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References
Knippers, R.A. (1998). Coordinate systems and Map projections. Non-published notes, Enschede, ITC.
Knippers, R.A. (1999). Geometric Aspects of Mapping. Non-published notes, Enschede, ITC.
Stefanovic, P. (1996) Georeferencing and Coordinate Transformations. Non-published notes. Enschede, ITC.
2) Scale distortions on a Map
The transformation from the curved reference surface of the earth to the flat plane of the map is never completely successful. Look at the diagram below. By flattening the curved surface of the sphere onto the map the curved surface is stretched in a non-uniform manner.
It appears that it is impossible to project the Earth on a flat piece of paper without any locational distortions, therefore without any scale distortions.
Projection plane tangent to the reference surface
The distortions increase as the distance from the central point of the projection increases. Placing the map plane so that it intersects the reference surface will reduce and mean out the scale errors.
Since no map projection maintains correct scale throughout the map, it may be important to know the extent to which the scale varies on a map.
On a world map, the distortions are evident where landmasses are wrongly sized or out of shape and the meridians and parallels do not intersect at right angles or are not spaced uniformly. Some maps have a scale reduction diagram, which indicates the map scale at different locations, helping the map-reader to become aware of the distortions.
On maps at larger scales, maps of countries or even city maps, the distortions are not evident to the eye. However, the map user should be aware of the distortions if he or she computes distances, areas or angles on the basis of measurements taken from these maps.
Scale distortions can be measured and shown on a map by ellipses of distortion. The ellipse of distortion, which is also known as Tissot's Indicatrix, shows the shape of an infinitesimally small circle with a fixed scale on the earth as it appears when plotted on the map. Every circle is plotted as circle or an ellipse or, in extreme cases, as a straight line.
The size and shape of the ellipse shows how much the scale is changed and in what direction. On map projections where all indicatrices remain circles, but the sizes change, the scale change is the same in all directions at each location. These conformal projections represent angles correctly and have no local shape distortion ( e.g. the Mercator projection ).
The indicatrices on the diagram below are circles along the equator. There are no scale distortions along the equator. The indicatrices elsewhere are ellipses with varying degrees of flattening. The projection represents areas correctly - all ellipses have the same area - but angles and, consequently, shapes are not represented correctly.
Lambert Cylindrical equal-area projection with ellipses of distortion
Scale distortions can also be shown on a map by a scale factor. A scale factor smaller than 1 indicates that the scale is smaller than the nominal scale, the scale given on the map. A scale factor larger than one indicates that the scale is larger than the nominal scale.
For example, on the UTM projection a scale factor of 0.99960 has been given to the central meridian of a UTM zone. This means that 1000m measured on the ground becomes 999.6m on the map surface along the central meridian. E.g. the actual map scale along the central meridian will be 1:10,004 (10000 / 0.9996) at a nominal map scale of 1:10,000, so smaller than the nominal scale.
Note Scale distortions can remain within certain limits by choosing the right map projection
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