2/6/2001
Geodetic and Geocentric Latitude
ASEN 3200
George H. Born
z
North
Observer
Circumscribing
Circle

Equator
r
b
y 


x
a
Cross section
of ellipse
R
+
e
Figure 1
Reference Ellipsoid representing the Earth
   Geocentric Latitude – The acute angle measured perpendicular to the equatorial
plane and a line joining the center of the earth and a point on the surface of the reference
ellipsoid. (   / 2      / 2 )
  Geodetic Latitude – The acute angle between the equator and a line drawn
perpendicular to the tangent of the reference ellipsoid. Map coordinates are given as
longitude and geodetic latitude.
  Reduced Latitude – See figure 1 for definition
Reference :
P.R. Escobal, “Method of Orbit Determination”, John Wiley & Sons, NY,
1965 # Page 24-29 and 135-136.
1
Useful Equations
z
1  e 2 sin 

sin  = =
;
r
1  e 2 sin 2 
cos   =
sin  =
x
cos 
=
;
r
1  e 2 sin 2 
cos  =
sin  
1  e 2 cos 2  
1  e 2 cos 
1  e 2 cos 2  
In terms of geocentric latitude, we have
x =
z =
R 1  e 2 cos 
1  e cos  
2
2
,r =
x 2  z 2 = R(1  f sin 2  )
R 1  e 2 sin  
1  e 2 cos 2  
Where e is the eccentricity and f is the flattening of the ellipsoid.
For the World Geodetic Survey (WGS-84)
f 
ab
,
a
e2 = 2 f  f 2 ,
f =
1.0
,
298.25722
e = 0.0818191
In terms of geodetic latitude, we have
sin 2  =
(1  e 2 ) sin 2 
1  e 2 cos 2 
cos 2  =
(1  e 2 ) cos 2 
1  e 2 cos 2 
x =
R cos 
1  e sin 
2
2
,
z =
R(1  e 2 ) sin 
1  e sin 
2
2
,
r =
x2  z2
Note that if we wish to find x and z for a tracking station located at height H above the
reference ellipsoid we must add (H cos  ) to x and (H sin  ) to z (see Eqn. 2.8-7 in Bate
Mueller and White).
2
The relation between  and   is given by
tan 
=
tan  
(1  f ) 2
3