wo values for k. The 
uch that, for a positive 
to that shown in fig.(1) 
  
  
aoc (15) 
aol 
r ago > 0 
16 
o) ago « O (16) 
late the value of 1, k' 
and (11) respectively. 
of K, L, K' and L’ are 
vith a 
at K' and L' coincide 
. K and L respectively. 
are centrally collinear. 
the collineation center 
athematically, this is a 
  
PE 
ce of T with à 
nes to a common carte- 
id K’ are chosen to be 
s, while the axes X and 
L' K' respectively. The 
ular to them as shown 
; will coincide with each 
cide as shown in fig.(5). 
ormed through several 
Vienna 1996 
  
  
  
Figure 6: Relationship between Homogeneous and 
rectangular coordinates 
1. For any finite points, let zo&z, be taken — 1, 
then the transformation from the homogeneous 
coordinate (1 : z; : z2) to rectangular Ç, (fig.6) 
is as follows: 
nr. sind 
Ç= 11 +n cotô (17a) 
and conversely 
z2 = n/sin® (17b) 
11 =(—n cot0 
2. From fig.(6) we get the relation between the 
space coordinate (X, Y) and (6, 5) as follows: 
Y =" "cos + Csnvy + Yi 
X Sauna” +" Ecosdt qoos ooo) 
and conversely 
é (Y -Yi)sinv + (X-Xi)cosv 
7) (Y -Yi)esy - (X-Xi)sinv 
(18b) 
where y 24 (X, A14) 7 arctan(Y —Y1)/(Xo — 
X1). 
uod 
3. From equation(17) (18), the space coordinates of 
K(1:k:0) and L(1:0:1) are found as follows: 
XK = X1 + k cos VY (19a) 
YK = Y; + ksin y 
and 
Xp 2 X, +1cos (y +0) (19b) 
YL = Y, +1sin (v +0) 
4. Let x 2 4 (X, LK), then 
x 2 arctan(Yk — Yz)/(Xk — Xr) (20) 
  
9. 
10. 
. Finally we get the relation between (X,Y) and 
x yy 
X =ukk + X cos x — Y sinx (21a) 
Vo UmY*R n oXsny u- Yecosy 
and 
X = (X-Xxk)cox + (Y-Yx)sinx 
Yo. = 
—(X — Xr )sin + (Y -Yx)cosx 
( K)sinx ( K) fo) 
. Similar formulas can be deduced for the im- 
age plane, except for Y' having opposite direc- 
tion(see fig.(4b)), hence: 
N S SiO 
QU m X + y cot? (22) 
[ler =a Xi hs kf! cosy 
K 1 
Yi: =. Y cb EF sing (23) 
Xp = Xi + ll cos(y' +6) (24) 
Ya — y! + I sin(y’ + 0’) 
where 
Hs: Y! Y/) 
= A A’ AT = t 2 1 
v XY ( yt] 2) arc n(x; — x) 
— BÓ ————À y? P. Y1) 
! zXOU IR tant cette 
x. (X. L'K Jerstan( re zy 
X 
Y 
(X' — X&.) cos x' t- (Y' = Yi.) sin x’ 
(X' — Xg.)sin x' - (Y' = Yi.) cos x’ 
(25 
lo 
Using (21b) and (25), the coordinates of each 
point A; and its image point A} relative to the 
common system (X — X', Y — Y") can be calcu- 
lated. 
. The common point of intersection between the 
lines A;A; (à = 1...4)can now be easily 
found. It is the center of perspective collineation 
So(X so, Y so); fig(?). 
. The space Coordinate of So( X so, Yso) are calcu- 
lated from (21a). 
From (5) : V,(1 : Ha : 0), hence its coordinates 
in space Xvi, Yv1 can be found from (17a) and 
(18a). The equation of the line v through Vi 
parallel to t will be: 
(Ÿ — Yv1) = (X — Xvi)tanx. (26a) 
Similarly the equation of t will be : 
(Y — Yk) =(X — Xk) tan. (26b) 
The distance r = Sov = IY s, — Yv,| can now be 
found from (26a) to be: 
r= (Ys, = Yy,) cos x 7 (Xs, = Xy, ) sin x| 3 
(27) 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996