VERTICAL PERSPECTIVE PROJECTION OF THE ROTATIONAL ELLIPSOID 
Prof. Wagih N. Hanna 
Faculty of Engineering Ain Shams University , Abbassia, Cairo, EGYPT 
Comission IV , Working Group 2 
KEY WORDS: Cartography, Space, Mapping, Analytical, Geometric. 
ABSTRACT 
The part of the earth visible from a certain outer point, which is the perspective center is illustrated in the photo from 
the space camera as a perspective projection. We consider the camera axis to be truly vertical through the center of 
projection and normal to the surface of the ellipsoid. In fact there are two real normals through any point in the plane 
of the ellipse. One normal belongs to the nearest part of latitude ¢, , another normal to the other part of latitude à. 
These latitudes are of different values and signs. The location of any point is given by the geographic coordinates ¢, 
À as well as its altitude h above the reference ellipsoid. In this paper we deduce analytically the corresponding map 
coordinates of the perspective projection of the points on the surface of the rotational ellipsoid onto the tangent plane 
through the two points of intersection of these normals with the ellipsoid. The coordinates are deduced in terms of the 
geodetic and geocentric coordinates. 
1 INTRODUCTION 
1.1 Notations Fig.(1), Fig.(2) 
L 
H 
LO, LO" 
ó 
À 
h 
P(®, A,h) 
P. (OA) 
PDA Be) 
T 
T, 
8 
P,(&,.Y1) 
X,-Xp 
Y,-Y, 
LZ» 
LP, 
$,- $', 
o 
: The perspective center. 
: The altitude of L. 
: The perpendiculars on the ellipsoid 
from L. 
: Latitude. 
: Longitude 
: Altitude above the reference ellipsoid. 
: Any Point on the earth= P(X,,Y,Z,). 
: Center of map (minimal distance). 
: Center of map (maximal distance). 
: Tangent plane at P,. 
: Tangent plane at P',. 
: The geocentric latitude w.r.t the center 
of the earth O. 
: The map point corresponding to P 
AY 
: AZ 
:D 
: Ab 
"dX 
1.2 The Geometry of the Rotational Ellipsoid 
  
Consider the earth as a rotational ellipsoid (Reference 
Ellipsoid) with semi-major axis a and semi minor axis c 
(axis of rotation). The equation of the rotational ellipsoid 
in the system shown in (Fig.2) is given by: 
X1«y?..2? 
eor 
a? ce? 
  
(1) 
The following elliptical parameters (Fig.3) are frequently 
used in the analytical calculations: 
Eccentricity : n. (a? -c?) 
a? 
Q) 
332 
Radius of curvature: at a point on the ellipsoid: 
a 
V1 - e? sin) 
1.3 Coordinates 
N- (3) 
The position of a point on the surface of the rotational 
ellipsoid is expressed by either the geocentric cartesian 
coordinates (X,Y,Z) or by the two dimensional geodetic 
coordinates (¢,A), where : 
X= coscosA (4) 
Y1-e?sind 
Y= cos®sinÀ (5) 
1-e?sin$ 
Z- a(1-e?) sind © 
1-e%sind 
¢=arctan[ Z ] (7) 
  
A -arctan(-. -arccos[ ] (8) 
The equation of the tangent plane at any point ($2) is 
given by: 
2 
xcosdcosA *ycosdsinA *sin$-^A (9) 
Sometimes the points are expressed by the geocentric 
latitude dg, which is the angle subtended by the 
geocentric line OP and the equator. The relation between 
the geocentric latitude ¢g and the geodetic latitude ¢ is 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996