(GAMES101) 1 - Review && Transformation (2D/3D)

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#1. Review of Linear Algebra

Vectors Normalization & Cartesian Coordinates

magnitude (length) : \(\Vert\vec a\Vert\)

unit vector : \(\hat a = \cfrac{\vec a}{\Vert {\vec a} \Vert}\)

\(\boldsymbol A = \begin{bmatrix} ~x~ \\ y \end{bmatrix} ~~~~~ \boldsymbol A^T = \begin{bmatrix} ~x , ~y~ \end{bmatrix} ~~~~~ \Vert \boldsymbol A \Vert = \sqrt{x^2 + y^2}\)

Dot (Scalar) Product

\(\vec a \cdot \vec b = \Vert \vec a \Vert \Vert \vec b \Vert \cos \theta ~~~~~ \cos \theta = \cfrac{\vec a \cdot \vec b}{\Vert \vec a \Vert \Vert \vec b \Vert}\)

for unit vectors : \(\cos \theta = \hat a \cdot \hat b\)

in cartesian coordinates : \(\vec a \cdot \vec b = \begin{bmatrix} ~x_a~ \\ y_a \\ z_a \end{bmatrix} \cdot \begin{bmatrix} ~x_b~ \\ y_b \\ z_b \end{bmatrix} = x_ax_b + y_ay_b + z_az_b\)

\[i.e. ~~\boldsymbol a \cdot \boldsymbol b = \boldsymbol a ^ T \boldsymbol b \]

projection of \(\vec b\) onto \(\vec a\) : \(\Vert\vec b_{\perp}\Vert = \Vert \vec b \Vert \cos \theta\)

usage : intersection angle, decompose, determine for/backward

Cross (Vector) Product

construct coordinate systems

cartesian formula : \(\vec a \times \vec b = \begin{bmatrix} ~y_az_b - y_bz_a ~\\~ z_ax_b - x_az_b ~\\~ x_ay_b - y_ax_b ~\end{bmatrix}\)

\[i.e. ~~ \vec a \times \vec b = A^*\boldsymbol b = \mathop{\begin{bmatrix} ~ 0 & -z_a & y_a ~\\ ~z_a & 0 & -x_a~ \\ ~-y_a & x_a & 0~\end{bmatrix}}\limits_{\text{dual matrix of a}} \begin{bmatrix} ~x_b~ \\ ~y_b~ \\ ~0~\end{bmatrix} \]

usage : determine left / right, determine inside / outside

Matrix

array of numbers ($m \times n = $ m rows, n columns)

Matrix-Matrix Multiplication

require # columns in A = # rows in B

\[i.e. ~~(m \times n)(n \times p) = (m \times p) \]

\((i, j)\) is dot product of \(\mathcal{row~ i~ from~ A}\) and \(\mathcal{column~ j~ from~ B}\)

• non-commutative (\(AB\) and \(BA\) are different in general)

• associative and distributive :

\[(AB)C = A(BC) \\ A(B + C) = AB + AC \\ (A + B)C = AC + BC \]

Matrix-Vector Multiplication

treat vector as a column matrix \(m \times 1\)

key for transforming points

Transpose of a Matrix

switch rows and columns

property : \((AB)^T = B^TA^T\)

Identity Matrix and Inverses

\(AA^{-1} = A^{-1}A = I ~~~~~~~~~ (AB)^{-1} = B^{-1}A^{-1}\)

 

 

#2. Transformation (2D)

Scaling

\[\begin{bmatrix}~ x' ~\\~ y' ~\end{bmatrix} = \begin{bmatrix} ~s_x & 0 ~\\~ 0 & s_y ~\end{bmatrix} \begin{bmatrix}~ x ~\\ ~ y ~ \end{bmatrix} \]

Reflection

horizontal reflection

\[\begin{bmatrix}~ x' ~\\ y'\end{bmatrix} = \begin{bmatrix} ~-1 & 0 ~ \\~ 0 & 1~\end{bmatrix} \begin{bmatrix} ~x~ \\ y\end{bmatrix} \]

Shearing

horizontal shift

\[\begin{bmatrix}~ x' ~\\ y'\end{bmatrix} = \begin{bmatrix} ~1 & \tan\phi ~ \\~ 0 & 1~\end{bmatrix} \begin{bmatrix} ~x~ \\ y\end{bmatrix} \]

