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KITSAT-1 
ast mean 
cof GCP 
ips. Fig. 1 
correction 
neters 
Ps 
  
sm 
ablish the 
as. Three 
nter of the 
M be the 
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as shown 
Satellite Orbit 
  
        
    
  
Satellite C2 
z- (8 
The Earth North Pole 
  
  
0? Longitude 
  
  
  
: f : 3 
Figure 3. The Establishment of Four Coordinate Systems 
Let the coordinates of a point on the CCD sensor denoted 
in C1 system be (xy) and the longitude and latitude 
values of image appeared in the point be (o,D). If T is the 
function determined by the geometric model, it satisfies 
the relationships in that 
T(x,y)=(0.,B) (1) 
T (0, B)=(x.y) - "(23 
3.1 Non-inverse Geometric Model 
The following procedures are to derive the function T. 
Since the coordinates of the point A is expressed in C2 
system as shown Eq.(3), the vector L can be denoted as 
Eq.(4). 
Az(x,y)ci = (EX,Y)e2 (3) 
Lez = (0,0,0) - (f,x,y) = (-f,-x,-Y)e2 (4) 
If the rotation matrix between the C2 and C3 system is 
denoted as Res and the rotation matrix between the C3 
and C4 system Ras, the vector L is expressed in C4 
system as 
L 
Les = f = RuR Lez 
J, 
CaCs c7 San oT CSS CiC2 C1$S2$3 + SıC3 015103 + S1S3 —f 
= SiC (Ca as] SICH ZI CIE SC CS NA 
Ss 0 Cs $2 - C153 C203 ay 
where c1=cos64, S1=sin6;, C2=c0s02, S2=SIN02, C3=C0OS%:, 
S3=SIN03, C4-COS 0, S4=SING, Cs=COSN, And Ss=sinn. 
The position of the point B is nearly the same as the 
location of the satellite in C4 system. The position of the 
satellite is found from 
Bes=((R+h)esc4, (R+h)ess4, (R+h)ss) (6) 
An arbitrary point C on the earth with certain longitude 
and latitude have the coordinates like 
C = (Rcosfcosa, Rcosfsina, Rsinf) (7) 
The vector M that is from the point B to C is expressed as 
M = C - B = (Rcosficoso-(R- h)css,, 
Rcosfsino-(R+h)csS4, Rsinf-(R+h)ss) (8) 
The condition that the vector M is parallel to the vector L 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
makes the following three relationships; 
Rcosfcoso-(R+h)esc4 = kl, 
Rcosfsino-(R+h)css4 = kl, (9) 
Rsinß-(R+h)ss = kl, 
where k is an arbitrary real number. 
If cosß=0, it implies the position on the north or the south 
pole and then the longitude value « is meaningless. If 
cosB=0, the second order equation about k is derived 
from Eq.(9) and described as 
KA eM) 2kCf Qe h)esca- LU h)ess;- 
l, (R+h)ss) + (R+h)’ - R*=0 (10) 
The value of A never come to zero because the 
point A is the location coordinates of the satellite, the Æ 
can be evaluated by solving the second order equation of 
Eq.(10). At last, the o and B are derived from 
p ss +(R ce 
  
R 
kl, +(R+h kl.+(R+h a 
cst +(R+h)csss kL+(R+h)cscs 
Rcosp ti: RcosB 
by substituting the k evaluated above. If the second order 
equation in Eq.(10) may produce two different solutions of 
k and then also of « and p, the proper one between two 
solution is one of the point with the shorter distance from 
the satellite. The function of T was then determined by 
the values derived in Eqs(3- 11). 
3.2 Inverse Geometric Model 
The inverse function of the geometric model, T' is 
derived reversibly by using Eqs.(3-11). The vector Lcs is 
derived from Eq.(5) as 
l. 
Lo [ = 
L 
(12) 
RcosBcos — (R + h)cscs 
where W =| Rcosfsina — (R + h)css: 
RsinB - (R * h)s: 
Since Re. and Ras are rotation matrices between 
coordinate systems, they are orthogonal 
matrices.(Hughes) The vector Lc2 is expressed from 
Eq.(5) as 
-zf 
Loz —X ER RD Ra Ru W (13) 
=v 
The k, x and y can derived from Eq.(13) and described as 
1 
k = —— RR: Ru W 
i | 
X 1 T. T 
| | Ru Ww] 
(14) 
y 
431 
des