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A2.8 The 4×4 TransformationMatrix Notation
A 4×4 matrix encodes the orientation, scale, and location of an object. An unscaled, unrotated, unmoved object is represented by the identity matrix. The last column of the 4×4 matrix is unused and should be set to {0, 0, 0, 1}.
In[52]:={ { r11 , r12 , r13 , 0 }
, { r21 , r22 , r23 , 0 }
, { r31 , r32 , r33 , 0 }
, { x , y , z , 1 } }
Out[52]=
The top left-hand corner is the 3×3 matrix previously discussed, except that it now includes scaling information. The elements x, y, and z indicate the displacement of the object.
If the orientation matrix is normalized properly, the scaling factors along each axis " ,  ,  " may be represented as
In[53]:=
Out[53]=
For example, the following 4×4 matrix represents a scaling factor of 2 along the y axis.
In[54]:={ { 1,0,0,0 }
, { 0,2,0,0 }
, { 0,0,1,0 }
, { 0,0,0,1 } }
Out[54]=
This change in scale could also be written in the following form.
In[55]:=
Out[55]=
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