225 
(Th-Tux* l )x + (T^ -T^x x )y + {Tl l -Tl A x* l )z + 
+ (T4i -5 44 x “ 0 
(T{ 2 - T i\y l ) x + ( T 22 - TW)y + (T32 - Tj 4 y* 1 )z + 
+ (2i-J&/ 1 ) = 0 
(7i 2 ,-T? 4 x' 2 )x + {T 2 X -T 2 4 x* 2 )y + (Tg ~T 2 4 x* 2 )z + 
+ (T 4 \ -T^x* 2 ) = 0 
The transformation 7’ 1 is different, in general, from the 
T 2 . In matricial form: 
(7i) 
where: 
A = 
T l n~T} 4 x' 1 
Tii-Thy 1 
T\\ ~T\ 4 x* 2 
Til-T&y 2 
J’l 'T’l *1 rp 1 rp\ *1 
i 21 _7 24 x •‘31 ~ •'34 x 
A-tW ri-rV 
^■2 t"2 *2 7*2 T>2 *2 
I 2l~ 1 24 x i 31~ i 34 x 
T 2 2 -T} 4 y 2 T? 2 -T? 4 y' 2 
(72) 
[jf = [x;y;r] (73) 
Pf = [T^* 1 -TlyjUy 1 -Tl 2 -Tlx' 2 + -Ti V ,T 2 4 y 2 -Tl 2 \ 
(74) 
We have four equations in three unknowns. The problem is 
iperdetermined and can be solved ad example by least 
squares: 
44I4M4W 
A third made of consider the couple of equations is that of 
know the real and image co-ordinates of some points present 
in the two projections. It becomes possible to determine the 
elements of the transformation 77. Known these elements, 
also points of unknown real co-ordinates can be transformed. 
Let’s re-write the couple of equations: 
(7ji - T\ 4 x* )x + (T 2 \ -T 24 x*)y + 
+ (B31 -Tt, 4 x )z+(T 4 \ -T m x ) = 0 
, * O 6 ' 
(T12 ~ 7)4 y )x + {J22 ~ 7^24 y )y + 
+ (B32 ~T’3 4 y )z + (T 42 -T^y ) = 0 
* * 
Let both X , y and x, y, z be known quantities. We have 
then two equations in twelve unknowns, that are the elements 
77 . By applying these equations a six points not coplanar, 
with known model and plate co-ordinates, the 77 are 
obtained. The transformation producing the perspective 
projection, remains then determined for example a 
photograph. Note that none ‘a priori’ information on the 
transformation is needed. For example, in the case of 
photograph, it is not necessary to know' the point from where 
the photograph has been taken neither the orientation of the 
camera. In matricial form: 
*1 
0 
-*1*1 
y\ 
0 
-V1*1* 
*1 
0 
-*1*1* 
1 
0 -*j 
'Til 
0 
*1 
-•w* 
0 
Vi 
-VlVi* 
0 
*1 
-*1V1* 
0 
1 ->7 
712 
*2 
0 
~ X 2 X 2 
V2 
0 
-y 2 *2 
*2 
0 
— *2*2 
1 
0 — * 2 
714 
0 
*2 
-JC2V2 
0 
V2 
-V2V2 
0 
*2 
“*2V2 
0 
1 -y\ 
7-21 
*3 
0 
-■*3*3 
V3 
0 
-V3*3 
23 
0 
— *3*3 
1 
0 —*3 
T 22 
0 
*3 
-*3V3 
0 
V3 
-V3V3 
0 
*3 
-*3V3 
0 
1 ->-3 
T 2 A 
*4 
0 
-*4*4 
V4 
0 
-V4*4 
*4 
0 
— *4*4 
1 
0 — *4 
7-31 
0 
*4 
— *4+4 
0 
V4 
-V4V4 
0 
*4 
-*4V4 
0 
1 -y 4 
732 
*5 
0 
~ *5*3 
Vs 
0 
~V5*5 
*5 
0 
—*5*5 
1 
0 -*; 
7-34 
0 
*5 
-*5>’5 
0 
V5 
-V5V5 
0 
*5 
-*sVs 
0 
1 -y s 
7-41 
*6 
0 
-*6*6 
Ve 
0 
-V6*6 
*6 
0 
-*6*6 
1 
0 
1 
7-42 
0 
*6 
-*6V6 
0 
V6 
-V6V6 
0 
26 
-*6Ve 
0 
1 -V6. 
7*44 
(77) 
where the low index indicates the point of known co 
ordinates. In a compact form: 
[A’][T\ = 0 (78) 
When the equations are homogeneous, they contain an of 
arbitrary scale factor. For example we could divide by the 
term T u , so that the unknowns would reduce to eleven and 
the system would be normalised. If the transformation is 
normalised, the last column of the [A ] is moved to the first 
term, and the non-homogeneous system is solved. 
19. Notes of rendering 
In order to get a good simulation of the reality, the good 3D 
virtual reconstruction by drawings is not yet enough, also the 
lights, the colours, and the shades must be simulated and 
generated. When the object has already modelled, we 
proceed: 
Generation of a axonometric or perspective projection 
Removing hidden lines and surfaces 
Shading and radiosity of the body 
20. Clipping 
To solve the problem of removing hidden lines and surfaces, 
a solution is to order the objects according to their distance to 
projection plane. An other solution is to depict the objects 
one over the other beginning from the farest one (the so- 
called the painter algorithm). The second method is less 
accurate bud quicker. 
21. The shading 
The shading and the computation of the chromatic qualities 
or radiosity are the core of the rendering procedure. Th 
simplest algorithm is the law of cosine of Lambert. The light 
intensity reflected by the surface, is expressed as function of 
the cosine of the angle formed by the light with the normal 
vector to the surface. Such a model is: 
I r = 11 cos 9 (79) 
where 6 = angle between the normal vector and the 
incident light ray 
I r = intensity of the reflected light 
7 ■ = intensity of the mcident light 
The model can be improved taking into account the different 
tvrc of surface 
I r =K Ii cosO (80) 
where K is a coefficient for the surface. One can consider the 
luminositv of the ambient 
I = I r /d + K a I a (81) 
The limit of the model is that when the surface is curved, the 
computing time for the normal vector is rather long. When