Rotation

about the origin \((0,0)\), CCW by default

\[\begin{bmatrix}~ x' ~\\ y'\end{bmatrix} = \begin{bmatrix} ~\cos\phi & -\sin\phi ~ \\~ \sin\phi & \cos\phi \end{bmatrix} \begin{bmatrix} ~x~ \\ y\end{bmatrix} \]

Linear Transforms = Matrices (of the same dimension)

\[\mathbf x' = \mathcal M \mathbf x \]

Translation

In order to solve such the problem

\[\begin{bmatrix} ~ x' ~ \\ ~ y' ~ \end{bmatrix} = \mathcal M \begin{bmatrix} ~ x ~ \\ ~ y ~ \\ \end{bmatrix} = \begin{bmatrix} ~ x + t_x ~ \\ ~ y + t_y ~ \end{bmatrix} \]

2D point \(= (x, y, 1)^T\)

2D vector \(= (x, y, 0)^T\)

Matrix representation of translation

\[\begin{bmatrix} ~ x' ~ \\ ~ y' ~ \\ ~ w' ~ \end{bmatrix} = \begin{bmatrix} ~ 1 & 0 & t_x ~\\ ~ 0 & 1 & t_y ~ \\ ~ 0 & 0 & 1 ~ \end{bmatrix} \begin{bmatrix} ~ x ~ \\ ~ y ~ \\ ~ 1 ~ \end{bmatrix} = \begin{bmatrix} ~ x + t_x ~ \\ ~ y + t_y ~ \\ ~ 1 ~ \end{bmatrix} \]

Valid operation if w-coordinate of result is \(1\) or \(0\)

• vector + vector = vector
• point - point = vector
• point + vector = point
• point + point = midpoint

In homogeneous coordinates, \(\begin{bmatrix} ~ x ~ \\ y \\ w \end{bmatrix}\) is the 2D point \(\begin{bmatrix} ~ x/w ~ \\ y/w \\ 1 \end{bmatrix},~w \neq 0\)

Affine map = linear map + translation (Linear Transforms First)

\[\begin{bmatrix} ~ x' ~ \\ ~ y' ~ \end{bmatrix} = \begin{bmatrix} ~ a & b ~ \\ ~ c & d ~ \end{bmatrix} \begin{bmatrix} ~ x ~ \\ ~ y ~ \\ \end{bmatrix} + \begin{bmatrix} ~ t_x ~ \\ ~ t_y ~ \end{bmatrix} \]

Using homogeneous coordinates

\[\begin{bmatrix} ~ x' ~ \\ ~ y' ~ \\ ~ w' ~ \end{bmatrix} = \begin{bmatrix} ~ a & b & t_x ~\\ ~ c & d & t_y ~ \\ ~ 0 & 0 & 1 ~ \end{bmatrix} \begin{bmatrix} ~ x ~ \\ ~ y ~ \\ ~ 1 ~ \end{bmatrix} \]

2D Transformations with homogeneous coordinates

Scale

\[\mathbf S(s_x, s_y) = \begin{bmatrix} ~ s_x & 0 & 0 ~\\ ~ 0 & s_y & 0 ~ \\ ~ 0 & 0 & 1 ~ \end{bmatrix} \]

Rotation

\[\mathbf R(\phi) = \begin{bmatrix} ~ \cos\phi & -\sin\phi & 0 ~ \\ ~ \sin\phi & \cos\phi & 0 ~ \\ ~ 0 & 0 & 1 ~ \end{bmatrix} \]

Translation

\[\mathbf T(t_x, t_y) = \begin{bmatrix} ~ 1 & 0 & t_x ~\\ ~ 0 & 1 & t_y ~ \\ ~ 0 & 0 & 1 ~ \end{bmatrix} \]

Inverse Transform

\[\mathbf x = \mathcal {M^{-1}} \mathbf x' \]

Composite Transform

Transform Ordering Matters !

Cause matrix multiplication is not commutative, matrices are applied right to left.

