Geodetic Control Network
Lecture 3.
The positioning and orientation of the horizontal control
network on the ellipsoid.
Outline
The positioning and orientation of the network on
the ellipsoid
• astronomical observations;
• Laplace stations and „deflection of vertical” stations;
• Centring excentric observations;
Computations on the ellipsoid
• Ellipsoidal curves (normal sections, curve of alignment,
the geodesic);
• The computation of ellisoidal triangles;
• The differential equations of the geodesic;
The positioning and orientation of the HCN
The necessity of the application of surface
coordinate systems.
Geoid -> not a mathematical surface
Ellipsoid fits the geoid well, and is a
mathematical surface.
At least the ellipsoidal position of one
point of the network, and the orientation
of at least one side of the network must
be determined.
Astronomical observations
Astronomical observations
Astronomical positioning: the determination of the coordinates
of terrestrial points using observations to the stars.
Observations: horizontal and vertical angles; timing; <-> GPS
The „old-fashioned way”:
Since the standing axis of the instrument is vertical, therefore
astronomical latitudes, longitudes and azimuths are measured and
NOT ellipsoidal ones.
The types of astronomical observations:
As already discussed, every 120-150 km a pair of astronomical
stations should be created. The astronomical observations are
important for the positioning and orientation of the network, as
well as:
• the azimuth and latitude observations can be used to minimize
Geodesy
the angular distortions in the network;
• in case of large number of observations, the parameters of a
best fitting ellipsoid can be determined;
• the determination of the geoid – ellipsoid distances can be
computed (geoid undulations)
Types of astronomical points
Laplace points:
Points, where longitude and azimuth values are measured.
Here the Laplace equation can be written:
       sin
where:
• , are the astronomical azimuth and longitude values;
• , are the ellipsoidal azimuth and longitude values;
•  is the ellipsoidal latitude (which can be approximated by the
astronomical latitude as well).
By reformatting the above equation, the ellipsoidal azimuth can
be computed:
       sin
Since the astronomical latitude is not necessary for the
Laplace equation, therefore it is not mandatory to
measure the latitude at the Laplace points.
Types of astronomical points
„Deflection of the vertical” points:
Only latitude and longitude, or latitude and azimuth is observed.
What is the deflection of the vertical?
The angle between the local vertical and the ellipsoidal normal (Q).
The direction of the deflection has an angle e with the local meridian.
Thus the deflection of the vertical in the meridian direction and in the
orthogonal direction is:
  Q cose
  Q sine
The astronomical observations can be used to compute these values:
   
     cos
   A   cot
,  or
,  should be
measured
Choosing the loc. of astronomical points
Usually both ends of the extended baselines are used as the
location of the astronomical points:
• one of them is a „master station”: (, , )
• the other one is the „slave station”: (,  for the Laplace eq.)
But! Due to the size of the instruments, the chosen point
should have a good transportation connection.
And good visibility to the sky as well.
Reducing excentric observations
In many cases the observations are taken at an excentric station.
Reducing the latitude and longitude values:
   ' '
r
cos 
R
and
 
r sin   ' '
T sec
R cos  15
Reducing the azimuth observations:
A  Aobserved     , where
r
t
r sin 
  ''
tgQ
R
   ' ' sine
 – the convergence between
the two meridian
Outline
The positioning and orientation of the network on
the ellipsoid
• astronomical observations;
• Laplace stations and „deflection of vertical” stations;
• Centring excentric observations;
Computations on the ellipsoid
• Ellipsoidal curves (normal sections, curve of alignment,
the geodesic);
• The computation of ellisoidal triangles;
• The differential equations of the geodesic;
Basic principles
Ellipsoid of rotation: A 3D surface, which created by ellipse,
rotated around its minor axis.
Geometric parameters:
a
b
-
semimajor axis
semiminor axis
e2 
a2  b2
a2
-
the square of the first excentricity
e2 
a 2  b2
b2
-
the square of the second excentricity
-
flattening
-
inverse flattening
f 
a b
a
f 1 
a
a b
Two of these should be specified to identify the ellipsoid,
one of them should be the size.
Definition of ellipsoidal coordinates
 – ellipsoidal latitude
 – ellipsoidal longitude
 – ellipsoidal azimuth
The reduced latitude:
Thank You for Your Attention!