World Transform

World transform is quite useful for the users. Consider the following scenario: a rotated cube, You may go through all troubles to calculate the coordinations manually; or create an ``normal'' cubic, and then apply a set of world transformation to move it to the right position.

The world transform matrix consists five matrices:

$$\mathbf{T} = \left[ \begin{array}{llll} 1 & 0 & 0 & x \\ 0... ... 0 & 0 & 1 \end{array} \right ] \textrm{- translation matrix}$$

$$\mathbf{R_{x}} = \left [ \begin{array}{llll} 1 & 0 & 0 & 0 ... ...nd{array} \right ] \textrm{- rotation about the \emph{x}-axis}$$

$$\mathbf{R_{y}} = \left [ \begin{array}{llll} cos\beta & 0 &... ...nd{array} \right ] \textrm{- rotation about the \emph{y}-axis}$$

$$\mathbf{R_{z}} = \left [ \begin{array}{llll} cos\gamma & -s... ...nd{array} \right ] \textrm{- rotation about the \emph{z}-axis}$$

$$\mathbf{S} = \left [ \begin{array}{llll} S_{x} & 0 & 0 & 0... ...\ 0 & 0 & 0 & 1 \end{array} \right ] \textrm{- scale matrix}$$

We take the OpenGL convention for the world transform, the world transformation matrix is defined as:

$$\mathbf{W} = \mathbf{T} \times \mathbf{R_{x}} \times \mathbf{R_{y}} \times \mathbf{R_{z}} \times \mathbf{S}$$ (1)

Note: The order of the transfrom matrix is arbitary

And we apply the world transform matrix on the left side of the corordination $({x}, {y}, {z})$ to get the new coordination $({x'}, {y'}, {z'})$:

$$\left [ \begin{array}{l} {x'} \\ {y'} \\ {z'} \\ c \end{... ...begin{array}{l} {x} \\ {y} \\ {z} \\ 1 \end{array} \right ]$$ (2)