[Computer Graphics] 02. Basic Mathmetics

Vector

• 1D: [x]
• 2D: [x,y]
• 3D: [x,y,z]
• 4D: [x,y,z,w]
• nD: [x,y,z,w,…]

Typical notation

• column vector
  $\begin{bmatrix}1\\2\\3\\ \end{bmatrix} = \begin{bmatrix}1&2&3\\ \end{bmatrix}^T$
• row vector
  $\begin{bmatrix}1&2&3\\ \end{bmatrix}$

Meaning

• position (in space)
• Direction (with length)

Dot product

Cross Product

Matrix

$m \times n$ matrix
$\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\\ \end{pmatrix}$

Matrix multiplication

$M\times N$

Identity Matrix

$\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\\ \end{pmatrix}$
$M\times I = I\times M = M$

If $AB = I$, $B$ is called the inverse of A,
and $B = A^{-1}$

Line, Ray and Linear Interpolation

line : $p(t) = p_0 + t(p_1 - p_0 )$

if t is in $[-\infty, \infty]$,

→ infinite line

if t is in $[0, \infty]$,

→ ray