131 lines
2.9 KiB
Markdown
131 lines
2.9 KiB
Markdown
# Mathematics for Graphics
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### Vectors
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- The n-dimensional Euclidean Space is $\mathbb{R}^n$
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- $\mathbb{R}^n = \{(v_0, v_1, ... v_{n-1}) | v_0, v_1, ...v_{n-1} \in \mathbb{R}\}$
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- A vector is an n-turple
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- $v\in \mathbb{R}^n \Longleftrightarrow v=(v_0, v_1, ...v_{n-1}) | v_0, v_1, ...v_{n-1} \in \mathbb{R}$
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- In computer graphics we normally deal with 3-Dimensional Euclidean space $\mathbb{R}^3$
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- vec3 notation:
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- $v=(v_0, v_1, v_2)$
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- $v = \begin{pmatrix} {v_0}\\{v_1}\\{v_2} \end{pmatrix}$
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- Where $v_0$ represents x, $v_1$ represents y, and $v_2$ represents z axis
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##### Vector Scaling
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Each element of $v$ is scaled independently by $s$. Only the length is changed
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$$
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v\cdot s = \begin{pmatrix} {v_0\cdot s}\\{v_1\cdot s}\\{v_2\cdot s} \end{pmatrix}
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$$
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##### Vector Addition
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$$
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v + u = \begin{pmatrix} {v_0+u_0}\\{v_1+u_1}\\{v_2+u_2} \end{pmatrix}
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$$
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##### Vector Length
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$$
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||v|| = \sqrt{v_0^2 + v_1^2 + v_2^2}
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$$
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##### Vector Normalisation
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To change the length of the vector to 1.
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$$
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\frac{1}{||v||} \cdot v
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$$
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$\hat{v}$ is the notation for a normalised vector
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##### Dot Product
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$$
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u\cdot v = \sum_{i=0}^{n-1} u_i \times v_i
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$$
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or $u\cdot v = (u_0 * v_0) + (u_1 * v_1) + (u_2 * v_2)$
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The dot product is also defined in $\mathbb{R}^2$ and $\mathbb{R}^3$ as:
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$$
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u\cdot v = ||u||\times||v||cos\theta
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$$
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where $\theta$ is the smallest angle between $u$ and $v$
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If the dot product is **0**: the two vectors are **perpendicular**
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If the dot product is **positive**: $0 \leq \theta \leq \frac{\pi}{2}$
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If the dot product is **negative**: $\frac{\pi}{2} \leq \theta \leq \pi$
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##### Cross Product
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In $\mathbb{R}^3$ cross product is defined as follows:
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$$
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u \times v = \begin{pmatrix}
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(u_1 * v_2)-(u_2*v_1)\\
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(u_2 * v_0)-(u_0*v_2)\\
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(u_0 * v_1)-(u_1*v_0) \end{pmatrix}
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$$
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##### Matrices
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**Identity Matrix**
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$$
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\begin{pmatrix}
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1 \quad 0 \quad 0 \quad 0 \\
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0 \quad 1 \quad 0 \quad 0 \\
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0 \quad 0 \quad 1 \quad 0 \\
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0 \quad 0 \quad 0 \quad 1
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\end{pmatrix}
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$$
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###### Transpose of a Matrix
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Turns each row into a column
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###### Matrix addition
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###### Matrix Multiplication
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Two matrices can only be multiplied if they both have the same number of columns and rows.
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To get the resulting matrix, for each $(x,y)$ pair, is the cross product of the $x^{th}$ column and the $y^{th}$ row.
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- Matrix multiplication is not communative
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- $MN \neq NM$
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###### Matrix-Vector Multiplication
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A matrix multiplied by vector gives new vector
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Each row of the resulting vector is that row of the vector, dot producted with that row on the matrix.
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##### Trigonometry
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If $p=(p_x, p_y)$ is a unit vector, we can write them as:
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$$
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p_x = cos \space \alpha \\
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p_y = sin \space \alpha
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$$
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$$
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sin \space \alpha = \frac{opp}{hyp} \\
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cos \space \alpha = \frac{adj}{hyp} \\
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tan \space \alpha = \frac{opp}{adj} \\
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hyp^2 = opp^2 + adj^2
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$$
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