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

Exercise 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 56o and longitude 10o, 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: 0o, 45o, 56o, 60o and 80o ?

1.4 We now define a Mercator-projection from a sphere with radius 6371000 m, where the point with latitude 56o and longitude 10o 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 9o longitude as contact meridian. 500000 m are added to all East-going coordinates (Eastings). The point with latitude 0o and longitude 9o will then get the coordinates (E,N) = (500000 m, 0 m).

Calculate the image (E,N) of the point having 56o latitude and 10o 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 56o latitude and 10o 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 10.0o, base-parallel 56.0o , Lambert conform-conical with 2 cutting parallels at 54o and 56o. Central meridian 10o, 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.