Calculate world position, rotation and scale

index.tex

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\begin{document}

Let RS(denoted as $R$) be "rotation then scale". \\

RS then translation(denoted as $t$):

$$

\begin{bmatrix}
1 & t\\
0 & 1
\end{bmatrix}
\times
\begin{bmatrix}
R & 0\\
0 & 1
\end{bmatrix}
=
\begin{bmatrix}
R & t\\
0 & 1
\end{bmatrix}

$$

Two RS:

$$

\begin{bmatrix}
R_1 & 0\\
0 & 1
\end{bmatrix}
\times
\begin{bmatrix}
R_2 & 0\\
0 & 1
\end{bmatrix}
=
\begin{bmatrix}
R_1 \times R_2 & 0\\
0 & 1
\end{bmatrix}

$$

Two TRS transforms:

$$

\begin{bmatrix}
R_2 & t_2\\
0 & 2
\end{bmatrix}
\times
\begin{bmatrix}
R_1 & t_1\\
0 & 1
\end{bmatrix}
=
\begin{bmatrix}
R_2 \times R_1 & R_2 \times t_1 + t_2 \\
0 & 1
\end{bmatrix}

$$

Conclusion:

\begin{itemize}
    \item The global translation of a node can be calculated as successively(from direct parent to root) apply the local transforms to the local translation of the node.
    
    \item The global rotation(or scale) of a node can be calculated as successively(from direct parent to root) apply the local rotation and local scale to the local rotation(or scale) of the node.
\end{itemize}

\end{document}