Projections Systems
Spheroid, Datum, Projections
Projections
As you know now, the earth is round, and I don\'t know if you have tried to flatten a ball but it is rather difficult without creating permanent damage. It is the same case with projections. Projection are a way to represent the earth on a flat piece of paper. You project features from the Earth to the Paper. As the earth gets deformed in the process (permanent damage), we have to choose which caracteristics we want to keep: surfaces correct, angles correct...
This defines the way the earth will be flattened or projected onto the paper. Think about shadows.
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Robinson and Azimuth Equidistant Projections
Graticule
The graticule is the representation of Longitude and Latitude on the paper.
Projection Types
There are mainly three type of projections. Cylindrical, Conical and Hemispherical. There are some other projections but they are rarely used. The problem of a projection of a spheroid is that the user has to choose which parameters should be conserved: Angles, Distance or Surfaces. Each projection system will coserved one of this parameter while trying to minimize the error on the other parameters.
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Cylindrical
A cylindrical projection is the common type of projection used for navigation. It keeps angles correct.
Conical
A conical projection will minimise the distortion created by the projection. However this projection is not optimal for survey work.
Hemispherical
This projection is mainly used for representing only half hemispheres. It is used to display Earth poles.
Non-Projected (MapInfo for Long/Lat)
This last projection is not a real projection, but it is the way for MapInfo? to represent any dataset which is in Geodetic (Long/Lat) coordinates. It is however recommended to always use a Longitude/Latitude projection associated with a datum to facilitate coordinates conversion when using the table with several other tables in different systems.
Areas of Accuracy
For each type of projection only a small area of the map is accurate to a scale factor to the area on the Earth.
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The red line has accurate distance
Mercator and Transverse Mercator
A Mercator projection is of type cylindrical, while a transverse mercator is a cylindrical projection but with the cylinder axe not oriented South/North. The main characteristic of a mercato projection is the fact that the radius of the cylinder is smaller that the spheroid radius. It allows the cylinder to intersect the spheroid on to circles. Any object located on the earth on these two circles won\'t be distorded by the projection, while the area between the two circles will be of minimum distortion.
This specificty allows that over small earth area the projection is very similar to the real earth and that distances on the map are identical at a scale factor to the distance on the earth. This is interesting for surveying as the projection graticule is very similar to the datum grid.
To increase the accuracy of the distance on the area between the circles we usally adopt a scale factor for the distances on the circle. We then obtain a scale factor of 1 in the middle of the two circles. Usually the scale factor is defined as:
Scale Factor=1-1/(Map Scale)
The projection is specified by the point of connection between the geodetic coordinates to the false coordinates generally expressed in meters.
Universal Transverse Mercator
The fact that the graticule is very similar to the grid makes the Transverse Mercator projection one of the most used. However to be of world wide used the parameters of the projection have been standardised as the term universal. In Univeral Transverse Mercator or UTM is the earth is divided in 60 zones of 6 degrees. UTM zone 1 to UTM zone 60.
The point of origin Longitude is spaced every 6 degrees between zones starting at 177 W or -177 to 177 E or 177.
| Zone 1 | Zone 2 | Zone 3 | Zone 31 | Zone 58 | Zone 59 | Zone 60
|
| -180/-174 | -174/-168 | -168/-162 | 0/6 | 162/168 | 168/174 | 174/180
|
| -177 | -171 | -165 | 3 | 165 | 171 | 177 |
UTM Grid
In the Case of MapInfo? for the Pacific, we decided not to cut the Pacific in half and we used the range 0 to 360 degrees instead of the normal cartographic range -180 to 180 degrees. There fore we define 30 extra zones from Zone 61 to zone 90. It is to be understood that these zones are equivalent to Zone 1 to Zone 30 except for MapInfo?. It is not uncommon to see people superposing tables created in Zone 1 with tables created in Zone 61. If the False coordinates will be identical MapInfo? won\'t treat the projections are identical, and strange displays will result.
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The formula to convert False Coordinates into Geodetic Corrdinates is given:
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ko is scale on the central meridian
Mo is M for Oo
a and e are specified by the ellipsoid
Phio and lambda0 are the UTM Origin in degrees
Phi and lambda are the coordinates to convert in Easting/Northing
When x and y are found, the user must add the origin Easting and Northing to obtain the real X and Y Easting and Northing, then the user can compute k.
Next...
Franck Martin 1998
With thanks to Mike Poidevin for his inputs.
Books to read:
- Map Projections- A working Manual, John P. Snyder, USGS Professional Paper 1385
- Geodesy: the Concepts, Petr Vanicek, Edward J. Krakiwsky, North Holland
