223 
space is transformed in the semi-space included in [0;l/r]. 
All the straight lines originally parallel to the axis z pass 
through the vanishing point. 
In a completely similar manner we can write the 
transformation matrix with projection centre and vanishing 
T = 
If the projection centre is on the y-axis, then we have: 
1 0 0 0' 
a change of scale simulating the effect for which a far object 
is seen in reduced scale respect to a close one. We can 
substitute to the translation a rotation, for example about the 
axis y by an angle (p: 
'1 
0 
0 
p 
COS if) 
0 
-sirup 
0' 
'1 
0 
0 
0' 
0 
1 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
1 
0 
(48) 
sirup 
0 
COS (p 
0 
0 
0 
0 
r 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
1 
T = 
0 1 0 
0 0 1 
0 0 0 
(49) 
cos<p 
0 
0 
sirup 
0 
1 
0 
0 
sirup 
0 
0 
-cos <plz t 
0 
0 
0 
1 
(55) 
In a similar manner the rotation can be about the axis x: 
Perspective projections more complex are more able to 
produce the three-dimensional perception of the object. 
Adding a vanishing point can complicate the things. The 
transformation relative to this type of perspective projection, 
indicated as two vanishing points perspective, is the 
following: 
10 0/7 
0 10 q 
0 0 10 
0 0 0 1 
(50) 
By applying it to an arbitrary point we have: 
[x;y;z;l] [x;y;z;(px + qy +1)] (51) 
By dividing as usual by the fourth component we get the co 
ordinates expressed in the normal manner. Such a 
transformation can be obtained by concatenating two simple 
perspective transformations, those with one vanishing point 
onlv: 
W-W-fcbW-W < 52 > 
In a similar manner we can pass to a perspective 
transformation with three vanishing points: 
10 0/? 
0 1 0 q 
0 0 1 r 
0 0 0 1 
(53) 
15. Techniques for the generation perspective views 
For an observer, to perceive the three-dimensional form of an 
object from a single point of view is difficult when more than 
one side are not visible. Selected a projection centre, and a 
plane of projection orthogonal to the direction of view the 
vision of multiple sides can be obtained by applying a 
rotation or a translation to the object. Let’s consider a 
translation followed by a perspective transformation with one 
vanishing point only, with projection centre Z — Z c on the z 
= 0 plane: 
T-iTVlfob 
'1 
0 
0 
0' 
'1 
0 
0 
o' 
'1 
0 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
1 
0 
0 
0 
0 
1 
0 
0 
0 
0 
r 
0 
0 
0 
-l/z c 
/ 
m 
n 
1 
0 
0 
0 
1 
/ 
m 
0 
1 -n/z e _ 
where r - -l/z c . 
The translation along x and y shows further sides of the object 
besides the frontal one. The translation along the axis z brings 
10 0 0 
_ 0 cos 9 0 -sinO/z c 
T = c (56) 
0 -sinG 0 -cos#/z c 
0 0 0 1 
A single rotation about an axis orthogonal to the one where 
the projection centre lies, is equivalent to a perspective with 
two vanishing points. Note that for a single rotation the term 
of the perspective for the axis of the rotation remains 
unchanged. 
Normally it is not sufficient a rotation to show the minimum 
of three sides needed for an adequate three-dimensional 
representation. In general to the rotation the translation must 
be added to obtain a good visual effect. The transformation 
assumes the following form: 
T = [R x HT r HP rz ) (57) 
In a similar manner a perspective with three vanishing points 
can be obtained by rotating about two or more principal axes 
and by operating a simple perspective transformation (with 
one vanishing point) in cascade: 
T = [RrHRxHP K ] (58) 
The terms of the fourth column are all not null. When the 
translation follows the rotation, then the matrix becomes: 
T = [R Y ][Rx][Tr][PRz] = 
cos <p 
sirupsinO 
0 
(sirupcosG)/Zq 
0 
cos 0 
0 
-sinO/z c 
(59) 
sirup 
-cos (psinO 
0 
(- COS 9 COS <P)I Z C 
l 
m 
0 
l-n/z C 
the term is the responsible of the change of scale. If the 
order of the rotations is inverted, because of the non 
commutativity of the product raws by columns, the result is 
different. 
From these results it is clear that the double or triple 
perspective projection can be build by using rotations and 
translations about and along the principal axes, followed by a 
simple perspective projection with projection centre on one 
of the principal axes. This holds when the rotation is about an 
arbitrary of die space. 
16. Vanishing points 
When a perspective view of an object is created, a horizontal 
line is traced. The vanishing points are pomts on this line, on 
which straight lines before parallel to the not transformed 
principal axis converge. In general different series of parallel 
straight lines have different vanishing points. Normally, two 
methods for the determination the position of the vanishing 
points are used. The first one computes simply the point of 
intersection of two transformed straight lines, before parallel.