Comparison of Algorithm

Map Projection
 Maximum Distortion for Data Sets

238 world cities
 400 grid point
 30 points near the poles
Albers
 5.6
 -
 432.4
Bonne
 6.1
 -
 3.2
Cassini
 9.3
 -
 10.2
Cylindrical
 10.2
 144.1
 12.9
Gnomonic
 7.4
 97.2
 113.5
Lambert Azimuthal
 4.7
 -
 124.2
Mercator
 4.2
 140212
 210342
Miller Cylindrical
5.5
 56273
 91231
Sinusodial
 18.4
 -
 24.3
Stereographic
 9.6
 1478
 2375

Our Implementation
6.7
54.8
7.8

This map projection was compared with several other commonly used map projections over a variety of data sets. Matlab's Mapping Toolbox was used to generate the different projections and the maximum distortion was calculated in a similar manner as described above. The table shows that this implementation performs favorably with most of the common map projections used. One can also note that the distortion obtained from this implementation are always within a factor 3.5 from the best map projection for that data set. Theorem 3 showed that there existed a 3.512 approximation algorithm under the Euclidean distance.

Some general remarks about these map projections are:
1) The Albers projection is an equal-area conic projection where the parallels are unequally spaced arcs of concentric circles which become more closely spaced at the north and south edges of the map. The Albers projection works better for east-west regions as opposed to north-south regions.

2) The Bonne projection is an equal-area projection with true scale along the central meridian and along each parallel. The shape distortion increases away from the central meridian.

3) The Cassini projection is a cylindrical projection that is neither equal-area nor conformal. The Cassini projection works better for north-south regions as opposed to east-west regions.

4) The Cylindrical projection is an orthographic projection of the sphere onto a cylinder. There is substantial distortion at 900 from the central line.

5) The Gnomonic projection is azimuthal and perspective and it is neither conformal nor equal area. Disortion and scale increase rapidly away from the center.

6) The Lambert Azimuthal projection is equal-area but not a perspective projection. The antipodal point from the center is shown as a circle surrounding the map.

7) The Mercator projection has very high distortion for points near the poles since it maps the poles to infinity.

8) The Miller Cylindrical projection is a compromise between the Mercator and other cylindrical projections but still suffers from the poles distortion.

9) The Sinusoidal projection is an equal-area, pseudo-cylindrical projection. The scale is true along the central meridian and all parallels. There is no distortion along the Equator and central meridian but distortion becomes pronounced near the outer meridians, especially in the polar regions.

10) The Stereographic projection is azimuthal and conformal and is a perspective projection. The point opposite the center of the projection cannot be plotted and therefore if the North Pole is the center of the map, the South Pole cannot be plotted since it is projection at an infinite distance in the plane of the map.


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