given by: 
$, -arctan[(1-e?)tan] (10) 
2 CENTRAL PROJECTION 
2.1 Vertical Projection onto a Tangent Plane T 
(Minimal Distance) 
From Fig.(4), we have for the point P: 
y,=LP tanA =H tan(A-A,) 
So we get: 
AXsin®, -AZsin®, 
AZsinó, * AXcosó, 
Yı (11) 
Also 
NY y, Y» 
PK  LPsinA, 
  
X; 
where LP=AX/sinA. So we get 
DH TITI 
AZsino, *AXcosó, 
(12) 
To derive the mapping equation in terms of the 
geocentric coordinates (¢,A,h) of the point P, P, and L 
we have: 
For the point P: 
X,=PJ= (N+h)cosdpcosdA 
Y "PS -(N * h)cosásindA (13) 
Z -JO-[N( -e?) -h]sinó 
For the point P, : 
X, -MO-(N,*h,) cos, 
y 0 (14) 
Z, -P,M-IN, -e^) *h]sinó, 
X, -RO-RM*MO"[H-*h,*N ]cosó,-Rcoso, 
Y,-0 (45) 
Z,=LR=[H+h,+N (1-e?]sind,=R cos, 
Substituting from the above geocentric coordinates of the 
points P, P,, and L into equations (11) and (12), we 
obtain after some trignometrical and algebraic reductions 
the mapping equations: 
334 
CsindA 
G -Ssino, - Ccos$,cosdA 
  
(16) 
F+Scos®, -Csin®,cosdÀ 
G -Ssin, -Ccosd,cosdÀ 
  
y,=H 
where 
c= 0s 
a 
d 
s= N(1-e Los id 
a 
$ 
(17) 
x ?N sin,cos®, 
a 
p Ht * N, 1 - e?sin?$) 
a 
  
After expanding the forms of sine and cosine of (A-A) 
and substitiuting into equations (16), we get the general 
mapping formulas of the vertical perspective projection of 
the rotational ellipsoid onto the tangent plane of minimal 
distance: 
CcosA sinA -CsinA cosA 
G-Ssind, -Ccos,cosÀ, cosÀ -Ccos®, sin, sind 
  
x, =H 
(18) 
F+Scos®, -Csin®,cosÀ,cosÀ -Csin®,sinÀ,sinÀ 
G-Ssinh, -Ccos, cos, cosÀ -Ccos,sinA sinA 
  
» (19) 
2.2 The mapping equations using geocentric latitude 
of the perspective center 
The mapping equations (18) and (19) can also be 
modified if the geocentric latitude ¢, of the perspective 
center L is given as in the case of some satellite 
problems. So we have: 
cos, 
  
R,=(H+h,+N, 
cos, 
e R cos($, - 6 ie) 
a 
(20) 
FS R,cos(®, -®,) 
a 
e?N sind cos 
0, 7d, arcein | onte 
0 
The value of ¢, may be found by a rapidly converging 
iteration, with an initial value of ¢,=¢, and using 
equation (20) to obtain Ro. The new value R. is used to 
obtain the next approximation of d, . The values of R 
and 6, are iterated until the change in d) is considered 
negleigible. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996