2.3 Vertical Projection onto a Tangent Plane T, 
(Maximal Distance) 
In this case, the perspective center L is at a distance 
LP’ =D from the tangent plane T, , the relation between 
$, and $', is given by the equation: 
i . e?sin(20,) 
¢’,=-¢,+arcs wt Q1) 
N 
0 
The normal distance LP’ =D between the center L and the 
plane T, is given by: 
cos, 
D-N' *[N,*H] 
  
cos’, Ga) 
The mapping equations can be obtained by the same 
method explained above. The analogy of equations (16) 
gives the mapping coordinates relating to the new system: 
, CsindA 
x,=D 
G-Ssin®', -Ccosd®',cosdA 
  
(23) 
F'+Scos®', -Csin®' cosdA 
G /-Ssind", - CcosQ", cosdA 
where 
  
yi^ 
_e*N sin cosd’, - (H + h, + N )sinAd 
a 
F' 
  
Q4) 
Sd -e?N sino, sin $ + (H + h, + N.) cosA® 
  
a 
C and S are determined from equation (17). 
2.4 Inverse Mapping Equations 
The inverse mapping equations for the vertical 
prespective projection of the ellipsoid permit the 
calculation of geoditic latitude d and longitude À for a 
selected point which has rectangular coordinates x, , and 
yi: 
2.4.1 Geoditic Latitude ¢ : Since the mapping equations 
(18),(19) are linear in sinA and cosA, then they can be 
arranged as follows: 
A, C sin) + A, C cos) = A, S + A, 
A, C sind + A, C cosÀ = A, S + As 25) 
By elliminating A from equations (25) we get a quadratic 
equation in S, which has two solutions. One of them 
gives the latitude of the nearest point on the ellipsoid, the 
other of the hidden point on the other side. Since Z>0 
then the positive root is considered. So we have : 
sind = a (26) 
336 
Then by using an itteration method we get ¢. 
2.4.2 Geoditic Longitude À: After some trignometric 
calculations equations (25) can be reduced to: 
Tsi = 
A=A, an i So co (27) 
  
S (y,sino, + Hcos®,) - 9, G - HF) 
where: T=[H+h,+N,(1-e")]/a 
References 
[1] Abbas, Y. Triple projection of a topographic surface 
from an external persepective center, Ph.D Thesis, Assuit 
University, Egypt (1993). 
[2] Alfred, L. GPS Satellite Survying, John Wiley and 
Sons, Inc., New York (1990). 
[3]Beate, M., Kartenprojektionen des dreiacbsigin 
Ellipsoid Geodatishes Institut, Stuttgart (1991). 
[4] Dragomir, V. Theory of Earth Shape., Elsevier 
Scientific Publishing Company, Amsterdam, Oxford, New 
York (1982). 
[5] Dumetrescu T.V., Kosmographishe Persepective 
Projektionen. Allgemeine Vermeasugs Nachrichten pp. 
91-104 (1974). 
[6] Fredenick P. , Map Projection Methods, Sigma 
Scientific Inc., Blacksburg, Virginia (1984) 
[7] Kahle H.G., Einführung in die Hóhere Geodasie 
Verlag der Fachverein, Zürich, (1988). 
[8] Lauf G.B. Geodesy and Map Projection , Tafe Pub. 
Unit (1983). 
[9] Peno P., Transformation of Rectangular Coordinates 
into Geographical by Closed Formulas, Mapping and 
Photogrammetry Vol. 20, No. 3, pp 175-177, (1978). 
[10] Snyder P.J., The perspective Map Projection of the 
Earth, The American Cartographer Vol. 8, No. 2, pp 149- 
160 (1981). 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996 
  
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