219 
x* — X + 1 
y* = y+m (5) 
z*-z + n 
where 4 m, n, are the factors of translation, respectively 
along the axis x, y, z. 
3. Scale transformations 
They produce in the object a change of the linear dimensions. 
They can have global effect or local effect. They are 
characterised by a transformation matrix as (5): 
'a 0 0 0' 
0 0 0 1_ 
The scale factors are obviously a, e , j, acting respectively on 
x, y, z. In the case with three different factors, the scale 
transformation are said local. On the contrary', in a global 
scale transformation we have: 
a = e=j 
When the transformation is global, the whole transformation 
matrix is normally divided by the scale factor, so that the sole 
element different from the unity is the 4,4 term the only one 
that assumes a value equal to the inverse of the scale factor. 
4. Shear transformations 
Their effect is to act on the angles of the solid object where 
they are applied. They can be regarded as complementary to 
the scale transformations. 
'a 0 0 0' 
S- 0 e 0 0 (7) 
0 0 j 0 
0 0 0 1 
[x + yd + gz,bx + y + iz,cx + fy + z, l] (8) 
5. Rotations 
Before investigate a rotation about an arbitrary axis on the 
three-dimensional space, let's examine the rotations about the 
co-ordinate axes. So it will be possible to combine them in a 
series of 
'1 
0 
0 
0' 
0 cos# 
sin 9 
0 
0 - 
sin 9 
cos 9 
0 
0 
0 
0 
1 
COS Y Sin \ff 0 
0 
— sin 
Y cos y 0 
0 
0 
0 1 
0 
0 
0 0 
1 
COS0 
0 
-sin<f> 
o' 
0 
1 
0 
0 
sin<f> 
0 
COS <t> 
0 
0 
0 
0 
1 
transformations in cascade. By rotating an object about the x- 
axis, the abscise of such an object does not change. The same 
holds for the axes y and z. That means that any of these 
transformations is plane. Writing the transformation matrices 
respectively for the rotations about the axes x, y, z, all that 
becomes very clear. 
The negative sign position is due to the choice of the right 
hand convention, or positive rotations when counter 
clockwise. As we will see later, to obtain a rotation about an 
arbitrary axis, multiple transformations will be needed, more 
matrices in cascade. It has to be reminded that the product 
raws by columns is not commutative, so in general the 
execution order of the transformations will be fundamental. 
6, Transformation of reflection 
For an object it is not always possible to obtain the required 
configuration, by means of translations and rotations. The 
transformation of reflection cannot be obtained as 
composition of rotations and translations. Similarly to the 
rotation, the reflection takes place respect to a geometrical 
entity, which in the case is the plane. For now we take into in 
consideration the reflections respect to the planes co-ordinate 
only. In the chapter dedicated to the multiple transformations, 
we will see that, as for the rotation, the reflection respect to 
an arbitrary plane, can be obtained by composition of simple 
reflections. The determinant of the reflection matrix values 
always -1. Consider for example a reflection respect to a 
plane xy, only the co-ordinate along z is modified; it changes 
the sign. All that suggests the form of the transformation 
matrix: 
'1 0 0 0' 
0 0 0 1 
respect to xy 
-1 0 0 0' 
0 0 0 1 
respect to yz 
'1 0 0 0' 
0 0 0 1_ 
respect to xz 
7. The multiple transformations 
As anticipated, many transformations can be obtained with a 
series of matricial products in cascade. The most general 
transformations are obtained as matrix concatenation of more 
basic transformations. 
• Arbitrary translation 
• Rotation about an arbitrary axis 
• Reflection through an arbitrary plane 
The closer matrix generates the first transformation, the 
farther the last one. 
[Xym = [X]-[T\)-[T2\-...-[Tn\ (is) 
The [77] are combinations of translations, rotations, 
reflections etc. Among the multiple transformations there are 
the perspective and projective transformations also. Such 
transformations produce a lost of information, so that when 
present they must be executed as last. 
8. Rotation about an axis parallel to the co-ordinate axes 
Often a rotation about an axis different from the co-ordinate 
ones must be performed. The paragraph considers the case 
when such an axis is parallel to one of the co-ordinate axes 
only. We take into consideration first the parallelism with the