Transform

Translate

code:latex

\bf T = \begin{bmatrix}

1 & 0 & 0 & t_x \\

0 & 1 & 0 & t_y \\

0 & 0 & 1 & t_z \\

0 & 0 & 0 & 1

\end{bmatrix}

Rotation

いろいろあるので省略

$ \bf R

Scale

code:latex

\bf S = \begin{bmatrix}

s_x & 0 & 0 & 0 \\

0 & s_y & 0 & 0 \\

0 & 0 & s_z & 0 \\

0 & 0 & 0 & 1

\end{bmatrix}

TRS

$ \bf M = \bf T \bf R \bf S

Compose

上のを掛け算しただけ

code:latex

\bf M = \begin{bmatrix}

{\bf R}_{11} s_x & {\bf R}_{12} s_y & {\bf R}_{13} s_z & t_x \\

{\bf R}_{21} s_x & {\bf R}_{22} s_y & {\bf R}_{23} s_z & t_y \\

{\bf R}_{31} s_x & {\bf R}_{32} s_y & {\bf R}_{33} s_z & t_z \\

0 & 0 & 0 & 1

\end{bmatrix}

Decompose

上のComposeを見たら自明

code:latex

s = \begin{pmatrix}

\| ({\bf M}_{11}, {\bf M}_{21}, {\bf M}_{31}) \| \\

\| ({\bf M}_{12}, {\bf M}_{22}, {\bf M}_{32}) \| \\

\| ({\bf M}_{13}, {\bf M}_{23}, {\bf M}_{33}) \|

\end{pmatrix}

code:latex

{\bf R} = \begin{bmatrix}

\frac{{\bf M}_{11}}{s_x} & \frac{{\bf M}_{12}}{s_y} & \frac{{\bf M}_{13}}{s_z} \\

\frac{{\bf M}_{21}}{s_x} & \frac{{\bf M}_{22}}{s_y} & \frac{{\bf M}_{23}}{s_z} \\

\frac{{\bf M}_{31}}{s_x} & \frac{{\bf M}_{32}}{s_y} & \frac{{\bf M}_{33}}{s_z}

\end{bmatrix}

code:latex

t = \begin{pmatrix}

{\bf M}_{14} \\

{\bf M}_{24} \\

{\bf M}_{34}

\end{pmatrix}

code:glsl

code:glsl

Normal Matrix