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docs/lectures/graphics/03_graphic_math.md

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+# 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-turple
+- $v\in \mathbb{R}^n \Longleftrightarrow v=(v_0, v_1, ...v_{n-1}) | v_0, v_1, ...v_{n-1} \in \mathbb{R}$
+- 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 x, $v_1$ represents y, and $v_2$ represents 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 communative
+  - $MN \neq NM$
+
+###### Matrix-Vector Multiplication
+
+A matrix multiplied by vector gives 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 them as:
+
+$$
+p_x = cos \space \alpha \\
+p_y = sin \space \alpha
+$$
+
+$$
+sin \space \alpha = \frac{opp}{hyp} \\
+cos \space \alpha = \frac{adj}{hyp} \\
+tan \space \alpha = \frac{opp}{adj} \\
+hyp^2 = opp^2 + adj^2
+$$