[r] = [/M-[Äx].[Pz] 
Executing the products: 
cos^ siruf>-sin9 
[T] = 0 cos 0 
sinp -cos <p-sin9 
0 1 
0 1 
0 1 
(31) 
(32) 
The versor transformed: 
b*J 
-M-M- 
'10 0 1' 
cos^ sin<p-sin9 0 1 
0 10 1 
•M- 
0 cos# 0 1 
0 0 11 
sm<fi -cosip-sin9 0 1 
The square of the lenght of the transformed versor let us to 
determine the entity of the rotations <p and 9: 
f Y 2 =(X? + Y?) = cos 2 9 (34) 
f 2 = (X* 2 + Y* 2 ) = sin 2 <f> + cos 2 <p • sin 2 <p (35) 
f 2 = (X* 2 + Y* 2 ) = cos 2 </> + sin 2 (p ■ sin 2 9 (36) 
Equating the factors of scale along x and y we get an equation 
in the unknowns <p and 9. 
cos 2 (f> + sin 2 (f> ■ sin 2 9 = cos 2 9 (37) 
By using the trigonometric identities cos 2 ^ = 1 -sin 2 <j> and 
cos 2 <9 = 1- sin 2 9 we obtain: 
sin 2 (f> = sin 2 9 /(1 - sin 2 9) (38) 
A second relationship is obtained using the factor along the 
direction z: 
2 sin 4 9 - (2 + f 2 ) • sin 2 9 + f?=0 (39) 
Setting u=sin 2 9 we have: sin 2 9 = f 2 12 
Finally : 9 = arcsin{± /, / yfl) ; <p = arcsin{± f 2 /-J2- f 2 ) 
(40). 
This shows haw the range of the scale factor is [0; 1] and 
furthermore, that for any scale factor four dimetnc 
projections are got. 
The isometric axonometric projection is a particular case of 
the dimetric axonometric projection with <f> = ±45° and 
9 = ±35.26° . 
14. Perspective transformations 
As already said they are transformations with three- 
dimensional dominio and codominio. A difference of the 
parallel projections with the perspective the projective lines 
converge in one point and the dimensions of the object image 
diminishes to the increase of the distance of the projection 
P r = 
1 0 0 0' 
0 10 0 
0 0 1 r 
0 0 0 1 
(41) 
centre. All that helps the perception of the depth and makes 
more realistic the view. The perspective transformation from 
a single point is given by (41): 
[x\y\z\\]-[T] = [x\y,z\rz+\] (42) 
that applied to an arbitrary point supplies (42). 
The fourth component is generally different from 1. Dividing 
by h=rz^l we get: 
[x*;/;z*;l]= 
X y z 
/•z + l’rz + l’rz+1 
;1 
(43) 
Concatenating a perspective transformation and an 
orthographic transformation we obtain the image that we see 
looking for example the computer screen. An example can 
be: 
T = [P,\[P=b 
'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 
r 
0 
0 
0 
0 
0 
0 
0 
r 
_0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
1 
The transformed co-ordinates: 
[x;y;0;/z + 1] 
expressed in the normal manner, dividing 
component: 
rz+Ÿ rz+Y 
(45) 
by the fourth 
(46) 
To clarify the effect of the perspective transformation lets 
consider the following figure: 
The image shows the perspective projection of a segment AB 
on the z = 0 plane. The transformed segment is A B . The 
projection takes place through the projection centre placed on 
the z-axis with co-ordinates [0;0;-l/r\. It can be regarded 
made in two phases. The first one produces the already three- 
dimensional image A B . The second one consists in the 
projection of A B on the plane z = 0, with a projection 
centre to the infinity on the z-axis. From the figures the 
extension of the segment A B intersects the plane z = 0 in 
the same point of the extension of the segment AB. It 
intersects also the axis z in the point [0;0;+l/r\. Such a point 
is called vanishing point, and it lies at an equal distance of the 
projection centre, from the projection plane. Effectively by 
applying the transformation matrix to the point to the infinity 
on the axis Z + we have: 
[0;0;1;0] 
10 0 0 
0 10 0 
0 0 0 /- 
0 0 0 1 
[0;0;l;r] 
(47) 
The point [x ;y ;z*;l] = [0;0;l/r;l] corresponds to the 
point to the infinity transformed now in a real point of the 
positive semi-axis of the z. This means that the positive semi