $R_y$, leaves $y$ unchanged while it rotates $x$ and $z$. And the third matrix, $R_z$, leaves $z$ unchanged while it rotates $x$ and $y$.

$$ R_x(\theta) = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & \cos{\theta} & -\sin{\theta} & 0 \\ 0 & \sin{\theta} & \cos{\theta} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] $$

$$ R_y(\theta) = \left[ \begin{array}{rrrr} \cos{\theta} & 0 & -\sin{\theta} & 0 \\ 0 & 1 & 0 & 0 \\ \sin{\theta} & 0 & \cos{\theta} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] $$

$$ R_z(\theta) = \left[ \begin{array}{rrrr} \cos{\theta} & -\sin{\theta} & 0 & 0 \\ \sin{\theta} & \cos{\theta} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] $$

#### Pitch, Roll, and Yaw

The terms *pitch*, *roll* and *yaw* are often used to describe the motion of vehicles like aircraft, marine craft, and spacecraft. Pitch is $R_x$, rotation about the $x$-axis.

Roll is $R_y$, rotation about the $y$-axis.

And yaw is $R_z$, rotation about the $z$-axis.

#### Translations

Translations are described by matrices with values in the fourth column. Multiplying a vector by one of these matrices produces a translation in the direction of the corresponding axis.

$$ T_x(\delta) = \left[ \begin{array}{rrrr} 1 & 0 & 0 & \delta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] $$

$$ T_y(\delta) = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \delta \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] $$

$$ T_z(\delta) = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \delta \\ 0 & 0 & 0 & 1 \end{array} \right] $$

While it is true that translations could be accomplished simply by adding the increment to the specified coordiate, the use of matrix multiplication allows translations to be combined in a uniform way with rotations and other operations. The arithmetic units on today's Graphics Processing Units, GPUs, are designed to do 4-by-4 matrix multiplications at speeds hundreds of times faster than general purpose Central Processing Units, CPUs.

#### Horizontal and Vertical

Inspired by David Singmaster's notation for Rubik's cubes, L, R, B, F, U, and D, we can use the descriptive terms *left* and *right* for horizontal motion in the $x$ direction; *back* and *forth* for horizontal motion in the $y$ direction; and *up* and *down* for vertical motion in the $z$ direction.

$T_x$, left and right.

$T_y$, back and forth.

$T_z$, up and down.

#### Scalings

This matrix applies a single scaling factor to all three axes.

$$ S(\sigma) = \left[ \begin{array}{rrrr} \sigma & 0 & 0 & 0 \\ 0 & \sigma & 0 & 0 \\ 0 & 0 & \sigma & 0 \\ 0 & 0 & 0 & 1 \end{array} \right] $$

#### Larger and Smaller

$S$

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