On-Line Computer Graphics Notes
TRANSLATION

Overview

Translation is one of the simplest transformations. A translation moves all points of an object a fixed distance in a specified direction. It can also be expressed in terms of two frames by expressing the coordinate system of object in terms of translated frames.

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Development of the Transformation in Terms of Frames

Translation is a simple transformation. We can develop the matrix involved in a straightforward manner by considering the translation of a single frame. If we are given a frame ${\cal F} =( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$, a translated frame would be one that is given by ${\cal F} '=( {\vec u} , {\vec v} , {\vec w} , {\bf O} ')$ - that is, the origin is moved, the vectors stay the same.

If we write ${\bf O} '$ in terms of the previous frame by

$${\bf O} ' = a {\vec u} + b {\vec v} + c {\vec w} + {\bf O}$$

then we can write the frame ${\cal F} '$ in terms of the frame ${\cal F}$ by

$$\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ ... ...ray}{c} {\vec u} \\ {\vec v} \\ {\vec w} \\ {\bf O} ' \end{array}\right]$$

So a $4 \times 4$ matrix implements a frame-to-frame transformation for translated frames, and any matrix of this type (for arbitrary $a,b,c$) will translate the frame ${\cal F}$. We call any matrix

$$T_{a,b,c} \: = \: \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ a & b & c & 1 \\ \end{array}\right]$$

a translation matrix and utilize matrices of this type to implement translations.

Applying the Transformation Directly to the Local Coordinates of a Point

Given a frame ${\cal F} = ( {\vec u} , {\vec v} , {\vec w} , {\bf O} )$ and a point ${\bf P}$ that has coordinates $(u, v, w)$ in ${\cal F}$, if we apply the transformation to the coordinates of the point we obtain

$$\left[ \begin{array}{cccc} u & v & w & 1 \end{array}\right] \left... ... = \: \left[ \begin{array}{cccc} u + a & v + b & w + c & 1 \end{array}\right]$$

That is, we can translate the point within the frame ${\cal F}$. An illustration of this is shown in the following figure

Summary

Translation is a simple transformation that is calculated directly from the conversion matrix for two frames, one a translate of the other. The translation matrix is most frequently applied to all points of an object in a local coordinate system resulting in an action that moves the object within this system.

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