3D Transformations

• Transformation matrix

 

Rotation and parallel translation only in 3D Transformations(not like a 2D Transformations).

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A matrix that rotates points (X, Y, Z) in a 3D space by θ radians based on the X, Y, and Z axes.

 

$$\begin{array}{c}{R}_x(\theta )=[\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & -\sin \theta \\ 0& \sin \theta & \cos \theta \end{array}]\\ R_y(\theta )=[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}]\\ R_z(\theta )=[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}]\end{array}$$

 

$$R=R_z({\theta }_3)R_y({\theta }_2)R_x({\theta }_1)$$

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- 3D Transformation matrix(Rigid Transformation)

 

 

$$\left[\begin{array}{c}{X}^{\prime }\\ Y^{\prime }\\ Z^{\prime }\end{array}\right]=R\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]+\left[\begin{array}{c}{t}_x\\ t_y\\ t_z\end{array}\right]$$

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- Convert to homogeneous coordinate system

 

$$\left[\begin{array}{c}{X}^{\prime }\\ Y^{\prime }\\ Z^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{c}{r}_{11}{r}_{12}{r}_{13}{t}_x\\ r_{21}{r}_{22}{r}_{23}{t}_y\\ r_{31}{r}_{32}{r}_{33}{t}_z\\ 0001\end{array}\right]\left[\begin{array}{c}X\\ Y\\ Z\\ 1\end{array}\right]$$

 

https://darkpgmr.tistory.com/81