# Mathematics for Graphics

### Vectors

- The n-dimensional Euclidean Space is $\mathbb{R}^n$
    - $\mathbb{R}^n = \{(v_0, v_1, ... v_{n-1}) | v_0, v_1, ...v_{n-1} \in \mathbb{R}\}$
- A vector is an n-tuple
- $v\in \mathbb{R}^n \Longleftrightarrow v=(v_0, v_1, ...v_{n-1}) | v_0, v_1, ...v_{n-1} \in \mathbb{R}$
- In computer graphics we normally deal with 3-Dimensional Euclidean space $\mathbb{R}^3$
    - vec3 notation:
        - $v=(v_0, v_1, v_2)$
        - $v = \begin{pmatrix} {v_0}\\{v_1}\\{v_2} \end{pmatrix}$
    - Where $v_0$ represents the x axis, $v_1$ represents the y axis, and $v_2$ represents the z axis

##### Vector Scaling

Each element of $v$ is scaled independently by $s$. Only the length is changed

$$
v\cdot s = \begin{pmatrix} {v_0\cdot s}\\{v_1\cdot s}\\{v_2\cdot s} \end{pmatrix}
$$

##### Vector Addition

$$
v + u = \begin{pmatrix} {v_0+u_0}\\{v_1+u_1}\\{v_2+u_2} \end{pmatrix}
$$

##### Vector Length

$$
||v|| = \sqrt{v_0^2 + v_1^2 + v_2^2}
$$

##### Vector Normalisation

To change the length of the vector to 1.

$$
\frac{1}{||v||} \cdot v
$$

$\hat{v}$ is the notation for a normalised vector

##### Dot Product

$$
u\cdot v = \sum_{i=0}^{n-1} u_i \times v_i
$$

or $u\cdot v = (u_0 * v_0) + (u_1 * v_1) + (u_2 * v_2)$

The dot product is also defined in $\mathbb{R}^2$ and $\mathbb{R}^3$ as:

$$
u\cdot v = ||u||\times||v||cos\theta
$$

where $\theta$ is the smallest angle between $u$ and $v$

If the dot product is **0**: the two vectors are **perpendicular**

If the dot product is **positive**: $0 \leq \theta \leq \frac{\pi}{2}$

If the dot product is **negative**: $\frac{\pi}{2} \leq \theta \leq \pi$

##### Cross Product

In $\mathbb{R}^3$ cross product is defined as follows:

$$
u \times v = \begin{pmatrix} 
(u_1 * v_2)-(u_2*v_1)\\
(u_2 * v_0)-(u_0*v_2)\\
(u_0 * v_1)-(u_1*v_0) \end{pmatrix}
$$

##### Matrices

**Identity Matrix**

$$
\begin{pmatrix}
1 \quad 0 \quad 0 \quad 0 \\
0 \quad 1 \quad 0 \quad 0 \\
0 \quad 0 \quad 1 \quad 0 \\
0 \quad 0 \quad 0 \quad 1
\end{pmatrix}
$$

###### Transpose of a Matrix

Turns each row into a column

![1645633205.png](img/1645633205.png)

###### Matrix addition

![1645633407.png](img/1645633407.png)

###### Matrix Multiplication

Two matrices can only be multiplied if they both have the same number of columns and rows.

To get the resulting matrix, for each $(x,y)$ pair, is the cross product of the $x^{th}$ column and the $y^{th}$ row.

- Matrix multiplication is not commutative
    - $MN \neq NM$

###### Matrix-Vector Multiplication

A matrix multiplied by a vector gives a new vector

Each row of the resulting vector is that row of the vector, dot producted with that row on the matrix.

##### Trigonometry

If $p=(p_x, p_y)$ is a unit vector, we can write its components as:

$$
p_x = cos \space \alpha \\
p_y = sin \space \alpha
$$

$$
sin \space \alpha = \frac{opp}{hyp} \\
cos \space \alpha = \frac{adj}{hyp} \\
tan \space \alpha = \frac{opp}{adj} \\
hyp^2 = opp^2 + adj^2
$$