© 2015-2020 Jacob Ström, Kalle Åström, and Tomas Akenine-Möller

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Chapter 1: Introduction

This chapter was intentionally meant to be as short as possible, so that you (the reader) can start with the real content (starting in Chapter 2) swiftly. The math presented in this chapter is meant as a refresher for what is necessary to follow the rest of the book, not as an educational material. Hence there will be no explanations of the formulae in this chapter. To get that, we recommend you turn to books on trigonometry. To navigate in this book, you can use the menu button in the upper left corner, and click on links for chapters, sections, figures, etc.

In the next section, we will introduce our book's basic notation, which will not be introduced later in this book. Hence, linear algebra specific notation, e.g., vectors, points, matrices, etc., will be introduced throughout the book. Finally, this introductory chapter ends with a recap of some trigonometry (Section 1.2), which may be good to review before starting with the rest of the book.
1.1 Brief Note on Notation

This section describes the notation used in this book. The reader may skip this now, and check back when needed, but it is a quick and simple read, so it may be preferable to do it before diving into the other chapters.

Note that everything is numbered in this book. This is to make it simpler to discuss the different equations, expressions, etc with another person. That is, you can always refer to Equation 3.12, which means that the equation/expression is in Chapter 3, and it is the 12:th one there.

A number, in the common sense, may be positive, negative, it may have decimals, and it may even be rational or irrational. The set of all such numbers are denoted by $\R$, and the numbers are said to be real numbers. We often use the term a scalar or a scalar value instead of saying a real number, and those are always denoted by lower-case letters, e.g., $j, s, t, a, m$, in this book. They may also have subscripts, which means that $k_1$, $k_2$ and $k_3$ are different numbers. One may also write $\sum_{i=1}^3k_i$ in order to add $k_1$, $k_2$ and $k_3$. The notation $k\in \R$ means that $k$ is a real number, or more precisely, that $k$ belongs to the set of all real numbers.

The absolute value of a scalar, $k$, is denoted $\abs{k}$, and is defined as
\begin{equation} \abs{k} = \left\{ \begin{matrix} k, & \mathrm{if\ } k \geq 0\\ -k, & \mathrm{if\ } k < 0 \end{matrix} \right. , \end{equation} (1.1)
i.e., if the number is negative, the minus sign is removed.

A set is a small collection of, for example, integer numbers. A set with the numbers $1$, $2$, and $4$ is denoted: $\{1,2,4\}$. When it is desirable to have a variable, $i$, for example, that can take on any of the members in a set, then we write: $i\in\{1,2,4\}$.

Real numbers can often take on values in a certain range. As an example if we know $x$ can take on any value from zero to one, including zero and one, we can write $x \in [0, 1]$. A soft parenthesis marks that the end point is not included. As examples,
\begin{gather} x \in [-1, 2) \ \ \ \mathrm{denotes} \ \ \ -1 \leq x < 2 \\ \theta \in (-\frac{\pi}{2}, \frac{\pi}{2}] \ \ \ \mathrm{denotes} \ \ \ -\frac{\pi}{2} < \theta \leq \frac{\pi}{2}. \end{gather} (1.2)
1.2 Some Trigonometry

This section provides a summary of some useful trigonometry concepts that will be used in this book, and that said, it is by no means a thorough treatment of the subject.

The cosine, sine, and tangent functions are always good to have fresh in mind. Refer to Figure 1.1.
For right triangles, the following holds:
\begin{align} \cos \theta &= \frac{a}{c}, \\ \sin \theta &= \frac{b}{c}, \\ \tan \theta &= \frac{\sin \theta}{\cos \theta} = \frac{b/c}{a/c} = \frac{b}{a}. \end{align} (1.3)
The Pythagorean theorem for Figure 1.1 is $c^2 = a^2 + b^2$.

Next, a set of useful trigonometric identities will be presented. The trigonometric unity is
\begin{equation} \cos^2\theta + \sin^2\theta = 1. \end{equation} (1.4)
The law of cosines is a very useful formula to know, and the geometrical setup is shown in Figure 1.2.
The law of cosines is then
\begin{equation} c^2 = a^2 + b^2 - 2ab\cos \gamma. \end{equation} (1.5)
The law of sines is related, and its geometrical setup is shown in Figure 1.3.
The law of sines is simply
\begin{equation} \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}. \end{equation} (1.6)
We omit the proofs to all trigonometry in this section. Some of these formulas will be proved in later chapters though.

The inverse functions for the cosine and sine are also important to know about. For example, $\theta = \arcsin a$ means that $\sin \theta = a$. The function $\arcsin a$ can take on values in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Likewise $\theta = \arccos a$ means that $\cos \theta = a$, and $\arccos a \in [0, \pi]$.

Note however that just because $\sin \theta = a$ we cannot conclude that $\theta = \arcsin a$. As an example, $\sin 4\pi = 0$, but $\arcsin 0 = 0$, not $4\pi$. In fact $\sin \theta = a$ has an infinite number of solutions, and $\arcsin$ is only the inverse to $\sin$ within the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$. However we can list all the solutions to $\sin \theta = a$ as $\theta = \arcsin(a) \pm 2 k \pi$, where $k = 0, 1, 2, 3, \ldots$ and so on.

At this point, it is time to start with the first real chapter of this book. Click here to go to Chapter 2, or navigate using the menu at the left top corner.

Chapter 0: Preface (previous) Chapter 2: Vectors (next)