H:\EXCERC\GEOD98e.EX1.wpd

**Department of Geophysics, Juliane Maries Vej 30, 2100
København Ø.**

E**xercise 1. Geodesy Course.**

The purpose of the exercise is to illustrate the transformation between different coordinate-systems and illustrate map-projections from the sphere to the plane and from the
ellipsoid to the plane.

1.1. A point has geographical latitude 56^{o} and longitude 10^{o},
as well as the height 0 above the ellipsoide in the GRS80
reference system. Compute geocentric latitude, the Cartesian
co-ordinate (X,Y,Z), as well as the length of the radius-vectors. What is the reduced latitude ?

1.2 The point in exercise 1.1 is now supposed to be given in
ED1950, i.e. on the International (Hayford) Ellipsoid. What
are the Cartesian Coordinates in this system ?

1.3 What are the isometric latitudes for points on a spherical Earth with radius 6371000 m having latitudes: 0^{o}, 45^{o}, 56^{o},
60^{o} and 80^{o} ?

1.4 We now define a Mercator-projection from a sphere with
radius 6371000 m, where the point with latitude 56^{o} and longitude 10^{o} are mapped into (X,Y) = (0,0). The scale of the
projection is 1:1 .

What are the plane co-ordinates for the points

Latitude Longitude

2 55.90 10.0

4 56.0 10.2

6 55.9 10.2

Give a mathematical expression for the inverse mapping. Check
the mapping by reverse transformation of point no. 6.

1.5 We now define a transverse cylindric projection with 9^{o}
longitude as contact meridian. 500000 m are added to all East-going coordinates (Eastings). The point with latitude 0^{o} and
longitude 9^{o} will then get the coordinates (E,N) = (500000 m, 0
m).

Calculate the image (E,N) of the point having 56^{o} latitude and
10^{o} longitude, as well as for the points 2, 4 og 6 in exercise
1.4.

1.6 The program **TRANS13** must be used to transform coordinates
for points given by their geographical coordinates on the
international ellipsoid (in ED1950) to coordinates in a
projection:

As zero-point (origin) use the point with co-ordinate 56^{o }
latitude and 10^{o} longitude. The point is located on map-sheet
1314 III.

The other points have the coordinates

Latitude Longitude

1 55.95 10.0

2 55.90 10.0

3 56.0 10.1

4 56.0 10.2

5 55.95 10.1

6 55.9 10.2

Transform the coordinates to coordinates in UTM zone 32,
Mercator with central meridian 10.0^{o}, base-parallel 56.0^{o } ,
Lambert conform-conical with 2 cutting parallels at 54^{o} and
56^{o}. Central meridian 10^{o}, scale-factor 1.0, as well as System
34, Jylland.

Calculate then distances and directions from the basis point, to the 6 points in the 3 map projections.

Compare distances and directions calculated in the different
projections with each other and with the distance measured in
the map.

Explain the differences between the distances and directions,
which depends on the map-projection.

In the exercise are used programs, which are found on the
unix-system disc disk1, in directory /disk1/cct/dgravsoft.

Using trans13 it is a good idea to create a file and write the
coordinates of the basis-point followed by the coordinates of
the other points into the file. This file can then be used as
input to all the runs with trans13. The program runs
interactively.

The file can be created and data input using the program **edit**
on a PC.

You access a unix-computer from a PC using telnet or X-win32.
The computers seth, geb or mani can be used. Under Windows
click on the Telnet or X-win32 icons. Then you will be
prompted for a user id and passwd.

PS: If you are in a hurry use the program **DIDR** (didr) for the
calculation of distances and directions.