Back to Notes_on_Map_Geometry_and_Projections

**How to flatten Earth onto a plane map?**

__This cannot be done without distorting the result__

**!**

**Properties we would like a map to
have**

- Showing area shapes correctly: Angle-true
- Showing distance true to scale: Distance-true
- Showing areas correctly: Area-true
- Showing points at correct position: Position-true

All of these cannot be achieved throughout the entire map for true projective (simple) Map Projections.

But any one, singly, can be achieved through implementing conditions (differential conditions), however all of the other properties will be violated.

As a compromise restricting the area mapped and how it is mapped one can produce maps with distortions no larger than is acceptable for a particular application scenario.

Mapping authorities all around the World have chosen the latter approach (more about this later).

**Examples of simple projective Map
Projections**

**1. Sterographic (planar)
Projections**

Imagine a huge plane (flat) projection screen being placed
against Earth touching in a single point (of the selected Ellipsoid
at MSL) and a projection source (e.g. light source) placed at a
point exactly opposite. Tracing all features of interest on the
projection screen will produce a **Stereographic (planar)
Projection** map of Earth.

The only place where this kind of map will meet all of the desirable map properties (true to ..) can be found in the point where the imaginary projection screen touches the mapping surface ==> a single point, the point of tangancy. Mathematically speaking, a point has a position, but no extent, no size, no area. That means this kind of map is distorted through out its entire extent.

Note however, map distortions grow with the distance away from the distortion free feature (here the point of tangancy) slowly at an increasing rate.

That means, that distortions up to a certain distance away from the point of tangency may still be managable (small enough to be neglected for specific applications).

In this resepect, the shear size of Earth is helpful. Near circular countries like Switzerland may use this type of Map Projection when the point of tangency is situated at the centroid of the country.

This type of projection can be modified to enlarge the area of
acceptably small distortions by placing the imaginary projection
screen slightly below the surface of the selected ellipsoid. This
would produce a (circular) line of intersection between the two
with no distortions on the map. (A line has a length, but no width
and thus no area). By adjusting the depth of intersection, the area
inside the intersection circle can be held to acceptable distortion
magnitudes and at the same time expanding the reach of acceptable
distortions out side this line (circle).

This type of projection can further be modified by placing the projection (light) source closer (larger distortions) or further away (smaller distortions) from the imaginary projection screen.

**2. Conical Projections**

Imagine the projection screen rolled up to form a 'witches hat' (cone) and being forced against Earth's ellipsoid to produce a single touching line (line of tangency).

The distortion free feature no is a line of considerable length, but no width and no area.

Distortions grow with the distance away from the line of tangency. Using a similar trick as before, intersecting the ellipsoid by pushing the conical screen below the ellipsoid surface again enlarges the width of acceptable distortions about the initial line of tangency,

This kind of projection is suitable for countries like Chile or Austria.

**3. Cylindric Projections: Mercator**

Imagine the projection screen being rolled up into a cylinder touching the ellipsoid along its equator creating a single line of tangency. Modifications as explained above apply here also.

Distortions are extrem for pole near locations. The poles (single points) will be mapped as a line in length equaly at scale to the circumference of Earth.

However, this kind of map has one desirable property: (magnetic)
compass bearings are (reasonable) true as map angle between
meridian and a course line drawn as a straight line. Mercator maps
have long been in use for maritime and other navigation
purposes.

Mercator maps are however otherwise distorted in all respects with
distortions growing with the distance away from the equator.

**4. Cylindric Projections: Transverse
Mercator**

A very similar projection type to the one above with the modification of wrapping the mapping cylinder around Earth so that it touches along a single meridian, the 'Central Meridian'. The Central Meridian in this case is the line of tangeny. It can also be called a line of 'identity' because it will not distort the map along its entire length. Unfortunately, as any line, it has no width and area.

Similar modifications as described above, reducing the diameter of the mapping cylinder creates two lines where the cylinder will intersect Earth equal distant from the original Central Meridian. These intersection lines now become 'lines of identity' with no distortions. Map content between these two lines will be shown at reduced size (reduced scale factor, less than 1), along theese lines the map has a scale of exactly one and outside it shown Earth enlarged (scale factor larger than 1). Because the map scale varies with the distance of points away from the Central Meridian, one speaks of a variable 'Point Scale Factor'.

This kind of map looses the property of correct angular bearings and needs to be dealt with mathematically (see below).

This style of map is one of the major map projections in use today with additonal modifications as described below.

**5. The good compromise: UTM, Universial
Transverse Mercator projection**

UTM is a Tranverse Mercator projection as briefly described above with two major modifications:

- It is 'conformal': Angle-Truth is enforced by implementing
differential conditions causing that angles between intersecting
lines on Earth are shown correct when compatible lines on the map
are used.

In general terms, straight lines on the map have their compatible counter part in curved lines on Earth and visa versa (see link below for Azimuth and Arc-to-Chord correction). - International Agreements exists about mapping 'Zone' width, location and numbering (see the second link below).

For the GIS practioner it is important to be familiar with the principles shown in the link below, which is a good example about UTM, as used in Australia and applies in broad terms to any country where UTM maps are in use.

For a very comprehensive description see [1].

You find a complete map of all UTM mapping Zones here [2].

The information about a GIS map layers projection is coded in the layers *.prj file:

PROJCS["**UTM Zone 55, Southern Hemisphere**",GEOGCS["GRS
1980(IUGG,
1980)",DATUM["D_unknown",SPHEROID["GRS80",6378137,298.257222101]],PRIMEM["Greenwich",0],UNIT["Degree",0.017453292519943295]],**PROJECTION["Transverse_Mercator"],PARAMETER["latitude_of_origin",0],PARAMETER["central_meridian",147],PARAMETER["scale_factor",0.9996],PARAMETER["false_easting",500000],PARAMETER["false_northing",10000000],UNIT["Meter",1]]**

In this example, the layer is a UTM projection based on GDA (=GRS80) for UTM Zone 55.

The map projection obviously is MGA with Zone 55 centred on Central Meridian at 147 degree East.

The Zone has a Central Scale Factor of 0.9996.

Map coordinates are referring to a False Origin which has been shifted by 500000 m west and 1000000 meters south of the True Origin (intersection of equator and Central meridian).

Coordinate units are set to Meter.

*This page may contain inconsistencies and errors. Please correct or complain to --Gngdowid 04:53, 25 December 2009 (UTC)*