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Solutions/3D Modelling Part II

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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