notes/docs/lectures/graphics/03_graphic_math.md

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

Matrix addition

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