What is 3D transformation?

Software Development
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#1

3-D Transformation is the process of manipulating the view of a three-D object with respect to its original position by modifying its physical attributes through various methods of transformation like Translation, Scaling, Rotation, Shear, etc

Properties of 3-D Transformation :

  • Lines are preserved,
  • Parallelism is preserved,
  • Proportional distances are preserved.

Types of Transformations :

  1. Translation
  2. Scaling
  3. Rotation
  4. Shear
  5. Reflection

Translation :
It is the process of changing the relative location of a 3-D object with respect to the original position by changing its coordinates. Translation transformation matrix in the 3-D image is shown as –

\ \newline\hspace{4.48cm} \Large \mathbf{ T[x, y, z]= \left [ \begin{matrix} 1 &0&0& 0\ 0 & 1&0&0&\ 0 & 0&1&0\ D_x&D_y&D_z&1\ \end{matrix}\right]} \newline \hspace{3.08cm}\\

Where Dx, Dy, Dz are the Translation distances, let a point in 3D space is P(x, y, z) over which we want to apply Translation Transformation operation and we are given with translation distance [Dx, Dy, Dz] So, new position of the point after applying translation operation would be –

, \hspace{4.5cm} \textbf{P'[x', y', z', 1] = P[x, y, z, 1].T[x, y, z]}

Problem : Perform translation transformation on the following figure where the given translation distances are Dx = 2, Dy = 4, Dz = 6.

Solution : On applying Translation Transformation we get corresponding points –

Fig.1

\large \mathbf{ O'[x, y, z, 1]= [0, 0, 0, 1]\left [ \begin{matrix} 1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 2&4&6&1\ \end{matrix}\right]=[2, 4, 6, 1]}\\ \hspace{4cm}\mathbf{A'[x, y, z, 1]= [0, 4, 0, 1]\left [ \begin{matrix} 1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 2&4&6&1\ \end{matrix}\right]=[2, 8, 6, 1]}\\ \hspace{4cm} \mathbf{B'[x, y, z, 1]= [0, 4, 4, 1]\left [ \begin{matrix} 1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 2&4&6&1\ \end{matrix}\right]=[2, 8, 10, 1]}\\ \hspace{4cm} \mathbf{C'[x, y, z, 1]= [4, 4, 0, 1]\left [ \begin{matrix} 1&0&0&0\ 0&1&0&0\ 0&0&1&0\ 2&4&6&1\ \end{matrix}\right]=[6, 8, 6, 1]}\\ \hspace{4cm} \mathbf{D'[x, y, z, 1]= [4, 4, 4, 1]\left [ \begin{matrix} 1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 2 & 4 & 6 & 1\ \end{matrix}\right]=[6, 8, 10, 1]}\\ \hspace{4cm} \mathbf{E'[x, y, z, 1]= [4, 0, 0, 1]\left [ \begin{matrix} 1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 2 & 4 & 6 & 1\ \end{matrix}\right]=[6, 4, 6, 1]}\\ \hspace{4cm}\mathbf{ F'[x, y, z, 1]= [0, 0, 4, 1]\left [ \begin{matrix} 1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 2 & 4 & 6 & 1\ \end{matrix}\right]=[2, 4, 10, 1]}\\ \hspace{4cm} \mathbf{G'[x, y, z, 1]= [4, 0, 4, 1]\left [ \begin{matrix} 1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 2 & 4 & 6 & 1\ \end{matrix}\right]=[6, 4, 10, 1]}\\

After performing translation transformation over the Fig.1, it will look like as below –

The steps involved in projection transformation?

A projection transformation also prepares for these follow-on tasks: Clipping - the removal of elements that are not in the camera’s line of sight. Viewport mapping - convert a camera’s viewing window into the pixels of an image. Hidden surface removal - determining which objects are in front of other objects.
3D Transformations take place in a three dimensional plane. 3D Transformations are important and a bit more complex than 2D Transformations. Transformations are helpful in changing the position, size, orientation, shape etc of the object.