What is 3D transformation?

3D transformation in homogeneous coordinates manipulates the position, orientation, and scale of 3D objects in three-dimensional space.

Homogeneous coordinates

Homogeneous coordinates extend the traditional Cartesian coordinates (X, Y, Z) with an additional coordinate (X, Y, Z, W), enabling the perspective projections using matrix multiplication.

The homogeneous coordinates (X, Y, Z, W) represents a 3D point as follows:

3D transformations using homogeneous coordinates are typically represented as $4\times4$ matrices, known as transformation matrices.

Translation

The translation moves an object in 3D space along the x, y, and z axes. Suppose $(x,y,z,1)$ is a point in 3D space and $(x',y',z',1)$ is a transformed point, then the translation equation would be:

Here, d $(t_{x},t_{y},t_{z},1)$represents the translation vector. The following figure visualizes the 3D translation:

Rotation

Rotation involves changing the orientation of an object around one or more axes. The rotation occurs along an axis. It also includes the angle of rotation $\theta$ that determines the extent to which the object will be turned about that axis. If $\theta$ is positive, the rotation would be counterclockwise.

Suppose $(x,y,z,1)$ is a point in 3D space and $(x',y',z',1)$ is a transformed point, then the rotation along the x-axis would be:

The rotation along the y-axis would be:

The rotation along the z-axis would be:

Here, $\theta$ represents the angle of rotation. The following figure visualizes the 3D rotation:

Scale

Scaling is the process of resizing an object in 3D space. Suppose $(x,y,z,1)$ is a point in 3D space and $(x',y',z',1)$ is a transformed point, then the scaling equation would be:

where, $S_{x}, S_{y} , \textnormal{and }S_{z}$ represent the scaling factors along the $x, y \textnormal{, and }z$ axes, respectively. The following figure visualizes the 3D scaling:

Shear

The shear transformation distorts the shape of an object along one or more axes. Suppose $(x,y,z,1)$ is a point in 3D space and $(x',y',z',1)$ is a transformed point, then the shearing equation would be:

where $s_{x}, s_{y} , \textnormal{and }s_{z}$ are the shear factors along the x, y, and z axes, respectively. The following figure visualizes the 3D shearing:

Conclusion

By understanding and using 3D transformations effectively, we can create complex and realistic 3D scenes in computer graphics. It plays a crucial role in various applications, from video games and animation to medical imaging and augmented reality.

Note: Learn about two dimentional transformation.