When we have a sequence of affine transforms \(A_1, A_2, A_3,~...\) :

Pre-multiply \(n\) matrices to obtain a single matrix for performance

\[A_n(...A_2(A_1(\mathbf x))) = \mathbf A_n \cdots \mathbf A_2 ~\cdot~ \mathbf A_1 ~\cdot~ \begin{bmatrix} ~ x ~ \\ ~ y ~ \\ ~ 1 ~ \end{bmatrix} \]

Decomposing Complex Transform

eg. rotate around a given point c

\[\mathbf T(\mathbf c) \cdot \mathbf R(\mathbf \phi) \cdot \mathbf T(\mathbf {-c}) \]

#3. Transformation (3D)

Using homogeneous coordinates again !

• 3D point = \((x, y, z, 1)^T\)
• 3D vector = \((x, y, z, 0)^T\)

In general, \((x, y, z, w)~(w != 0)\) is the 3D point \((\cfrac{x}{w}, \cfrac{y}{w}, \cfrac{z}{w})\)

Use \(4 \times 4\) matrices for affine transformations

\[\begin{bmatrix} ~ x' ~ \\ ~ y' ~ \\ ~ z' ~ \\ ~ w' \end{bmatrix} = \begin{bmatrix} ~ a & b & c & t_x ~\\ ~ d & e & f & t_y ~ \\ ~ g & h & i & t_z ~ \\ ~ 0 & 0 & 0 & 1 ~ \end{bmatrix} \begin{bmatrix} ~ x ~ \\ ~ y ~ \\ ~ z ~ \\ ~ 1 ~ \end{bmatrix} \]

Scale

\[\mathbf S(s_x, s_y, x_z) = \begin{bmatrix} ~ s_x & 0 & 0 & 0 ~\\ ~ 0 & s_y & 0 & 0 ~ \\ ~ 0 & 0 & s_z & 0 ~ \\ ~ 0 & 0 & 0 & 1 ~ \end{bmatrix} \]

Translation

\[\mathbf T(t_x, t_y, t_z) = \begin{bmatrix} ~ 1 & 0 & 0 & t_x ~\\ ~ 0 & 1 & 0 & t_y ~ \\ ~ 0 & 0 & 1 & t_z ~ \\ ~ 0 & 0 & 0 & 1 ~ \end{bmatrix} \]

Rotation around x-, y-, or z-axis

\[\mathbf R_x(\phi) = \begin{bmatrix} ~ 1 & 0 & 0 & 0 ~\\ ~ 0 & \cos\phi & -\sin\phi & 0 ~ \\ ~ 0 & \sin\phi & \cos\phi & 0 ~ \\ ~ 0 & 0 & 0 & 1 ~ \end{bmatrix} \]

\[\mathbf R_y(\phi) = \begin{bmatrix} ~ \cos\phi & 0 & \sin\phi & 0 ~\\ ~ 0 & 1 & 0 & 0 ~ \\ ~ -\sin\phi & 0 & \cos\phi & 0 ~ \\ ~ 0 & 0 & 0 & 1 ~ \end{bmatrix} \]

\[\mathbf R_z(\phi) = \begin{bmatrix} ~ \cos\phi & -\sin\phi & 0 & 0 ~\\ ~ \sin\phi & \cos\phi & 0 & 0 ~ \\ ~ 0 & 0 & 1 & 0 ~ \\ ~ 0 & 0 & 0 & 1 ~ \end{bmatrix} \]

Compose any 3D rotation from \(R_x, R_y, R_z\)

Euler angle (roll, pitch, yaw)

\[\mathbf R_{xyz}(\alpha, \beta, \gamma) = \mathbf R_x(\alpha)\mathbf R_y(\beta)\mathbf R_z(\gamma) \]

Rodrigues' Rotation Formula

Rotation by angle \(\alpha\) around axis \(\boldsymbol n\)

\[\mathbf R(\mathbf n,\alpha) = \cos(\alpha)\mathbf I + (1 - cos(\alpha))\mathbf n \mathbf n^T + sin(\alpha) \underbrace{ \begin{bmatrix} ~ 0 & -n_z & n_y ~\\ ~ n_z & 0 & -n_x ~ \\ ~ -n_y & n_x & 0 ~ \end{bmatrix} }_\mathbf N \]