220 
axis of the x, and then after the procedure will be extended to 
the other axes also. The procedure consists in bring such a 
case to the set of known transformations. In the particular 
case: 
1 translation of the body since the axis of rotation local 
(that for hypothesis is parallel to the global reference 
system) doesn’t becomes coincident with the x-axis. 
2 rotation about the x-axis. 
3 translation of the transformed body in the position of 
origin. 
Mathematically: 
[X’\ = [X][Ty[Rx\{TT' (16) 
Passing from this rotation to the other one, respect to the 
parallel ofy and z, means substitute the rotation [ifcc] with the 
[7?y] and [Rz] respectively. More interesting is to deal with 
the translation appearing in these products raws by columns. 
In fact the definition of the rotation axis will be necessary to 
define a transition point (the direction naturally will be the 
one of the axes of the global reference system). The 
translation must bring such a point on the axis co-ordinate 
chosen for the rotation. In the hypothesis that it is x, and that 
the transition point is [X 0 ,Y 0 ,Z 0 ], then the translation 
matrix will be: 
'1 0 0 0' 
.0 T 0 Zo 1. 
It is not necessary to translate the abscises, because the 
rotation takes place about an axis and not about a point. In a 
similar manner, if the rotation axis is parallel to y or to z, the 
translation Y 0 , X Q and Z Q will be null respectively. The 
last matrix is the inverse of the translation matrix, bringing to 
its place the rotation axis. Such a matrix must exist, because 
the determinant of the translation matrix is equal to 1 
(therefore different from zero). 
9. Rotation about an arbitrary axis in the space 
According to the logic of the previous paragraph, let’s make 
the rotation about an arbitrary axis, as product of a series of 
transformations. First it is necessary to define the straight line 
about which the rotation has to be performed. To define such 
a straight line in a unique manner in the space, we can use a 
transition point, both for example [A 0 ,T 0 ,Z 0 ], and the 
director cosines, Cx, Cy, Cz. 
The rotation of an angle 5 about this axis can be done using 
the following procedure: 
1 perform a translation, making the point [X 0 ,T 0 ,Z 0 ] to 
coincide with the origin of the 
2 reference system 
3 perform a rotation to make the rotation axis to coincide 
withz 
4 rotate about the z axis by the angle 5 
5 perform the inverse of the rotation of point 2 
6 perform the inverse of the translation of point 1 
In general to perform the point two rotations are necessary', 
where the parameters must be determined. To clarify let’s use 
the following figure: 
To make the axis a to coincide with the axis z, a rotation Rx is 
performed followed by a rotation Ry. To determine the 
rotation angle Rx, first let’s project the versor of the axis a on 
the plane yz. The co-ordinates along y and z of the projected 
vectors are Cy and Cz, that is the director cosines of the axis 
versor a. We have so: 
cos a = Cz/d 
sin a-Cy/d (18) 
where the quantity d is : d = *JCy 2 + Cz 2 . 
After the rotation about the x-axis, the component z of the 
direction vector is d while the x component values Cx. The 
rotation p about the y-axis is: 
cosp = d sin/3 = Cx (19) 
The complete transformation can then be written as: 
[M] = [fy [&]■ [fiy]-[K4 1 ■ [ft*]- 1 .[fy 1 (20) 
where the single matrices have the following form. 
' 1 0 0 0' 
-xo - yo - zo 1 
1 0 
0 
0' 
'1 
0 
0 
0' 
0 cosa 
sina 
0 
0 
Cz/d 
Cy/d 
0 
0 -sina 
cos a 
0 
0 
-Cy/d 
Cz/d 
0 
0 0 
0 
1 
0 
0 
0 
1 
cos(-/3) 
0 -sin(~p) 
0' 
' d 
0 Cx 
0' 
0 
1 
0 
0 
0 
1 0 
0 
siri-fi) 
0 cos (-/?) 
0 
-Cx 
0 d 
0 
0 
0 
0 
1 
0 
0 0 
1 
cos 6 
sinS 
0 
0' 
- sinS 
cosò 
0 
0 
0 
0 
1 
0 
0 
0 
0 
1 
The angles a and (3 are not explicit. 
10. Reflection about an arbitrary plane in the space 
Often it is necessary' to reflect an object about a plane not a 
co-ordinate one. This can be made using a procedure similar 
to what already seen by the rotation, say taking advantage of 
the reflections about one of the co-ordinate planes after 
having modified the position of the object to whom the 
transformation is applied. Come obvious must be univocally 
defined the reflection plane. This can be done by using a 
point P = [X 0 ,Y 0 ,Z 0 ] lying on the plane, and the director 
cosines of the normal versor of such a plane. A possible 
procedure: 
1. translate the point known P [A 0 ,T 0 ,Z 0 ] on 
the origin of the reference system