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title: Solutions for 3D Modelling
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# Stage Turntable
+ Position in the local co-ordinate system $\mathbf{x}=(x,y,1)$
+ Translation from the local to the global system $T=(0,10,0)$.
+ Rotation
$$R_{-\alpha} = \begin{bmatrix}
\cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha
\end{bmatrix}$$
+ Homogeneous transformation
$$H_{-\alpha} = \begin{bmatrix}
\cos\alpha & \sin\alpha & 0 \\ -\sin\alpha & \cos\alpha & 10 \\
0 & 0 & 1
\end{bmatrix}$$
$$
+ Global position of the actress
$$
H_{-\alpha}\mathbf{x} =
\begin{bmatrix}
x\cos\alpha + y\sin\alpha \\ -x\sin\alpha + y\cos\alpha + 10 \\ 1
\end{bmatrix}$$
+ Global position in heterogeneous co-ordinates
$$(x\cos\alpha + y\sin\alpha, -x\sin\alpha + y\cos\alpha + 10)^T$$
# The Crane