221 
2. rotate the normal versor since it is coinciding 
with the z positive semi-axis. This makes the 3. 
reflection plane coinciding with the z = 0 plane 
4. apply the reflection transformation about the z 
= 0 plane 
5. perform the mverse transformations of points 
one and two. 
The transformation in general can be written: 
M= [T\M-M-M-№ •[&]■’ -[Tt 1 (25) 
where the matrices [Rx], [Ry], [7] are the same seen for the 
rotation. 
11. Affine and perspective geometry 
The concept of parallelism is fundamental in this affme type 
of geometry. In the perspective, on the contrary' the straight 
lines are generally not parallel. 
An affine transformation is a combination of linear 
transformations. The last column is in general [0,0,0,l] r . 
The affine transformations and the perspective ones are three- 
dimensional, and then to observe the result on a plane surface 
is necessary an operation of projection called plane geometric 
projection. 
Due to the fact that the plane geometrical projection 
transforms 3D objects in two-dimensional objects, its 
transformation matrix contains always a column of zeros. 
Such projections are formed by the intersection of lines called 
projective with a plane called projection plane. The projective 
lines are lines that, going from a point called projection 
centre, pass thought a point of the object. For the projection 
of the whole object, a projective line per any object point is 
needed. Using some special points such as coins and vertices, 
the construction of the image requires on the contrary a 
limited number of projective straight lines. 
If the projection centre is to the infinity all the projective 
lines are parallel and the result is a parallel projection. The 
plane geometric projections supply the base for the 
descriptive geometry. In developing the different 
transformations two approaches can be used: 
1. The first one assumes fixed the projection centre or 
point of view, and the projection plane is orthogonal to 
any projective straight line. It results to be the object to 
be manipulated to obtained the required view. 
2. The about assume fixed the object, the centre of 
projection is free to move everywhere in the space. The 
projection plane is not necessarily orthogonal to the 
direction of view. 
The first type of approach is the one more likely to the 
condition of an operator placed in front of the screen of a 
computer, where such a method will be used. The first 
projection plane to be analysed will be the parallel ones. 
11. Orthographic projection 
The simplest parallel projection is the orthographic 
projection. It brings the correct dimension and the right form 
of any single side of the object. The orthographic projections 
are projections m one of the co-ordinate planes x = 0 , y = 0 , 
z = 0. The matrix for the projection on the z = 0 plane is: 
10 0 0 
0 10 0 
0 0 0 0 
0 0 0 1 
(26) 
Note that the third column, the one of the z is a column of 
zeros. In similar manner for the projections on the x = 0 and y 
- 0 plane the matrices are: 
0 0 0 0' 
, = 0 1 0 0 
* 0 0 10 
0 0 0 1 
'1 0 0 0' 
B _ 0 0 0 0 
y ~ 0 0 1 0 
0 0 0 1 
(27) 
(28) 
A single orthographic projection does not supply sufficient 
information enabling the object reconstruction. Consequently 
multiple orthographic projections are needed. A maximum of 
six projections, are used, all made with combinations of 
reflections, rotations and translations followed by the 
projection on the z = 0 plane. For objects with sides non 
parallel to the co-ordinate planes, the orthogonal projections 
do not show the real form and dimension. One uses then 
some auxiliary views. Rotation and translation of the object 
obtain such views, so that the normal versor to the auxiliary 
plane be coinciding with one of the co-ordinate axes. The 
result is then projected on the co-ordinate plane orthogonal to 
this axis. 
13. Axonometric projections 
A single orthographic projection does not show the three- 
dimensional form of an object. The axonometric projection 
cope with this limitation. Such a transformation is build by 
manipulating the object, using rotations and translations, so 
that at least three adjacent sides are visible. The result is 
projected onto one of the co-ordinate planes (the projection 
centre is always to the infinity). Apart when a side of the 
object is parallel to the projection plane, the axonometric 
projection doesn’t show' the real form. Never the less the 
relative lengths of lines before parallel remain constant, say 
parallel lines are scaled by the same factor. The scale factor is 
just defined as ratio between the transformed lenght and the 
original one. 
There are three types of axonometric projections of practical 
interest: 
1. the trimetric ; 
2. the dimetric; 
3. the isometric. 
The trimetric axonometric projection is the less limiting of 
the three. It is formed by an arbitrary rotation about some or 
all the co-ordinate axes, follow'ed by a projection on the z = 0 
plane. The scale factor for each of the principal axes is 
different. For principal axis we intend an axis or a border of 
the object originally parallel to one of the x, y, or z axes. For 
any projection the scale factor can be obtained by applying 
the matrices of transformation to the versor along the 
principal axis. In other words: 
'1 
0 
0 
1* 
X*x 
Yx 
0 
1 
0 
1 
0 
1 
•H- 
Xy 
K 
0 
1 
0 
0 
1 
1 
x: 
y: 
0 
1 
where [U] is the unity original vector matrix along the x, y or 
z-axis and [7] is the matrix concatenated of the trimetric 
projection. The scale factors along the principal axes are: 
f, = V-sr ? + rT 2 ; fy = -¡X? + r? ■ f-- = № + R 1 
(30) 
A dimetric projection is similar to the trimetric one with two 
of the three scale factors equal. It scale can be build by 
concatenation of a rotation about y by an angle <f>, about the x 
axis by an angle 0, and projecting respect to a projection 
centre to the infinity. The transformation can be written: