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